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GNU MP - The Institute of Mathematical Sciences

1. 61 Bup randseed i e bec ER ees 62 gmp randseed ui e o gn ese nn 62 PMP SCAN ito ge ace hic isa 70 gnp snprintf sene cede n eed nace bs 65 gnp sprintf o ie ed ues er mb E AER 65 GNU MP 4 1 3 PMP SSCANE Lom chee nda ea OE CLG D 70 EE 65 PMP EE TOM iia is 20 eg E KEE 65 e NEE 70 E eh WEE 65 EMP VSCANE EE 70 Ee EE 65 E KEE 65 Emp VSSCANE EE n REPRE EE AE 70 H WY POs o Aere Ee thie a 76 I o e dgan e ret eiee ran o Eeo te e res re 80 M t Vo Lo EEN 80 e EE 81 DAI ipea dade BOR se fas Hed NR RT cn eR ands Bok ea Rea ass 80 e EE 81 KEEN 81 e GEET 81 DOVER dae 4d See Oa ee aR Shee std 80 np bits per limb ed EES leaves 19 np LAMB EE 16 mp set memory functions esses 82 O Ae Deals 50 A e ao a a 50 npf add Wi fi sek es Sod aie e obe aves p edema 50 SS A eae eie Puta 51 Ipf cl sS el uc EE Pe reu e ete rre iret 12 ppf class fits sint p eren 76 pf classi fits slong EE 76 npf class fits sshort LP soccer ese ewe naa 76 npf cl ss fits uint pit insisted sven ERES 77 mpf class fits ulong p nets ee gen d TT npf class fits ushoft P vic ve emet 77 Mpf classi get d rra 76 pf classi rekt mpf t ke e REA 73 mpf class get preC occm ese s ne Rv 77 Mpf class GetoSl oe derer ta 76 mpf Class got iaa a 76 npf class impf case e ire one sea 76 mpf class operat r obis ce sede cn P eee 76 npf class iset prec EENS EIN bods dee eee 77 mpf_class set_prec_
2. zo y Yo is best calculated as an absolute value and the sign used to choose to add or subtract Notice the sum high zoyo low x1y occurs twice so it s possible to do 5k limb additions rather than 6k but in GMP extra function call overheads outweigh the saving Squaring is similar to multiplying but with y the formula reduces to an equivalent with three squares z b b x b a zo b 1 x The final result is accumulated from those three squares the same way as for the three multiplies above The middle term x zo is now always positive A similar formula for both multiplying and squaring can be constructed with a middle term x zo yi yo But those sums can exceed k limbs leading to more carry handling and additions than the form above Karatsuba multiplication is asymptotically an O N 9 algorithm the exponent being log 3 log 2 representing 3 multiplies each 1 2 the size of the inputs This is a big improvement over the basecase multiply at O N and the advantage soon overcomes the extra additions Karatsuba performs MUL KARATSUBA THRESHOLD can be as little as 10 limbs The SQR threshold is usually about twice the MUL The basecase algorithm will take a time of the form M N aN bN c and the Karatsuba algorithm K N 3M N 2 dN e which expands to K N 2aN 3bN 3c dN e The 88 GNU MP 4 1 3 factor 2 for a means per crossproduct speedups in the basecase code will
3. For example when using a FreeBSD Athlon system to build GNU Linux m68k binaries configure build athlon pc freebsd3 5 host m68k mac linux gnu Compiler tools are sought first with the host system type as a prefix For example m68k mac linux gnu ranlib is tried then plain ranlib This makes it possible for a set of cross compiling tools to co exist with native tools The prefix is the argument to host and this can be an alias such as m68k linux But note that tools don t have to be setup this way it s enough to just have a PATH with a suitable cross compiling cc etc Compiling for a different CPU in the same family as the build system is a form of cross compilation though very possibly this would merely be special options on a native compiler In any case configure avoids depending on being able to run code on the build system which is important when creating binaries for a newer CPU since they very possibly won t run on the build system In all cases the compiler must be able to produce an executable of whatever format from a standard C main Although only object files will go to make up libgmp configure uses linking tests for various purposes such as determining what functions are available on the host system Currently a warning is given unless an explicit build is used when cross compiling because it may not be possible to correctly guess the build system type if the PATH has only a
4. e mpz_add mpz_add_ui mpz_sub and mpz_sub_ui need room for the larger of the two operands plus an extra mp bits per limb e mpz mul mpz mul ui and mpz mul ui need room for the sum of the number of bits in their operands but each rounded up to a multiple of mp bits per limb e mpz swap can be used between two array variables but not between an array and a normal variable For other functions or if in doubt the suggestion is to calculate in a regular mpz init variable and copy the result to an array variable with mpz set mpz array init can reduce memory usage in algorithms that need large arrays of integers since it avoids allocating and reallocating lots of small memory blocks There is no way to free the storage allocated by this function Don t call mpz clear void mpz realloc mpz t integer mp size t new alloc Function Change the space for integer to new_alloc limbs The value in integer is preserved if it fits or is set to 0 if not The return value is not useful to applications and should be ignored mpz realloc2 is the preferred way to accomplish allocation changes like this mpz realloc2 and _mpz_realloc are the same except that _mpz_realloc takes the new size in limbs 5 2 Assignment Functions These functions assign new values to already initialized integers see Section 5 1 Initializing Integers page 28 void mpz set mpz t rop mpz t op Function void mpz set ui mpz t rop unsigned long int op Funct
5. basecase can lower DIV DC THRESHOLD a little by offering a ready made advantage over repeated mpn submul 1 calls Divide and conquer is asymptotically O M N log N where M N is the time for an NxN multiplication done with FFTs The actual time is a sum over multiplications of the recursed sizes as can be seen near the end of section 2 2 of Burnikel and Ziegler For example within the Toom 3 range divide and conquer is 2 68M NN With higher algorithms the M N term improves and the multiplier tends to log N In practice at moderate to large sizes a 2NxN division is about 2 to 4 times slower than an NxN multiplication Newton s method used for division is asymptotically OC M N and should therefore be superior to divide and conquer but it s believed this would only be for large to very large N 94 GNU MP 4 1 3 16 2 4 Exact Division A so called exact division is when the dividend is known to be an exact multiple of the divisor Jebelean s exact division algorithm uses this knowledge to make some significant optimizations see Appendix B References page 113 The idea can be illustrated in decimal for example with 368154 divided by 543 Because the low digit of the dividend is 4 the low digit of the quotient must be 8 This is arrived at from 4x7 mod 10 using the fact 7 is the modular inverse of 3 the low digit of the divisor since 3x7 1 mod 10 So 8x543 4344 can be subtracted from the dividend leaving 363810 Notice th
6. make install DESTDIR my staging area or the same with a prefix override to install to a temporary directory is not fully supported by current versions of libtool when building a shared version of a library which depends on another being built at the same time like 1ibgmpxx and libgmp The problem is that libgmpxx is relinked at the install stage to ensure that if the system puts a hard coded path to libgmp within libgmpxx then that path will be correct Naturally the linker is directed to look only at the final location not the staging area so if Libgmp is not already in that final location then the link will fail A workaround for this on SVRA style systems such as GNU Linux where paths are not hard coded is to include the staging area in the linker s search us ing LD_LIBRARY_PATH For example with prefix usr but installing under ny staging area LD_LIBRARY_PATH my staging area usr lib make install DESTDIR my staging area GNU binutils strip prior to 2 12 strip from GNU binutils 2 11 and earlier should not be used on the static libraries libgmp a and libmp a since it will discard all but the last of multiple archive members with the same name like the three versions of init o in libgmp a Binutils 2 12 or higher can be used successfully The shared libraries libgmp so and libmp so are not affected by this and any version of strip can be u
7. mon cray add_n c is an example of this it adds all limbs in parallel adds one set of carry bits in parallel and then only rarely needs to fall through to a loop propagating further carries On the x86s GCC as of version 2 95 2 doesn t generate particularly good code for the RISC style idioms that are necessary to handle carry bits in C Often conditional jumps are generated where adc or sbb forms would be better And so unfortunately almost any loop involving carry bits needs to be coded in assembler for best results 16 8 4 Cache Handling GMP aims to perform well both on operands that fit entirely in L1 cache and those which don t Basic routines like mpn_add_n or mpn_1shift are often used on large operands so L2 and main memory performance is important for them mpn mul 1 and mpn_addmul_1 are mostly used for multiply and square basecases so L1 performance matters most for them unless assembler versions of mpn mul basecase and mpn sqr basecase exist in which case the remaining uses are mostly for larger operands For L2 or main memory operands memory access times will almost certainly be more than the calculation time The aim therefore is to maximize memory throughput by starting a load of the next cache line which processing the contents of the previous one Clearly this is only possible if the chip has a lock up free cache or some sort of prefetch instruction Most current chips have both these features Prefetching sou
8. 20 Web page resect one lite dE 2 X KEE 12 124 Function and Type Index _ GNU_MP_VERSION cc4s0 0408 ENEE EEN 880004 19 GNU MP VERSION MINOR less 19 GNU MP VERSION PATCHLEVEL 19 Jnpz realloGc isia pte ci ph E ER Rb 29 allocate function deeper RE RS 82 reu p CEP 76 CMD wea o vue PCR ER 74 75 76 D deallocate function e os eed 82 F SG EE 76 G EE 81 gnp asprintf ie eee ch eee eae es 65 E EE 61 GMP ERROR INVALID ARGUMENT 61 CMD ERROR UNSUPPORTED ARGUMENT 61 PMP fprinbtf dtd adem heed sachet as eee Rees 65 EMP SCADE EE 70 GMP LIMB BT TS iii ed 59 GMP NAIG BIG vis Eben rj RM PR ES exd 59 GMP NATL MASK c cei ae 59 GMP NUMB BITS vesical 59 GMP NUMB MASK conocio ii ad 59 GMP NUMB MAX rogaa eima ra A E 59 gmp obstack Ek EE 66 gup obstack vprintf oa wie sos cee s RR REDE eR 66 gmp prj3ntf 22 1 15 5 00 DEus old aids 65 GMP RAND ALG DEFAULT otes arni merpet a dawni eR 61 GMP RAND ALG LC 2d ata naadi See dee aad 61 gnp randclas8 2 cd aariak wr a EEE e Pg Enp randclass Eet f cante RR 78 gnp randclass get z bits sse 78 Egnp randclass get z range e 78 gmp_randclass gmp_randclass ti TS pup randclass 8eed c2occecm e sont ta Sees 78 Emp Tandil ia EE 62 EMP TAO di o oo 61 pup randinit default ben pee ie 61 gu up randinit lcC Zeen ERT NEE nas 61 gmp randinit lc 2exp size
9. Complexity is not currently implemented It uses a rearrangement similar to the divide and conquer for normal division see Section 16 2 3 Divide and Conquer Division page 93 but operating from low to high A further possibility not currently implemented is Bidirectional Exact Integer Division by Krandick and Jebelean which forms quotient limbs from both the high and low ends of the dividend and can halve once more the number of crossproducts needed in a 2NxN division A special case exact division by 3 exists in mpn_divexact_by3 supporting Toom 3 multiplication and mpq canonicalizations It forms quotient digits with a multiply by the modular inverse of 3 which is OxAA AAB and uses two comparisons to determine a borrow for the next limb The multiplications don t need to be on the dependent chain as long as the effect of the borrows is applied Only a few optimized assembler implementations currently exist 16 2 5 Exact Remainder If the exact division algorithm is done with a full subtraction at each stage and the dividend isn t a multiple of the divisor then low zero limbs are produced but with a remainder in the high limbs For dividend a divisor d quotient q and b 2 P bits_per_limb then this remainder r is of the form a qd rb n represents the number of zero limbs produced by the subtractions that being the number of limbs produced for q r will be in the range 0 lt r lt d and can be viewed as a remainder
10. He has also made valuable suggestions and tested numerous intermediary releases Joachim Hollman was involved in the design of the mpf interface and in the mpz design revisions for version 2 Bennet Yee contributed the initial versions of mpz_jacobi and mpz_legendre Andreas Schwab contributed the files mpn m68k 1shift S and mpn m68k rshift S now in asm form The development of floating point functions of GNU MP 2 were supported in part by the ESPRIT BRA Basic Research Activities 6846 project POSSO POlynomial System SOlving GNU MP 2 was finished and released by SWOX AB SWEDEN in cooperation with the IDA Center for Computing Sciences USA Robert Harley of Inria France and David Seal of ARM England suggested clever improvements for population count Robert Harley also wrote highly optimized Karatsuba and 3 way Toom multiplication functions for GMP 3 He also contributed the ARM assembly code Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development His mathematical expertise helped improve several algorithms Paul Zimmermann wrote the Divide and Conquer division code the REDC code the REDC based mpz_powm code the FFT multiply code and the Karatsuba square root The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3 Linus Nordberg wrote the new configure system based on
11. It evaluates its argument multiple times 5 11 Logical and Bit Manipulation Functions These functions behave as if twos complement arithmetic were used although sign magnitude is the actual implementation The least significant bit is number 0 void mpz and mpz_t rop mpz t op1 mpz t op2 Function Set rop to op1 bitwise and op2 void mpz ior mpz rop mpz t opi mpz t op2 Function Set rop to op1 bitwise inclusive or op2 void mpz xor mpz rop mpz t opi mpz_t op2 Function Set rop to op1 bitwise exclusive or op2 void mpz com mpz_t rop mpz t op Function Set rop to the one s complement of op unsigned long int mpz popcount mpz t op Function If op gt 0 return the population count of op which is the number of 1 bits in the binary representation If op lt 0 the number of Is is infinite and the return value is ULONG MAX the largest possible unsigned long unsigned long int mpz hamdist mpz t opi mpz t op2 Function If op1 and op2 are both gt 0 or both lt 0 return the hamming distance between the two operands which is the number of bit positions where op and op2 have different bit values If one operand is gt 0 and the other lt 0 then the number of bits different is infinite and the return value is ULONG MAX the largest possible unsigned long unsigned long int mpz_scan0 mpz t op unsigned long int Function starting bit unsigned long int mpz scan1 mpz t op unsigned long int Fun
12. and Insubl to L 4 These functions are designed for calculating isolated Lucas numbers When a sequence of values is wanted it s best to start with mpz lucnum2 ui and iterate the defining L 4 La L4 or similar The Fibonacci numbers and Lucas numbers are related sequences so it s never necessary to call both mpz fib2 ui and mpz lucnum2 ui The formulas for going from Fibonacci to Lucas can be found in Section 16 7 4 Lucas Numbers Algorithm page 101 the reverse is straightforward too 5 10 Comparison Functions int mpz cmp mpz t opi mpz t op2 Function int mpz cmp d mpz t opi double op2 Function int mpz cmp si mpz t op1 signed long int op2 Macro int mpz cmp ui mpz t opi unsigned long int op2 Macro Compare op1 and op2 Return a positive value if op1 gt op2 zero if op1 op2 or a negative value if op1 lt op2 Note that mpz cmp ui and mpz cmp si are macros and will evaluate their arguments more than once Chapter 5 Integer Functions 37 int mpz_cmpabs mpz_t op1 mpz_t op2 Function int mpz_cmpabs_d mpz_t op1 double op2 Function int mpz_cmpabs_ui mpz_t op1 unsigned long int op2 Function Compare the absolute values of op1 and op2 Return a positive value if op1 gt op2 zero if op1 op2 or a negative value if op1 lt op2 int mpz_sgn mpz_t op Macro Return 1 if op gt 0 0 if op 0 and 1 if op lt 0 This function is actually implemented as a macro
13. as for the constructors above Note that operator only stores a new value it doesn t copy or change the precision of the destination instead the value is truncated if necessary This is the same as mpf set etc Note in particular this means for mpf_class a copy constructor is not the same as a default constructor plus assignment mpf class x y x created with precision of y mpf class x x created with default precision Xx y value truncated to that precision Applications using templated code may need to be careful about the assumptions the code makes in this area when working with mpf class values of various different or non default precisions For instance implementations of the standard complex template have been seen in both styles above though of course complex is normally only actually specified for use with the builtin float types mpf_class abs mpf_class op Function mpf class ceil mpf class op Function int cmp mpf class op1 type op2 Function int cmp type opi mpf_class op2 Function mpf class floor mpf class op Function mpf class hypot mpf class op1 mpf class op2 Function double mpf class get d void Function long mpf class get si void Function unsigned long mpf class get ui void Function bool mpf class fits sint p void Function bool mpf class fits slong p void Function Chapter 12 C Class Interface 77 bool mpf class fits sshort p void Function bool mpf class fits ui
14. gt size If successful the return value is non zero If size is bigger than the table data provides then the return value is zero The maximum size currently supported is 128 void gmp_randinit gmp_randstate_t state gmp_randalg_t alg Function This function is obsolete Initialize state with an algorithm selected by alg The only choice is GMP_RAND_ALG_LC which is gmp_randinit_lc_2exp_size described above A third parameter of type unsigned long is required this is the size for that function GMP_RAND_ALG_DEFAULT or 0 are the same as GMP_RAND_ALG_LC gmp_randinit sets bits in the global variable gmp_errno to indicate an error GMP_ERROR_ UNSUPPORTED_ARGUMENT if alg is unsupported or GMP_ERROR_INVALID_ARGUMENT if the size parameter is too big It may be noted this error reporting is not thread safe a good reason to use gmp randinit lc 2exp size instead 62 GNU MP 4 1 3 void gmp_randclear gmp randstate t state Function Free all memory occupied by state 9 2 Random State Seeding void gmp_randseed gmp_randstate_t state mpz_t seed Function void gmp_randseed_ui gmp_randstate_t state unsigned long int seed Function Set an initial seed value into state The size of a seed determines how many different sequences of random numbers that it s possible to generate The quality of the seed is the randomness of a given seed compared to the previous seed used and this affects the randomness of separate number
15. is outside the range 0 to mpz_size op 1 5 5 Arithmetic Functions void mpz_add mpz t rop mpz_t opi mpz_t op2 Function void mpz add ui mpz t rop mpz t opi unsigned long int op2 Function Set rop to opl op2 void mpz sub mpz rop mpz t opi mpz t op2 Function void mpz sub ui mpz t rop mpz t opi unsigned long int op2 Function void mpz ui sub mpz t rop unsigned long int op1 mpz t op2 Function Set rop to opl op2 void mpz mul mpz trop mpz t opi mpz t op2 Function void mpz mul si mpz t rop mpz t op1 long int op2 Function void mpz mul ui mpz t rop mpz t op1 unsigned long int op2 Function Set rop to opl x op2 void mpz addmul mpz t rop mpz t opi mpz_t op2 Function void mpz addmul ui mpz t rop mpz_t op1 unsigned long int op2 Function Set rop to rop opl x oni void mpz submul mpz t rop mpz t opi mpz_t op2 Function void mpz submul ui mpz t rop mpz_t op1 unsigned long int op2 Function Set rop to rop opl x oni void mpz mul 2exp mpz_t rop mpz_t op1 unsigned long int op2 Function Set rop to op1 x 2 This operation can also be defined as a left shift by op2 bits void mpz neg mpz_t rop mpz t op Function Set rop to op void mpz abs mpz t rop mpz_t op Function Set rop to the absolute value of op 32 GNU MP 4 1 3 5 6 Division Functions Division is undefined if the divisor is zero Passing a zero divisor to the division or modulo
16. lt __gmp_binary_expr lt __gmp_expr lt T gt __gmp_expr lt U gt Op gt gt eval mpf_t f unsigned long int precision declare two temporaries mpf class tempi expr vali precision temp2 expr val2 precision Op eval f tempi get mpf t temp2 get_mpf_t The expression is thus recursively evaluated to any level of complexity and all subexpressions are evaluated to the precision of f Appendix A Contributors 111 Appendix A Contributors Torbjorn Granlund wrote the original GMP library and is still developing and maintaining it Several other individuals and organizations have contributed to GMP in various ways Here is a list in chronological order Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early versions of the library Richard Stallman contributed to the interface design and revised the first version of this manual Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions John Amanatides of York University in Canada contributed the function mpz_probab_prime_p Paul Zimmermann of Inria sparked the development of GMP 2 with his comparisons between bignum packages Ken Weber Kent State University Universidade Federal do Rio Grande do Sul contributed mpz_gcd mpz_divexact mpn_gcd and mpn_bdivmod partially supported by CNPq Brazil grant 301314194 2 Per Bothner of Cygnus Support helped to set up GMP to use Cygnus configure
17. mp size t sin Function const mp_limb_t s2p mp size t s2n unsigned long int d This function puts the low d mp bits per limb limbs of q slp s1n 4s2p s2n mod 27 at rp and returns the high d mod mp bits per limb bits of q slp sln q s2p s2n mod 951n mp bits per limb y placed at slp Since the low d mp_bits_per_limb limbs of this difference are zero it is possible to overwrite the low limbs at slp with this difference provided rp lt slp This function requires that s1n mp bits per limb gt D and that s2p s2n is odd This interface is preliminary It might change incompatibly in future revisions mp limb t mpn_lshift mp_limb_t rp const mp_limb_t sp mp size t n Function unsigned int count Shift sp n left by count bits and write the result to rp n The bits shifted out at the left are returned in the least significant count bits of the return value the rest of the return value is zero count must be in the range 1 to mp bits per limb 1 The regions sp n and rp n may overlap provided rp gt sp This function is written in assembly for most CPUs Chapter 8 Low level Functions 57 mp_limb_t mpn_rshift mp_limb_t rp const mp_limb_t sp mp size t n Function unsigned int count Shift Zon n right by count bits and write the result to rp n The bits shifted out at the right are returned in the most significant count bits of the return value the rest of the retu
18. provided that you release the Modified Version under precisely this License with the Modified Version filling the role of the Document thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it In addition you must do these things in the Modified Version A Use in the Title Page and on the covers if any a title distinct from that of the Document and from those of previous versions which should if there were any be listed in the History section of the Document You may use the same title as a previous version if the original publisher of that version gives permission B List on the Title Page as authors one or more persons or entities responsible for authorship of the modifications in the Modified Version together with at least five of the principal authors of the Document all of its principal authors if it has fewer than five unless they release you from this requirement C State on the Title page the name of the publisher of the Modified Version as the publisher D Preserve all the copyright notices of the Document E Add an appropriate copyright notice for your modifications adjacent to the other copy right notices F Include immediately after the copyright notices a license notice giving the public permission to use the Modified Version under the terms of this License in the form shown in the Addendum below G Preserve in that license notice the full lists of Inva
19. s Please make an effort to produce a self contained report with something definite that can be tested or debugged Vague queries or piecemeal messages are difficult to act on and don t help the development effort It is not uncommon that an observed problem is actually due to a bug in the compiler the GMP code tends to explore interesting corners in compilers If your bug report is good we will do our best to help you get a corrected version of the library if the bug report is poor we won t do anything about it except maybe ask you to send a better report Send your report to bug gmp gnu org If you think something in this manual is unclear or downright incorrect or if the language needs to be improved please send a note to the same address 28 GNU MP 4 1 3 5 Integer Functions This chapter describes the GMP functions for performing integer arithmetic These functions start with the prefix mpz_ GMP integers are stored in objects of type mpz_t 5 1 Initialization Functions The functions for integer arithmetic assume that all integer objects are initialized You do that by calling the function mpz_init For example mpz_t integ mpz_init integ mpz_add integ mpz_sub integ Unless the program is about to exit do mpz clear integ As you can see you can store new values any number of times once an object is initialized void mpz_init mpz_t integer Function Initialize
20. the covers as long as they preserve the title of the Document and satisfy these conditions can be treated as verbatim copying in other respects If the required texts for either cover are too voluminous to fit legibly you should put the first ones listed as many as fit reasonably on the actual cover and continue the rest onto adjacent pages If you publish or distribute Opaque copies of the Document numbering more than 100 you must either include a machine readable Transparent copy along with each Opaque copy or state in or with each Opaque copy a computer network location from which the general network using public has access to download using public standard network protocols a complete Transparent copy of the Document free of added material If you use the latter option you must take reasonably prudent steps when you begin distribution of Opaque copies in quantity to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy directly or through your agents or retailers of that edition to the public It is requested but not required that you contact the authors of the Document well before redistributing any large number of copies to give them a chance to provide you with an updated version of the Document 4 MODIFICATIONS You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above
21. when compiling the mainline This is a good way to ensure minimum interference with normal operation The necessary symbol type and size information exists in most of the GMP assembler code The enable profiling build option can be used to add suitable compiler flags either for prof p or gprof pg see Section 2 1 Build Options page 4 Which of the two is available and what they do will depend on the system and possibly on support available in libc For some systems appropriate corresponding mcount calls are added to the assembler code too On x86 systems prof gives call counting so that average time spent in a function can be determined gprof where supported adds call graph construction so for instance calls to mpn_add_n from mpz_add and from mpz_mul can be differentiated On x86 and 68k systems pg and fomit frame pointer are incompatible so the latter is not used when gprof profiling is selected which may result in poorer code generation If prof profiling is selected instead it should still be possible to use gprof but only the gprof p flat profile and call counts can be expected to be valid not the gprof q call graph 3 14 Autoconf Autoconf based applications can easily check whether GMP is installed The only thing to be noted is that GMP library symbols from version 3 onwards have prefixes like __gmpz The following therefore would be a simple test AC CHECK LIB gmp gmpz in
22. with mp_limb_t etc but GMP 4 0 or up is best for this since prior versions needed certain D defines on systems using a long long limb The following would suit Autoconf 2 50 or up AC CHECK SIZEOF mp limb t include lt gmp h gt The optional mpfr functions are provided in a separate libmpfr a and this might be from GMP with enable mpfr or from MPFR installed separately Either way libmpfr depends on libgmp it doesn t stand alone Currently only a static libmpfr a will be available not a shared library since upward binary compatibility is not guaranteed AC CHECK LIB mpfr mpfr add AC MSG ERROR Need MPFR either from GNU MP 4 or separate MPFR package See http www mpfr org or http swox com gmpl 3 15 Emacs info lookup symbol is a good way to find documentation on C functions while editing see section Info Documentation Lookup in The Emacs Editor The GMP manual can be included in such lookups by putting the following in your emacs eval after load info look let mode value assoc c mode assoc symbol info lookup alist setcar nthcdr 3 mode value cons Cgmp Function Index nil gt nth 3 mode value The same can be done for MPFR with mpfr in place of gmp Chapter 4 Reporting Bugs 27 4 Reporting Bugs If you think you have found a bug in the GMP library please investigate it and report it We have made this library
23. 2 see Appendix B References page 113 or any number theory textbook See also the example program demos qcn c which uses mpz kronecker ui unsigned long int mpz remove mpz_t rop mpz_t op mpz t f Function Remove all occurrences of the factor f from op and store the result in rop The return value is how many such occurrences were removed void mpz fac ui mpz t rop unsigned long int op Function Set rop to op the factorial of op void mpz bin ui mpz t rop mpz t n unsigned long int k Function void mpz bin uiui mpz t rop unsigned long int n unsigned long int k Function Compute the binomial coefficient 7 and store the result in rop Negative values of n are supported by mpz bin ui using the identity 7 1 gru see Knuth volume 1 section 1 2 6 part G void mpz fib ui mpz t fn unsigned long int n Function void mpz fib2 ui mpz t fn mpz_t fnsub1 unsigned long int n Function mpz fib ui sets fn to to Fh the n th Fibonacci number mpz fib2 ui sets fn to F and fnsubl to F These functions are designed for calculating isolated Fibonacci numbers When a sequence of values is wanted it s best to start with mpz fib2 ui and iterate the defining F 41 F 4 EF 4 or similar void mpz lucnum ui mpz t ln unsigned long int n Function void mpz lucnum2 ui mpz t ln mpz_t 1nsubl unsigned long int n Function mpz lucnum ui sets In to to Ln the wth Lucas number mpz lucnum2 ui sets In to Ln
24. 25 B STEE EE 16 Berkeley MP compatible functions 80 Binomial coefficient functions 36 Bit manipulation functions esses ot Bit shift left argent Faden asda nis 31 Bit Shit EEN 32 Bits per limbs 2 624 ls be Iu ens uu pice rp 19 BSD MP compatible functions 80 B g reporting e peor oe eles Ae 27 Build notes for binary packaging 11 Build notes for particular systems 11 Bild opUOIs emma 4 Build problems known 13 Building GM it eo res 4 C CFE Interfaeez eu te erp viele Sac RE ais ed eng 72 CFP istream EENEG 70 C ostream Output 0 rnr eee eee 66 Comparison functions 0 36 44 50 Compatibility with older versions 20 Conditions for copying GNU MP 1 Configuring GMP ui PES 4 Constans ele be IPLE HH eR RUP EDS 19 Contributors rd RE EE TEE Conventions for parameters oooooococoooccoo 18 Conventions for variables o oooooo oooo 17 Conversion function 30 43 49 Copying Conditions vigrpiaran 343008 eee aks 1 CPUs supported serer iprrrrrr REIS ERR RS 2 Custom allocation ees eked m pere REP 82 121 D Deb gging c scr dae resPPRI PRECIO EEN 23 Demonstration Drogramsg sescrscsrseiisirese esis 20 DES TDGR EE 14 Digits in an integer M rp Ree 41 Division algorithms i esce neo eens eens d 92 Division functioNS o ooooococooccoo 32 43 50 E o EE 20 EMACS
25. 36 npz fdiv d hited eid Per ee rte door age 32 mpz fdiv dd ZE EE 32 mpz fdiv d uic His PRU Ren eee YER 32 npz fdiv EE 32 npz fdiv dr Ul ce EE 32 npz fdiv roo ilsias A na AE ERE 32 npz fdiv Y 26XD i ga Rh ee ek RR EE 92 npz fdiv RER 32 MZ fdiv ud eniesuenenes ph ter LE RO E RUPEE 32 mpz fibula ey ode or READER REOR 36 GNU MP 4 1 3 npz fib2 ui we e Resa semp dg 36 Pa fits Sint Pb osot piri ane 23 RUE e cus Al npz fits Slong WEE 41 mpz fits SsShort EE 41 MpZ FICS lnt Pierde 41 Xipz fits Ulong Picea ia ii 41 npz fits ushort Piscis 41 Wpz B6d ii a 35 mpz gcd ii seis eaan i hinia 35 mpz gcdext iee 9 Geeta sd RIEN D code PRU 35 mpz get iio seer eld eaa ted 30 mpz get d 20XD 2 ea RS dada EE dos 30 le GE EE 30 mp2 GE EE 30 mpz get LUT ci e a EE E 30 mpz PSCLIMBON Seene EES be Reds a 31 le KEE l MPZ IMPIT EE 39 WIES inita cheap ee iaa ded 28 ue A kaini nieee a EE eh 30 mpz init set d hoe utaina eiee 30 npz init OSA iii aria 30 npz init set SUE 4 EE E Ee 30 npz init Set Ui ec RR a 30 pZ init2 49 ae x eer rae RE DRE E RR 28 npz inp raW erss e boe Re RERO Re a 38 npz inp Str 22s ud o ena d e or ee ue e 38 npz invert i s e gg wa pace dad do dis oa te 223 MPZ d0X iii ii A A ARA 37 MPZ JACOD iii es 35 MPZ_KFODMECKEL iii bee ce nade des REESEN 35 mpz kronecker ei e pede een db oh rege 35 npz kron cker ui l 2 6629 ti 35 mpz lO 4cs ERRARE RESQUE XE 35 mpz lom EE 35 mpz legendre llc
26. B References ce cece eee eee 113 EE DEE 113 B 2 PRper8 cnl6994 ds ra SEA me ida ea D DUE eee 113 Appendix C GNU Free Documentation License 115 Cl ADDENDUM How to use this License for your documents 120 Concept Folge ge de green EN X RE Sd d us 121 Function and Type Index edu ww acr Dar ite CR Ree we E des 124 GNU MP Copying Conditions This library is free this means that everyone is free to use it and free to redistribute it on a free basis The library is not in the public domain it is copyrighted and there are restrictions on its distribution but these restrictions are designed to permit everything that a good cooperating citizen would want to do What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you Specifically we want to make sure that you have the right to give away copies of the library that you receive source code or else can get it if you want it that you can change this library or use pieces of it in new free programs and that you know you can do these things To make sure that everyone has such rights we have to forbid you to deprive anyone else of these rights For example if you distribute copies of the GNU MP library you must give the recipients all the rights that you have You must make sure that they too receive or can get the source code And you must tell them their rights Also for our own protection we must make certai
27. C Library Ref erence Manual 10 3 C Formatted Output The following functions are provided in libgmpxx which is built if C support is enabled see Section 2 1 Build Options page 4 Prototypes are available from lt gmp h gt ostreamk operator lt lt ostream amp stream mpz_t op Function Print op to stream using its ios formatting settings ios width is reset to 0 after output the same as the standard ostream operator lt lt routines do In hex or octal op is printed as a signed number the same as for decimal This is unlike the standard operator lt lt routines on int etc which instead give twos complement ostream amp operator lt lt ostream amp stream mpq t op Function Print op to stream using its ios formatting settings ios width is reset to 0 after output the same as the standard ostream operator lt lt routines do Output will be a fraction like 5 9 or if the denominator is 1 then just a plain integer like 123 In hex or octal op is printed as a signed value the same as for decimal If ios showbase is set then a base indicator is shown on both the numerator and denominator if the denominator is required ostream amp operator ostream amp stream mpf t op Function Print op to stream using its ios formatting settings ios width is reset to 0 after output the same as the standard ostream operator lt lt routines do The decimal point follows the current locale on syst
28. EE 30 SEET 41 MpZ 8IZeinbase c dv egg eel ads 41 HPZ SG Ci olii ote edad cbe ncaa 34 Npz EEN ee pubes dud eee 34 MPZ SUD EE 31 MP2 SUB U ues rao pon die Pado dann dace 31 npz SubmA u l oceereee Pardes Pr ea bee ee EUR 31 mpz SubMUL Ud ticked i geet ee aes 31 pz SWAP EE 29 Mp2 EE 16 npz CIO EE ET 32 Mp2 tdiv d 26XD 0250 x ns IER 32 npz tdiv q Ul cese A al 32 hpz E s ta dl 32 npz tdiv df ui mec ga kb Rn EE 32 npz tdiv E Ree ELE oe Ra 32 npz tdiv E 26XD pid e pe are ten RR 32 npz CAIVP UD esis ed o ev is 32 mpz tdiv iii da 32 Inpz tSUtbitj rne RP RESP es e 38 mpz i kroneckeE 2 9 de 9B hits p EFE aaa 35 mpz di pow Ulea 34 npz lui Sub iiizoxe REG en I EN eed eee nnd 31 npz urandodb eL un PERDE eus 39 npz ur ndomm 22 eg n ead Dae es 39 MPZ OM Ec EE de taa Eed 37 127 MET dG E Ee De 80 EE 80 IDbOX Eege NEE nee eh eae oe ee pees 81 DUI uiuis Rea RR IRR RR RUE E EUR aug OR Rs 80 operator ins d Rea p Ke AER EE ARA 73 Operator ivan ad ad 13 q coire O 66 Operator iia EE 70 75 P DIM 81 R reallocate function eee eee 82 A rbd pi abesnituehp uee npe 81 Sdi 554 sted ae Eis dane ERIS EES Re Ra ae TE EAS 80 BEBR iuc ea Sege 74 75 77 SOLE EE 74 77 T CHUNG M W A EE EE 80
29. Func tions page 53 are designed to facilitate a variety of implementations For example knowing mpn_add_n won t have partly overlapping sources and destination means reading can be done far ahead of writing on superscalar processors and loops can be vectorized on a vector processor depending on the carry handling 16 8 3 Carry Propagation The problem that presents most challenges in GMP is propagating carries from one limb to the next In functions like mpn_addmul_1 and mpn add n carries are the only dependencies between limb operations On processors with carry flags a straightforward CISC style adc is generally best AMD K6 mpn_addmul_1 however is an example of an unusual set of circumstances where a branch works out better On RISC processors generally an add and compare for overflow is used This sort of thing can be seen in mpn generic aors_n c Some carry propagation schemes require 4 instructions meaning at least 4 cycles per limb but other schemes may use just 1 or 2 On wide superscalar processors performance may be completely determined by the number of dependent instructions between carry in and carry out for each limb On vector processors good use can be made of the fact that a carry bit only very rarely propagates more than one limb When adding a single bit to a limb there s only a carry out if that limb was Chapter 16 Algorithms 103 OxFF FF which on random data will be only 1 in 2 P gt its per_limb
30. IEEE Symposium on Switching and Automata Theory October 1972 pp 90 96 Reprinted as Fast Modular Transforms Journal of Computer and System Sciences volume 8 number 3 June 1974 pp 366 386 Arnold Sch nhage and Volker Strassen Schnelle Multiplikation grosser Zahlen Comput ing 7 1971 pp 281 292 Kenneth Weber The accelerated integer GCD algorithm ACM Transactions on Mathe matical Software volume 21 number 1 March 1995 pp 111 122 Paul Zimmermann Karatsuba Square Root INRIA Research Report 3805 November 1999 http www inria fr RRRT RR 3805 html Paul Zimmermann A Proof of GMP Fast Division and Square Root Implementations http www loria fr zimmerma papers proof div sqrt ps gz Dan Zuras On Squaring and Multiplying Large Integers ARITH 11 IEEE Symposium on Computer Arithmetic 1993 pp 260 to 271 Reprinted as More on Multiplying and Squaring Large Integers IEEE Transactions on Computers volume 43 number 8 August 1994 pp 899 908 Appendix C GNU Free Documentation License 115 Appendix C GNU Free Documentation License Version 1 2 November 2002 Copyright 2000 2001 2002 Free Software Foundation Inc 59 Temple Place Suite 330 Boston MA 02111 1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document but changing it is not allowed 0 PREAMBLE The purpose of this License is to make a manual textbook or other func
31. MP Compatible Functions 81 void pow MINT base MINT exp MINT mod MINT dest Function Set dest to base raised to exp modulo mod Note that the name pow clashes with pow from the standard C math library see section Exponentiation and Logarithms in The GNU C Library Reference Manual An application will only be able to use one or the other void rpow MINT base signed short int exp MINT dest Function Set dest to base raised to exp void gcd MINT op1 MINT op2 MINT res Function Set res to the greatest common divisor of op and op2 int mcmp MINT op1 MINT op2 Function Compare op1 and op2 Return a positive value if op1 gt op2 zero if op1 op2 and a negative value if op1 lt op2 void min MINT dest Function Input a decimal string from stdin and put the read integer in dest SPC and TAB are allowed in the number string and are ignored void mout MINT src Function Output src to stdout as a decimal string Also output a newline char mtox MINT op Function Convert op to a hexadecimal string and return a pointer to the string The returned string is allocated using the default memory allocation function malloc by default It will be strlen str 1 bytes that being exactly enough for the string and null terminator void mfree MINT op Function De allocate the space used by op This function should only be passed a value returned by itom or xtom 82 GNU MP 4 1 3 14 Cu
32. Send feedback or alternate ideas to bug gmp gnu org Subclassing Subclassing the GMP C classes works but is not currently recommended Expressions involving subclasses resolve correctly or seem to but in normal C fashion the subclass doesn t inherit constructors and assignments There s many of those in the GMP classes and a good way to reestablish them in a subclass is not yet provided Templated Expressions A subtle difficulty exists when using expressions together with application defined template functions Consider the following with T intended to be some numeric type Chapter 12 C Class Interface 79 template lt class T gt T fun const T amp const T amp When used with say plain mpz class variables it works fine T is resolved as mpz class mpz class f 1 2 fun f g Good But when one of the arguments is an expression it doesn t work mpz_class f 1 g 2 h 3 fun f gth Bad This is because g h ends up being a certain expression template type internal to gnpxx h which the C template resolution rules are unable to automatically con vert to mpz_class The workaround is simply to add an explicit cast mpz class f 1 g 2 h 3 fun f mpz_class gth Good Similarly within fun it may be necessary to cast an expression to type T when calling a templated fun2 template lt class T gt void fun T f T g fun2 f Trei Bad Jj template class T gt v
33. a checksum of an array of limbs might have a load and an add but the load wouldn t be for that add rather for the one next time around the loop Each load then is effectively scheduled back in the previous iteration allowing latency to be hidden Naturally this is wanted only when doing things like loads or multiplies that take a few cycles to complete and only where a CPU has multiple functional units so that other work can be done while waiting A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage This is like juggling Within the loop some moves between registers may be necessary to have the right values in the right places for each iteration Loop unrolling can help this with each unrolled block able to use different registers for different values even if some shuffling is still needed just before going back to the top of the loop 16 8 8 Loop Unrolling Loop unrolling consists of replicating code so that several limbs are processed in each loop At a minimum this reduces loop overheads by a corresponding factor but it can also allow better register usage for example alternately using one register combination and then another Judicious use of m4 macros can help avoid lots of duplication in the source code Unrolling is commonly done to a power of 2 multiple so the number of unrolled loops and the number of remaining limbs can be calculated with a shift and
34. above for Karatsuba and Toom 3 A k parameter controls the split with an FFT k splitting into 2 pieces of M N 2 bits each N must be a multiple of 2 x mp bits per limb so the split falls on limb boundaries avoiding bit shifts in the split and combine stages The evaluations pointwise multiplications and interpolation are all done modulo 2 1 where NI is 2M k 3 rounded up to a multiple of 2 and of mp bits per limb The results of interpolation will be the following negacyclic convolution of the input pieces and the choice of N ensures these sums aren t truncated Wn 5 1 aiy i j b2k n b 0 1 The points used for the evaluation are ai for i 0 to 2 1 where g 22 7 g is a 2 th root of unity mod 2 1 which produces necessary cancellations at the interpolation stage and it s also a power of 2 so the fast fourier transforms used for the evaluation and interpolation do only shifts adds and negations The pointwise multiplications are done modulo 21 1 and either recurse into a further FFT or use a plain multiplication Toom 3 Karatsuba or basecase whichever is optimal at the size N The interpolation is an inverse fast fourier transform The resulting set of sums of x y are added at appropriate offsets to give the final result Squaring is the same but x is the only input so it s one transform at the evaluate stage and the pointwise multiplies are squares The interpolation is the same Chapte
35. are as follows p and will depend on support from the C library the rest are standard e character or characters d decimal integer e Ef g float i integer with base indicator n characters read so far o octal integer p pointer s string of non whitespace characters u decimal integer xX hex integer string of characters in a set e E f g and OI are identical they all read either fixed point or scientific format and either e or E for the exponent in scientific format x and X are identical both accept both upper and lower case hexadecimal o u x and X all read positive or negative values For the standard C types these are described as unsigned conversions but that merely affects certain overflow handling negatives are still allowed per strtoul see section Parsing of Integers in The GNU C Library Reference Manual For GMP types there are no overflows so d and w are identical Q type reads the numerator and optional denominator as given If the value might not be in canonical form then mpq_canonicalize must be called before using it in any calculations see Chapter 6 Rational Number Functions page 42 Qi will read a base specification separately for the numerator and denominator For example 0x10 11 would be 16 11 whereas 0x10 0x11 would be 16 17 n can be used with any of the types above even
36. as making it possible for a user to omit build and host so config guess will detect the CPU But a way to manually specify a build will be wanted for systems where config guess is inexact Note that gmp h is a generated file and will be architecture and ABI dependent 2 4 Notes for Particular Systems AIX 3 and 4 On systems aix 34 shared libraries are disabled by default since some versions of the native ar fail on the convenience libraries used A shared build can be attempted with configure enable shared disable static Note that the disable static is necessary because in a shared build libtool makes libgmp a a symlink to libgmp so apparently for the benefit of old ver sions of 1d which only recognise a but unfortunately this is done even if a fully functional 1d is available ARM 12 GNU MP 4 1 3 On systems arm versions of GCC up to and including 2 95 3 have a bug in unsigned division giving wrong results for some operands GMP configure will demand GCC 2 95 4 or later Compaq C Compaq C on OSF 5 1 has two flavours of iostream a standard one and an old pre standard one see man iostream_intro GMP can only use the standard one which unfortunately is not the default but must be selected by defining __USE_STD_ IOSTREAM Configure with for instance configure enable cxx CPPFLAGS D USE STD IOSTREAM Floating Point
37. autoconf and implemented the new random functions Kent Boortz made the Macintosh port 112 GNU MP 4 1 3 Kevin Ryde worked on a number of things optimized x86 code m4 asm macros parameter tuning speed measuring the configure system function inlining divisibility tests bit scanning Jacobi symbols Fibonacci and Lucas number functions printf and scanf functions perl interface demo expression parser the algorithms chapter in the manual gmpasm mode el and various miscellaneous improvements elsewhere Steve Root helped write the optimized alpha 21264 assembly code Gerardo Ballabio wrote the gmpxx h C class interface and the C istream input routines GNU MP 4 0 was finished and released by Torbjorn Granlund and Kevin Ryde Torbjorn s work was partially funded by the IDA Center for Computing Sciences USA This list is chronological not ordered after significance If you have contributed to GMP but are not listed above please tell tege swox com about the omission Thanks goes to Hans Thorsen for donating an SGI system for the GMP test system environment Appendix B References 113 Appendix B References B 1 Books Jonathan M Borwein and Peter B Borwein Pi and the AGM A Study in Analytic Number Theory and Computational Complexity Wiley John amp Sons 1998 Henri Cohen A Course in Computational Algebraic Number Theory Graduate Texts in Mathematics number 138 Springer Verlag 1993
38. available to you and it is not too much to ask you to report the bugs you find Before you report a bug check it s not already addressed in Section 2 5 Known Build Problems page 13 or perhaps Section 2 4 Notes for Particular Systems page 11 You may also want to check http swox com gmp for patches for this release Please include the following in any report e The GMP version number and if pre packaged or patched then say so e A test program that makes it possible for us to reproduce the bug Include instructions on how to run the program e A description of what is wrong If the results are incorrect in what way If you get a crash say so e If you get a crash include a stack backtrace from the debugger if it s informative where in gdb or C in adb e Please do not send core dumps executables or straces e The configuration options you used when building GMP if any e The name of the compiler and its version For gcc get the version with gcc v otherwise perhaps what which cc or similar e The output from running uname a e The output from running config guess and from running configfsf guess might be the same e If the bug is related to configure then the contents of config 1log e If the bug is related to an asm file not assembling then the contents of config m4 and the offending line or lines from the temporary mpn tmp lt file gt
39. be summed with r64 and r80 from the previous iteration For each loop then four 49 bit quantities are transfered to the integer unit aligned as follows I 64 bits 64 bits p00 r64 i00 p16 r80 i16 p32 r32 132 p48 r48 i48 The challenge then is to sum these efficiently and add in a carry limb generating a low 64 bit result limb and a high 33 bit carry limb 748 extends 33 bits into the high half 16 8 6 SIMD Instructions The single instruction multiple data support in current microprocessors is aimed at signal pro cessing algorithms where each data point can be treated more or less independently There s generally not much support for propagating the sort of carries that arise in GMP SIMD multiplications of say four 16x16 bit multiplies only do as much work as one 32x32 from GMP s point of view and need some shifts and adds besides But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile Chapter 16 Algorithms 105 On the 80x86 chips MMX has so far found a use in mpn_rshift and mpn_1shift since it allows 64 bit operations and is used in a special case for 16 bit multipliers in the P55 mpn mul 1 3DNow and SSE haven t found a use so far 16 8 7 Software Pipelining Software pipelining consists of scheduling instructions around the branch point in a loop For example a loop taking
40. bit so that the function has something to return unsigned long int mpn_scani const mp_limb_t s1p unsigned long int Function bit Scan slp from bit position bit for the next set bit It is required that there be a set bit within the area at slp at or beyond bit position bit so that the function has something to return void mpn random mp_limb_t r1p mp size t rin Function void mpn_random2 mp_limb_t r1p mp_size_t rin Function Generate a random number of length rin and store it at rip The most significant limb is always non zero mpn random generates uniformly distributed limb data mpn_random2 generates long strings of zeros and ones in the binary representation mpn_random2 is intended for testing the correctness of the mpn routines unsigned long int mpn_popcount const mp_limb_t s1p mp size t n Function Count the number of set bits in s1p ny Chapter 8 Low level Functions 59 unsigned long int mpn hamdist const mp_limb_t s1p const mp_limb_t Function s2p mp size t n Compute the hamming distance between slp n and s2p n which is the number of bit positions where the two operands have different bit values int mpn perfect square p const mp_limb_t s1p mp size t n Function Return non zero iff slp n is a perfect square 8 1 Nails Everything in this section is highly experimental and may disappear or be subject to incompat ible changes in a future version of GMP Nails are an exper
41. but one shifted up by a factor of b Chapter 16 Algorithms 95 Carrying out full subtractions at each stage means the same number of cross products must be done as a normal division but there s still some single limb divisions saved When d is a single limb some simplifications arise providing good speedups on a number of processors mpn_bdivmod mpn_divexact_by3 mpn modexact 1 odd and the redc function in mpz_powm differ subtly in how they return r leading to some negations in the above formula but all are essentially the same Clearly r is zero when a is a multiple of d and this leads to divisibility or congruence tests which are potentially more efficient than a normal division The factor of b on r can be ignored in a GCD when d is odd hence the use of mpn_bdivmod in mpn gcd and the use of mpn modexact 1 odd by mpn_gcd_1 and mpz kronecker ui etc see Section 16 3 Greatest Common Divisor Algorithms page 95 Montgomery s REDC method for modular multiplications uses operands of the form of zb and yb and on calculating x6 yb uses the factor of b in the exact remainder to reach a product in the same form zy b see Section 16 4 2 Modular Powering Algorithm page 97 Notice that r generally gives no useful information about the ordinary remainder a mod d since b mod d could be anything If however b 1 mod d then r is the negative of the ordinary remainder This occurs whenever d is a factor of
42. by taking GCDs with the original inputs gcd u gcd v g using the binary algorithm Since g is almost always small this takes very little time At small sizes the algorithm needs a good implementation of find_a At larger sizes it s domi nated by mpn_addmul_1 applying n and d 16 3 3 Extended GCD The extended GCD calculates gcd a b and also cofactors x and y satisfying ax by gcd a b Lehmer s multi step improvement of the extended Euclidean algorithm is used See Knuth section 4 5 2 algorithm L and mpn generic gcdext c This is an O N algorithm The multipliers at each step are found using single limb calculations for sizes up to GCDEXT_ THRESHOLD or double limb calculations above that The single limb code is faster but doesn t produce full limb multipliers hence not making full use of the mpn_addmul_1 calls When a CPU has a data dependent multiplier meaning one which is faster on operands with fewer bits the extra work in the double limb calculation might only save some looping overheads leading to a large GCDEXT THRESHOLD Currently the single limb calculation doesn t optimize for the small quotients that often occur and this can lead to unusually low values of GCDEXT_THRESHOLD depending on the CPU Chapter 16 Algorithms 97 An analysis of double limb calculations can be found in A Double Digit Lehmer Euclid Algo rithm by Jebelean see Appendix B References page 113 The code in GMP was develope
43. canonicalize ooooooo o 75 npq Class get Gian iaa e Ae Aere 75 npq Classi Get don oe obe PRG EE T5 pq class get den mpz t i ic oy leet 75 Mpq class get mpq t geese Sie 73 Mpq class get mmm oo oe d e rere eee 75 mpq class get num mpz t esee 75 npq class zmpq Classo cess uou iube Rd 74 75 A le acad sda teases DUIS 42 MPO CHpoicicideemeeT eR x Dhu P iren 44 MPALCMP Sia id ra OR aerie iterare 44 lee e JU EE 44 mpd denref l ienisssg c e RR rep EHE 44 Sasae Ps E E E E EEE EE 43 mpd div 26K Ds rerepi eog Aas oe EE ee E 43 MPG qua iia 44 MPALE EE 43 ue TEE 44 MPG get MUM cid ae tad Sete Ga vd teed lee dao 44 o ner aa aS 43 lipq E n h EE 42 pg inp Sti a dees hee ded us 45 11n 310 EE 44 PP 43 mnpq mul Zeep EE rrap since eR Ries 43 el ii ii a ERE 43 MPAA aaa 44 MPQLOUCLSED iii ici 45 ue E diia ais 42 npa set DEE 43 npq set iia e 44 npq Set f exe PRU ext Re RR Reo 43 MPG Set MUM e eta peak eo Rei RINGS 44 p Seb 15 45 03 eee a ei ee ned 42 mpg Set Sti 2 heb dead deg n EUR ER EE PE 42 MPG Set E EE 42 MPO SCC er RENE PERDRE RP YR seas 42 126 e BEE 44 Mpa SUD EE 43 npq SWapcic e kem jm be m bee nde dea capers ele Ri 43 pq t ssnsidbioceeTe4eeee mag aaa 16 npz abS i2arecicoee eR IRE PPP ERR DERI RUE 31 MP2 add et aa 31 npz add Ui yids cerca weed ne did Cae re ag 31 Luet OM iii ia 31 npz addinul Uri e gn ia ei 31 A E 37 npz EE ir wee een ee 28 npz bin EE 36 npz bin Miso at eden
44. char initial value Function Allocate an integer consisting of a MINT object and dynamic limb space Initialize the integer from initial value a hexadecimal null terminated C string Return a pointer to the MINT object void move MINT src MINT dest Function Set dest to src by copying Both variables must be previously initialized void madd MINT src 1 MINT src 2 MINT destination Function Add src 1 and src 2 and put the sum in destination void msub MINT src 1 MINT src 2 MINT destination Function Subtract src 2 from src 1 and put the difference in destination void mult MINT src 1 MINT src 2 MINT destination Function Multiply src 1 and src 2 and put the product in destination void mdiv MINT dividend MINT divisor MINT quotient MINT Function remainder void sdiv MINT dividend signed short int divisor MINT quotient Function signed short int remainder Set quotient to dividend divisor and remainder to dividend mod divisor The quotient is rounded towards zero the remainder has the same sign as the dividend unless it is zero Some implementations of these functions work differently or not at all for negative argu ments void msqrt MINT op MINT root MINT remainder Function Set root to op like mpz_sqrt Set remainder to op root i e zero if op is a perfect square If root and remainder are the same variable the results are undefined Chapter 13 Berkeley
45. destination variable Each function is defined to calculate with infinite precision followed by a truncation to the destination precision but of course the work done is only what s needed to determine a result under that definition The precision selected for a variable is a minimum value GMP may increase it a little to facilitate efficient calculation Currently this means rounding up to a whole limb and then sometimes having a further partial limb depending on the high limb of the mantissa But applications shouldn t be concerned by such details The mantissa in stored in binary as might be imagined from the fact precisions are expressed in bits One consequence of this is that decimal fractions like 0 1 cannot be represented exactly The same is true of plain IEEE double floats This makes both highly unsuitable for calculations involving money or other values that should be exact decimal fractions Suitably scaled integers or perhaps rationals are better choices mpf functions and variables have no special notion of infinity or not a number and applications must take care not to overflow the exponent or results will be unpredictable This might change in a future release Note that the mpf functions are not intended as a smooth extension to IEEE P754 arithmetic In particular results obtained on one computer often differ from the results on a computer with a different word size 7 1 Initialization Functions void mpf_set_
46. elementary linear algebra quickly isolates each w by subtracting multiples of one equation from another In the code the set of five values W 0 W oo will represent those certain linear combinations By adding or subtracting one from another as necessary values which are each w alone are arrived at This involves only a few subtractions of small multiples some of which are powers of 2 and so is fast A couple of divisions remain by powers of 2 and one division by 3 or by 6 rather and that last uses the special mpn divexact by3 see Section 16 2 4 Exact Division page 93 In the code the values wa wa and wy are formed in the destination with pointers E C and A and ws and w in temporary space D and B are added to them There are extra limbs tD tC and tB at the high end of w3 wa and w which are handled separately The final addition then is as follows high low D B tD tC tB The conversion of W t values to the coefficients is interpolation A polynomial of degree 4 like W t is uniquely determined by values known at 5 different points The points can be chosen to make the linear equations come out with a convenient set of steps for isolating the w In mpn generic mul_n c the interpolate3 routine performs the interpolation The open coded one pass version may be a bit hard to understand the steps performed can be better seen in the USE MORE MPN
47. ete AE A 98 16 6 1 Binary to Radix ere re a dedo 98 16 6 2 Radix to Binary 22304 Abee ei reet ia EAR UR oA Se ont eee aa 99 16 7 Other Algorithms eir xa ECRIRE LR lec DR RT A UR ade a 100 16 7 1 Factorlal erre RR ere rA RH E oad cobra RR E ae 100 16 7 2 Binomial Coefficient iiec eed a Reda eka n Red ERU d don a s 100 16 73 Fibonacci Numbers 3 203 ses pr ebe dats 101 1077 4 Lucas Numbers ae ere a eed axo nikon wide dance deine RO ins 101 16 8 Assembler Coding iua du debe ede teed eta did b eb eae etd D e done 102 16 8 Cod OreanisatiOn Lau sea nett d te a ORA Deeg RU Ue dn RA E 102 16 8 2 Assembler Basics 102 16 8 8 Carry Propagation asispa dria cece cnet hh 102 16 8 4 Cache Handling cargado hh eked ec dhe EE E eR 103 16 85 Floating Pot aria ue palati E Rp epu eed e tte epe 103 16 8 6 SIMD Instruction 104 16 8 7 Software Pipelining viii dd bd axi PRA rapta Pa dede 105 16 8 8 Loop Unroll g ur reete o eg cgo nete en Pt Pe Rae ee Roe 105 lr ARENAS serere EE too et eee Eur AAN 106 17 1 Integer Internals ii DRE RE Re ee dete ade ode Lea ee ERE 106 17 2 Rational Internals oi Ete a EE Dea nde CACHER E de UR ER RU d 106 1753 Float InternalS Rr ne by DERE Magee M THERE bee ec 107 lr4 Raw Output Internals viscosa td EN Ee Ea een PCR Bae 108 17 5 C Interface Internals ec eee ret ett nto Ron Re e CORR COR Deo dC Ries 109 Appendix A ConibiloESees4wwexkbew e bewtiker a xar 111 iv GNU MP 4 1 3 Appendix
48. http www math u bordeaux fr cohen Donald E Knuth The Art of Computer Programming volume 2 Seminumerical Algo rithms 3rd edition Addison Wesley 1998 http www cs faculty stanford edu knuth taocp html John D Lipson Elements of Algebra and Algebraic Computing The Benjamin Cummings Publishing Company Inc 1981 Alfred J Menezes Paul C van Oorschot and Scott A Vanstone Handbook of Applied Cryptography http www cacr math uwaterloo ca hac Richard M Stallman Using and Porting GCC Free Software Foundation 1999 avail able online http www gnu org software gcc onlinedocs and in the GCC package ftp ftp gnu org gnu gcc B 2 Papers Yves Bertot Nicolas Magaud and Paul Zimmermann A Proof of GMP Square Root Journal of Automated Reasoning volume 29 2002 pp 225 252 Also available online as INRIA Research Report 4475 June 2001 http www inria fr rrrt rr 4475 html Christoph Burnikel and Joachim Ziegler Fast Recursive Division Max Planck Institut fuer Informatik Research Report MPI I 98 1 022 http data mpi sb mpg de internet reports nsf NumberView 1998 1 022 Torbjorn Granlund and Peter L Montgomery Division by Invariant Integers using Multi plication in Proceedings of the SIGPLAN PLDI 94 Conference June 1994 Also available ftp ftp cwi nl pub pmontgom divcnst psa4 gz and psl gz Peter L Montgomery Modular Multiplication Without Trial Division in Math
49. in the extra limb of space and mp size becomes mp prec 1 Whenever mp prec 1 limbs are held in a variable the low limb is not needed for the intended precision only the mp prec high limbs But zeroing it out or moving the rest down is unnecessary Subsequent routines reading the value will simply take the high limbs they need and this will be mp prec if their target has that same precision This is no more than a pointer adjustment and must be checked anyway since the destination precision can be different from the sources Copy functions like mpf set will retain a full mp prec 1 limbs if available This ensures that a variable which has mp size equal to mp prec 1 will get its full exact value copied Strictly speaking this is unnecessary since only mp prec limbs are needed for the application s requested precision but it s considered that an mpf _ set from one variable into another of the same precision ought to produce an exact copy Application Precisions __GMPF_BITS_TO_PREC converts an application requested precision to an _mp_prec The value in bits is rounded up to a whole limb then an extra limb is added since the most significant limb of _mp_d is only non zero and therefore might contain only one bit __GMPF_PREC_TO_BITS does the reverse conversion and removes the extra limb from _mp_prec before converting to bits The net effect of reading back with mpf_get_ prec is simply the precision rounded up to a multiple o
50. increase the threshold since they benefit M N more than K N And conversely the 2 for b means linear style speedups of b will increase the threshold since they benefit K N more than M N The latter can be seen for instance when adding an optimized mpn sqr diagonal to mpn_sqr_basecase Of course all speedups reduce total time and in that sense the algorithm thresholds are merely of academic interest 16 1 3 Toom Cook 3 Way Multiplication The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom Cook and FFT algorithms A description of Toom Cook can be found in Knuth section 4 3 3 with an example 3 way calculation after Theorem A The 3 way form used in GMP is described here The operands are each considered split into 3 pieces of equal length or the most significant part 1 or 2 limbs shorter than the others high low X2 Ly Xo Ya Yi Yo These parts are treated as the coefficients of two polynomials X t Lot z t To Y t yo yit Yo Again let b equal the power of 2 which is the size of the xo 11 yo and y pieces ie if they re k limbs each then b 2 mp bits_per_limb With this x X b and y Y t Let a polynomial W t X t Y t and suppose its coefficients are W t wat wat wot wit wo The w are going to be determined and when they are they ll give the final result using w W b since zy X b Y b The
51. k 8 range somewhere between 3000 and 10000 limbs The way N is split into 2 pieces and then 2M k 3 is rounded up to a multiple of 2 and mp bits per limb means that when 2 gt mp bits per limb the effective N is a multiple of 22k 1 bits The k 3 means some values of N just under such a multiple will be rounded to the next The complexity calculations above assume that a favourable size is used meaning one which isn t padded through rounding and it s also assumed that the extra k 3 bits are negligible at typical FFT sizes The practical effect of the 2 constraint is to introduce a step effect into measured speeds For example k 8 will round N up to a multiple of 32768 bits so for a 32 bit limb there ll be 512 limb groups of sizes for which mpn_mul_n runs at the same speed Or for k 9 groups of 2048 limbs k 10 groups of 8192 limbs etc In practice it s been found each k is used at quite small multiples of its size constraint and so the step effect is quite noticeable in a time versus size graph The threshold determinations currently measure at the mid points of size steps but this is sub optimal since at the start of a new step it can happen that it s better to go back to the previous k for a while Something more sophisticated for MUL FFT TABLE and SQR FFT TABLE will be needed 16 1 5 Other Multiplication The 3 way Toom Cook algorithm described above see Section 16 1 3 Toom Cook 3 Way Multi plication
52. limbs but the return value and value stored through r2n indicate the actual number produced sip sin gt s2p s2n is required and both must be non zero The regions slp sin 1 and s2p s2n 1 are destroyed i e the operands plus an extra limb past the end of each The cofactor r1 will satisfy ros ks r The second cofactor k is not calculated but can easily be obtained from r ras1 s3 mp size t mpn_sqrtrem mp_limb_t r1p mp_limb_t r2p const Function mp_limb_t sp mp size t n Compute the square root of sp n and put the result at rlp n 2 and the remainder at r2p retval r2p needs space for n limbs but the return value indicates how many are produced The most significant limb of sp n must be non zero The areas rlp n 2 and tsp n must be completely separate The areas r2p n and Zon n must be either identical or completely separate If the remainder is not wanted then r2p can be NULL and in this case the return value is zero or non zero according to whether the remainder would have been zero or non zero 58 GNU MP 4 1 3 A return value of zero indicates a perfect square See also mpz perfect square p mp size t mpn get str unsigned char str int base mp_limb_t s1p Function mp size t sin Convert slp sln to a raw unsigned char array at str in base base and return the number of characters produced There may be leading zeros in the string The string is not in ASCII
53. mask But other multiples can be used too just by subtracting each n limbs processed from a counter and waiting for less than n remaining or offsetting the counter by n so it goes negative when there s less than n remaining The limbs not a multiple of the unrolling can be handled in various ways for example e A simple loop at the end or the start to process the excess Care will be wanted that it isn t too much slower than the unrolled part e A set of binary tests for example after an 8 limb unrolling test for 4 more limbs to process then a further 2 more or not and finally 1 more or not This will probably take more code space than a simple loop e A switch statement providing separate code for each possible excess for example an 8 limb unrolling would have separate code for 0 remaining 1 remaining etc up to 7 remaining This might take a lot of code but may be the best way to optimize all cases in combination with a deep pipelined loop e A computed jump into the middle of the loop thus making the first iteration handle the excess This should make times smoothly increase with size which is attractive but setups for the jump and adjustments for pointers can be tricky and could become quite difficult in combination with deep pipelining One way to write the setups and finishups for a pipelined unrolled loop is simply to duplicate the loop at the start and the end then delete instructions at the start which have no valid
54. might be predicted The formula given above for the Karatsuba algorithm has an equivalent for Toom 3 involving only five multiplies but this would be complicated and unenlightening An alternate view of Toom 3 can be found in Zuras see Appendix B References page 113 using a vector to represent the r and y splits and a matrix multiplication for the evaluation and interpolation stages The matrix inverses are not meant to be actually used and they have elements with values much greater than in fact arise in the interpolation steps The diagram shown for the 3 way is attractive but again doesn t have to be implemented that way and for example with a bit of rearrangement just one division by 6 can be done 16 1 4 FFT Multiplication At large to very large sizes a Fermat style FFT multiplication is used following Sch nhage and Strassen see Appendix B References page 113 Descriptions of FFTs in various forms can be found in many textbooks for instance Knuth section 4 3 3 part C or Lipson chapter IX A brief description of the form used in GMP is given here The multiplication done is zy mod 2 1 for a given N A full product xy is obtained by choosing N gt bits x bits y and padding x and y with high zero limbs The modular product is the native form for the algorithm so padding to get a full product is unavoidable The algorithm follows a split evaluate pointwise multiply interpolate and combine similar to that described
55. mpq add For example mpq increment mpz add mpq_numref q mpq numref q mpq denref q mpq unsigned long mpz_addmul_ui mpq numref q mpq denref q 123UL mpq mpz mpz submul mpq numref q mpq denref q z Number Sequences Functions like mpz fac ui mpz fib ui and mpz bin uiui are designed for calcu lating isolated values If a range of values is wanted it s probably best to call to get a starting point and iterate from there Text Input Output Hexadecimal or octal are suggested for input or output in text form Power of 2 bases like these can be converted much more efficiently than other bases like Chapter 3 GMP Basics 23 decimal For big numbers there s usually nothing of particular interest to be seen in the digits so the base doesn t matter much Maybe we can hope octal will one day become the normal base for everyday use as proposed by King Charles XII of Sweden and later reformers 3 12 Debugging Stack Overflow Depending on the system a segmentation violation or bus error might be the only indication of stack overflow See enable alloca choices in Section 2 1 Build Options page 4 for how to address this In new enough versions of GCC fstack check may be able to ensure an overflow is recognised by the system before too much damage is done or fstack limit symbol or fstack limit register may be able to add checking if the system itself do
56. mpz_t r mpz_t n unsigned long int b Function void mpz tdiv q mpz_t q mpz t n mpz_t d Function void mpz tdiv r mpz t r mpz t n mpz t d Function void mpz tdiv qr mpz_t q mpz_t r mpz_t n mpz_t d Function unsigned long int mpz tdiv q ui mpz_t q mpz t n Function unsigned long int d unsigned long int mpz tdiv r ui mpz_t r mpz_t n Function unsigned long int d unsigned long int mpz tdiv qr ui mpz td mpz_t r mpz_t n Function unsigned long int d unsigned long int mpz tdiv ui mpz_t n unsigned long int d Function void mpz tdiv q 2exp mpz t q mpz t n unsigned long int b Function void mpz tdiv r 2exp mpz_t r mpz t n unsigned long int b Function Divide n by d forming a quotient q and or remainder r For the 2exp functions d 2 The rounding is in three styles each suiting different applications e cdiv rounds q up towards 00 and r will have the opposite sign to d The c stands for ceil e fdiv rounds q down towards oo and r will have the same sign as d The f stands for floor Chapter 5 Integer Functions 33 e tdiv rounds q towards zero and r will have the same sign as n The t stands for truncate In all cases q and r will satisfy n qd r and r will satisfy 0 lt r lt d The q functions calculate only the quotient the r functions only the remainder and the qr functions calculate both Note that for qr the same variable cannot be passed for both q and r or r
57. op2 Function void mpf ui sub mpf_t rop unsigned long int opi mpf t op2 Function void mpf sub ui mpf_t rop mpf t op1 unsigned long int op2 Function Set rop to opl op2 void mpf mul mpf_t rop mpf t opi mpf_t op2 Function void mpf mul ui mpf_t rop mpf t op1 unsigned long int op2 Function Set rop to opl x op2 Division is undefined if the divisor is zero and passing a zero divisor to the divide functions will make these functions intentionally divide by zero This lets the user handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions void mpf div mpf_t rop mpf t opi mpf_t op2 Function void mpf ui div mpf_t rop unsigned long int opi mpf_t op2 Function void mpf div ui mpf_t rop mpf t op1 unsigned long int op2 Function Set rop to op1 op2 void mpf sqrt mpf_t rop mpf_t op Function void mpf sqrt ui mpf_t rop unsigned long int op Function Set rop to op void mpf pow ui mpf_t rop mpf t op1 unsigned long int op2 Function Set rop to op1 void mpf neg mpf_t rop mpf t op Function Set rop to op void mpf abs mpf_t rop mpf t op Function Set rop to the absolute value of op void mpf mul 2exp mpf t rop mpf t op1 unsigned long int op2 Function Set rop to op1 x 2 void mpf div 2exp mpf_t rop mpf_t op1 unsigned long int op2 Function Set rop to op1 2 7 6 Comparison Functions int mpf cmp mpf t opi mpf t o
58. python org GMPY http gmpy sourceforge net GNU Guile upcoming 1 8 http www gnu org software guile guile html RScheme http www rscheme org STklos http kaolin unice fr STklos GNU Smalltalk http www smalltalk org versions GNUSmalltalk html Axiom http savannah nongnu org projects axiom Computer algebra using GCL DrGenius http drgenius seul org Geometry system and mathematical programming language GiNaC nhttp www ginac de C computer algebra using CLN GOO http www googoogaga org Dynamic object oriented language Maxima http www ma utexas edu users wfs maxima html Macsyma computer algebra using GCL Q http www musikwissenschaft uni mainz de ag q Equational programming system Regina http regina sourceforge net Topological calculator Yacas http www xs4all nl apinkus yacas html Yet another computer algebra system 86 GNU MP 4 1 3 16 Algorithms This chapter is an introduction to some of the algorithms used for various GMP operations The code is likely to be hard to understand without knowing something about the algorithms Some GMP internals are mentioned but applications that expect to be compatible with future GMP releases should take care to use only the documented functions 16 1 Multiplication NxN limb multiplications and squares are done using one of four algorithms as the size N increases Algorithm Threshold Basecase none Karatsuba MUL_KARATSUBA_THRESHOLD Toom 3 MUL_T
59. sections Entitled Dedications You must delete all sections Entitled Endorsements Appendix C GNU Free Documentation License 119 10 COLLECTIONS OF DOCUMENTS You may make a collection consisting of the Document and other documents released under this License and replace the individual copies of this License in the various documents with a single copy that is included in the collection provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects You may extract a single document from such a collection and distribute it individually under this License provided you insert a copy of this License into the extracted document and follow this License in all other respects regarding verbatim copying of that document AGGREGATION WITH INDEPENDENT WORKS A compilation of the Document or its derivatives with other separate and independent documents or works in or on a volume of a storage or distribution medium is called an aggregate if the copyright resulting from the compilation is not used to limit the legal rights of the compilation s users beyond what the individual works permit When the Document is included in an aggregate this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document If the Cover Text requirement of section 3 is applicable to these copies of the Document then if the Document is less than
60. size t size gmp printf limb array Nx n ptr size For N the limbs are expected least significant first as per the mpn functions see Chapter 8 Low level Functions page 53 A negative size can be given to print the value as a negative All the standard C printf types behave the same as the C library printf and can be freely intermixed with the GMP extensions In the current implementation the standard parts of the format string are simply handed to printf and only the GMP extensions handled directly The flags accepted are as follows GLIBC style is only for the standard C types not the GMP types and only if the C library supports it 0 pad with zeros rather than spaces show the base with Ox OX or 0 always show a sign space show a space or a sign group digits GLIBC style not GMP types The optional width and precision can be given as a number within the format string or as a to take an extra parameter of type int the same as the standard printf The standard types accepted are as follows h and 1 are portable the rest will depend on the compiler or include files for the type and the C library for the output h short hh char j intmax_t or uintmax_t 1 long or wchar_t 11 long long L long double 64 GNU MP 4 1 3 q quad_t or u_quad_t t ptrdiff_t Z size_t The GMP types are mpf_t float conversions mpq_t integer conversions mp_limb_t
61. terms int mpz divisible p mpz_t n mpz td Function int mpz divisible ui p mpz_t n unsigned long int d Function int mpz divisible 2exp p mpz_t n unsigned long int b Function Return non zero if n is exactly divisible by d or in the case of mpz divisible 2exp p by 2 int mpz_congruent_p mpz n mpz t c mpz t d Function int mpz congruent ui p mpz n unsigned long int c unsigned long int d Function int mpz congruent 2exp p mpz_t n mpz t c unsigned long int b Function Return non zero if n is congruent to c modulo d or in the case of mpz congruent 2exp p modulo 2 34 GNU MP 4 1 3 5 7 Exponentiation Functions void mpz_powm mpz t rop mpz_t base mpz t exp mpz_t mod Function void mpz powm ui mpz_t rop mpz_t base unsigned long int exp mpz_t Function mod Set rop to base mod mod Negative exp is supported if an inverse base mod mod exists see mpz_invert in Section 5 9 Number Theoretic Functions page 34 If an inverse doesn t exist then a divide by zero is raised void mpz pow ui mpz t rop mpz_t base unsigned long int exp Function void mpz ui pow ui mpz t rop unsigned long int base unsigned long int Function exp Set rop to base The case 0 yields 1 5 8 Root Extraction Functions int mpz root mpz t rop mpz_t op unsigned long int n Function Set rop to 0p the truncated integer part of the nth root of op Return non zero if the computation wa
62. the GMP types to suppress assignment is allowed though the field would then do nothing at all Other conversions or types that might be accepted by the C library scanf cannot be used through gmp_scanf Whitespace is read and discarded before a field except for c and conversions For float conversions the decimal point character or string expected is taken from the current locale settings on systems which provide localeconv see section Locales and International ization in The GNU C Library Reference Manual The C library will normally do the same for standard float input The format string is only interpreted as plain chars multibyte characters are not recognised Perhaps this will change in the future 11 2 Formatted Input Functions Each of the following functions is similar to the corresponding C library function The plain scanf forms take a variable argument list The vscanf forms take an argument pointer see section Variadic Functions in The GNU C Library Reference Manual or man 3 va_start 70 GNU MP 4 1 3 It should be emphasised that if a format string is invalid or the arguments don t match what the format specifies then the behaviour of any of these functions will be unpredictable GCC format string checking is not available since it doesn t recognise the GMP extensions No overlap is permitted between the fmt string and any of the results produced int gmp_scanf const
63. there would be the cost of calculating a bigger radix power Another possible improvement for the sub quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced ie the highest power would be an b approximately equal to vt not restricted to a 2 factor That ought to smooth out a graph of times against sizes but may or may not be a net speedup 16 6 2 Radix to Binary Conversions from a power of 2 radix into binary use a simple and fast O N bitwise concatena tion algorithm Conversions from other radices use one of two algorithms Sizes below SET STR THRESHOLD use a basic O N method Groups of n digits are converted to limbs where n is the biggest power of the base b which will fit in a limb then those groups are accumulated into the result by multiplying by 6 and adding This saves multi precision operations as per Knuth section 4 4 100 GNU MP 4 1 3 part E see Appendix B References page 113 Some special case code is provided for decimal giving the compiler a chance to optimize multiplications by 10 Above SET_STR_THRESHOLD a sub quadratic algorithm is used First groups of n digits are converted into limbs Then adjacent limbs are combined into limb pairs with xb y where x and y are the limbs Adjacent limb pairs are combined into quads similarly with xb y This continues until a single block remains that being the result The advantage of this method is t
64. this document under the terms of the GNU Free Documentation License Version 1 2 or any later version published by the Free Software Foundation with no Invariant Sections no Front Cover Texts and no Back Cover Texts A copy of the license is included in the section entitled GNU Free Documentation License If you have Invariant Sections Front Cover Texts and Back Cover Texts replace the with Texts line with this with the Invariant Sections being list their titles with the Front Cover Texts being list and with the Back Cover Texts being list If you have Invariant Sections without Cover Texts or some other combination of the three merge those two alternatives to suit the situation If your document contains nontrivial examples of program code we recommend releasing these examples in parallel under your choice of free software license such as the GNU General Public License to permit their use in free software Appendix C Concept Index Concept Index e o ee neo ni eeano ne bia pais hieis 4 beet EE 4 ABlIG ien Bettie e dae ERU RU EH He 9 About this manual 2 E e 86 ALLOCA es suae pacte OE aug RUNE Rear iit Allocation of memory sees 82 Anonymous FTP of latest version 2 Application Binary Interface 9 Arithmetic functions sssuaeseses 31 48 50 Assignment functions oocooooooccooo o 29 48 Autoconf detections 0 000 eee cece eee
65. to convert it to printable format add the ASCII codes for 0 or A depending on the base and range base can vary from 2 to 256 The most significant limb of the input slp sin must be non zero The input s1p s1n is clobbered except when base is a power of 2 in which case it s unchanged The area at str has to have space for the largest possible number represented by a sin long limb array plus one extra character mp size t mpn set str mp_limb_t rp const unsigned char str size_t Function strsize int base Convert bytes str strsize in the given base to limbs at rp str 0 is the most significant byte and str strsize 1 is the least significant Each byte should be a value in the range 0 to base 1 not an ASCII character base can vary from 2 to 256 The return value is the number of limbs written to rp If the most significant input byte is non zero then the high limb at rp will be non zero and only that exact number of limbs will be required there If the most significant input byte is zero then there may be high zero limbs written to rp and included in the return value strsize must be at least 1 and no overlap is permitted between str strsize and the result at rp unsigned long int mpn scanO const mp_limb_t s1p unsigned long int Function bit Scan slp from bit position bit for the next clear bit It is required that there be a clear bit within the area at slp at or beyond bit position
66. 5 Integer Functions 35 Miller Rabin and similar tests can be more properly called compositeness tests Numbers which fail are known to be composite but those which pass might be prime or might be composite Only a few composites pass hence those which pass are considered probably prime void mpz nextprime mpz t rop mpz t op Function Set rop to the next prime greater than op This function uses a probabilistic algorithm to identify primes For practical purposes it s adequate the chance of a composite passing will be extremely small void mpz gcd mpz_t rop mpz t opi mpz_t op2 Function Set rop to the greatest common divisor of op and op2 The result is always positive even if one or both input operands are negative unsigned long int mpz gcd ui mpz_t rop mpz_t opi unsigned long int Function op2 Compute the greatest common divisor of op1 and op2 If rop is not NULL store the result there If the result is small enough to fit in an unsigned long int it is returned If the result does not fit O is returned and the result is equal to the argument opl Note that the result will always fit if op2 is non zero void mpz gcdext mpz t g mpz t S mpz_t t mpz_t a mpz t b Function Set g to the greatest common divisor of a and b and in addition set s and t to coefficients satisfying as bt g g is always positive even if one or both of a and b are negative If t is NULL then that value is not computed void mpz lc
67. 6 Rational Number Functions page 42 The base can be between 2 and 36 or can be 0 in which case the leading characters of the string determine the base Ox or OX for hexadecimal 0 for octal or decimal otherwise The leading characters are examined separately for the numerator and denominator of a fraction so for instance 0x10 11 is 16 11 whereas 0x10 0x11 is 16 17 46 GNU MP 4 1 3 7 Floating point Functions GMP floating point numbers are stored in objects of type mpf_t and functions operating on them have an mpf_ prefix The mantissa of each float has a user selectable precision limited only by available memory Each variable has its own precision and that can be increased or decreased at any time The exponent of each float is a fixed precision one machine word on most systems In the current implementation the exponent is a count of limbs so for example on a 32 bit system this means a range of roughly 2768719476768 t9 268719476736 or on a 64 bit system this will be greater Note however mpf_get_str can only return an exponent which fits an mp_exp_t and currently mpf_set_str doesn t accept exponents bigger than a long Each variable keeps a size for the mantissa data actually in use This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation even if the selected precision is high All calculations are performed to the precision of the
68. A EEN 59 9 Random Number Functions oo ooo oo o o o 61 HL Random State Initialization 5 ob sas Sandee A a a a 61 9 2 Random State Seeding ss errepre derete ei dead trae dx op dE dob eA EN ee as 62 10 Formatted Output eg tia riada RR 63 10 L Format Strings 424452 nenia Adee dd odin peste a dae dad Ged badd Pct ado Andee 63 10 2 UIC OMS EE 64 10 3 Gre Formatted Output isses ne eee ee EE RR RR REOR RUR ERROR E RO RR ded 66 ll Formatted Inpub vaes E xe RR E teens Rcs 68 11 1 Formatted Input SMS ve uen rtm ore Un tle te e eu ea Mt eer 68 11 2 Formatted Input Functons 0 000 ehh hh 69 MES CE Formatted Wi GE 70 12 C Class Inter a Gasaa dona REX dad 12 12 1 C Interface General ae T2 12 2 C Interface Integers viciado 73 12 3 C Interface Bat onals raa 74 12 4 C Interface Flota a E a A had eed eee ede dd 75 12 5 Cae Interface MPER eere n eee t oet esce dr acr Lh T 12 6 C Interface Random Number Var 12 7 C Interface Limitations a hes 78 13 Berkeley MP Compatible Functions 80 14 Custom Allocation oso rco chews ege ones 82 15 Language Bindings 000 AR 84 16 e eh EE 86 16 1 Multiplication e Rer dag ei Ae OC CERE RR eee ea deed dee eae ee 86 16 1 1 Basecase Multiplication 00oooocccccccccccnccr hh 86 16 1 2 Karatsuba Multpbcaton ees 87 16 1 3 Toom Cook 3 Way Multiplication 0 0 0 cece eect n 88 16 1 4 FET Multiplication cima cbe re
69. For example in some 32 bit ABIs GMP may support a limb as either a 32 bit long or a 64 bit long long By default GMP chooses the best ABI available for a given system and this generally gives significantly greater speed But an ABI can be chosen explicitly to make GMP compatible with other libraries or particular application requirements For example configure ABI 32 In all cases it s vital that all object code used in a given program is compiled for the same ABI Usually a limb is implemented as a long When a long long limb is used this is encoded in the generated gmp h This is convenient for applications but it does mean that gmp h will vary and can t be just copied around gmp h remains compiler independent though since all compilers for a particular ABI will be expected to use the same limb type Currently no attempt is made to follow whatever conventions a system has for installing library or header files built for a particular ABI This will probably only matter when installing multiple builds of GMP and it might be as simple as configuring with a special libdir or it might require more than that Note that builds for different ABIs need to done separately with a fresh configure and make each HPPA 2 0 hppa2 0 ABI 2 0w The 2 0w ABI uses 64 bit limbs and pointers and is available on HP UX 11 or up when using cc gcc support for this is in progress Applications must be compiled with
70. GNU MP The GNU Multiple Precision Arithmetic Library Edition 4 1 3 22 April 2004 by Torbjorn Granlund Swox AB tegeOswox com This manual describes how to install and use the GNU multiple precision arithmetic library version 4 1 3 Copyright 1991 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Free Software Foundation Inc Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 1 or any later version published by the Free Software Foundation with no Invariant Sections with the Front Cover Texts being A GNU Manual and with the Back Cover Texts being You have freedom to copy and modify this GNU Manual like GNU software A copy of the license is included in Appendix C GNU Free Documentation License page 115 Table of Contents GNU MP Copying Conditions ss d da zik io Ra moa 1 1 Introduction to GNU UE erg erg Srel a Beas 2 11 How to use this Manual 2 2 Installing GMP a we g ere cases er er error RES RR 4 2 1 Build Options iude a 4 2 2 AND Lad 194 ru obedit Sut dedidit qund cds oe he Ia 9 2 9 Notes for Package Builds sse vee rre peo sod ER eon C n Re 11 2 4 Notes for Particular Systems ssssseesseseeslee ehh 11 2 5 Known Buld Problems is 22352939 ld und dad eat e Y eurn ud esce den vds 13 3 GMP Basics 552236 ka Rm ra aa A A Ra RE EV 16 3 1 Headers and Libraries ac Zei id draw eke das Gade
71. Mode On some systems the hardware floating point has a control mode which can set all operations to be done in a particular precision for instance single double or extended on x86 systems x87 floating point The GMP functions involving a double cannot be expected to operate to their full precision when the hardware is in single precision mode Of course this affects all code including application code not just GMP Microsoft Windows On systems cygwin mingw and pw32 by default GMP builds only a static library but a DLL can be built instead using configure disable static enable shared Static and DLL libraries can t both be built since certain export directives in gmp h must be different enable cxx cannot be used when building a DLL since libtool doesn t currently support C DLLs This might change in the future Microsoft C A MINGW DLL build of GMP can be used with Microsoft C Libtool doesn t install 1ib and exp files but they can be created with the following commands where my inst dir is the install directory with a lib subdirectory lib machine IX86 def libs libgmp 3 dll def cp libgmp 3 lib my inst dir lib cp libs libgmp 3 dll exp my inst dir lib libgmp 3 exp MINGW uses the C runtime library msvcrt d11 for I O so applications wanting to use the GMP I O routines must be compiled with cl MD to do the same If one of the other C runtime librar
72. OOM3_THRESHOLD FFT MUL_FFT_THRESHOLD Similarly for squaring with the SQR thresholds Note though that the FFT is only used if GMP is configured with enable fft see Section 2 1 Build Options page 4 NxM multiplications of operands with different sizes above MUL_KARATSUBA_THRESHOLD are currently done by splitting into MxM pieces The Karatsuba and Toom 3 routines then operate only on equal size operands This is not very efficient and is slated for improvement in the future 16 1 1 Basecase Multiplication Basecase NxM multiplication is a straightforward rectangular set of cross products the same as long multiplication done by hand and for that reason sometimes known as the schoolbook or grammar school method This is an O N M algorithm See Knuth section 4 3 1 algorithm M see Appendix B References page 113 and the mpn generic mul_basecase c code Assembler implementations of mpn mul basecase are essentially the same as the generic C code but have all the usual assembler tricks and obscurities introduced for speed A square can be done in roughly half the time of a multiply by using the fact that the cross products above and below the diagonal are the same A triangle of products below the diagonal is formed doubled left shift by one bit and then the products on the diagonal added This can be seen in mpn generic sqr_basecase c Again the assembler implementations take essentially the same approach u0 u
73. P development but might find other uses Stack Backtraces On some systems the compiler options GMP uses by default can interfere with debugging In particular on x86 and 68k systems fomit frame pointer is used and this generally inhibits stack backtracing Recompiling without such options may help while debugging though the usual caveats about it potentially moving a memory problem or hiding a compiler bug will apply GNU Debugger A sample gdbinit is included in the distribution showing how to call some undoc umented dump functions to print GMP variables from within GDB Note that these functions shouldn t be used in final application code since they re undocumented and may be subject to incompatible changes in future versions of GMP 24 GNU MP 4 1 3 Source File Paths GMP has multiple source files with the same name in different directories For example mpz mpq mpf and mpfr each have an init c If the debugger can t already determine the right one it may help to build with absolute paths on each C file One way to do that is to use a separate object directory with an absolute path to the source directory cd my build dir my source dir gmp 4 1 3 configure This works via VPATH and might require GNU make Alternately it might be possible to change the c 1lo rules appropriately Assertion Checking The build option enable assert is available to add some consistency checks to the
74. PU The most significant nails bits of each word are unused and set to zero this can be 0 to produce full words The number of words produced is written to countp or countp can be NULL to discard the count rop must have enough space for the data or if rop is NULL then a result array of the necessary size is allocated using the current GMP allocation function see Chapter 14 Custom Allocation page 82 In either case the return value is the destination used either rop or the allocated block If op is non zero then the most significant word produced will be non zero If op is zero then the count returned will be zero and nothing written to rop If rop is NULL in this case no block is allocated just NULL is returned The sign of op is ignored just the absolute value is exported An application can use mpz_sgn to get the sign and handle it as desired see Section 5 10 Integer Comparisons page 36 There are no data alignment restrictions on rop any address is allowed When an application is allocating space itself the required size can be determined with a calculation like the following Since mpz sizeinbase always returns at least 1 count here will be at least one which avoids any portability problems with malloc 0 though if z is zero no space at all is actually needed or written numb 8 size nail count mpz sizeinbase z 2 numb 1 numb p malloc count size Chapter 5 Integer Functions 41 5 15 Misce
75. Stie perp ni ehh ie dda 48 npf TEE 48 MPL ENEE 48 Mpt SON EE 51 e SEE E E E A EEEE 50 npf sqrt Ul eean t ee cee tiaa eee et RERA 50 mpi SUD EE 50 mnpf SUD RTE 50 MPI SWAP EE 48 let Sti scone A ean RR EE 16 Dnpf EE 52 npf WF EE 50 mnpf ui Sub sie ls ee sie d buds EORR EE 50 Mpf urandoHb oii sss bids esas aa 52 MPL EE 77 MPN Ad ieee Sb aira ee Ree GR PER Ese PS 53 npn add 1 teri rerp secede ew kr aee I ed 53 npn add Miss elie ed epee EE 53 npn addinul DEE 54 npn bdivmnod ewe PERI vr EE US ES 56 npn EN eege as S 57 Mpndivexact D osc sis cce eae d et 56 npn divexact by36 2 wa eem eO RE OR 56 A eee e ee aa eei 56 npn divmod Li a D npn CAV YOM e a EE 55 npn divrem E oie sha 9 22 e 55 125 Npu ged coh pied su tbe 57 ste A 57 e EE 5T mpn get Stre seiere 022609 perai EepR ERIS 58 npn hamdist sso er etr RUE DEKOR RES 59 npu lshift seco e pirr ages See ERU 56 mpn mod A berer iniri CniE E E EEE ER 56 npn MU sie edhe gee ai a 55 mpn DEET EE 54 mpi mil TEE 54 mpn perfect square p RR 59 mpn ee E EE 58 npn randoh cerscoccGauuk ee pene cede ee panied 58 pn ER e EE 58 npn E EE 57 npn SCAM EE 58 ler id er rs EERE UE E i 58 Mp Set Stt ee Rb rra Aia 58 mpn Sqrtrem ieps vae A is 57 A 54 mpi sub TEE een aera aay wi aed a ERR 54 MPA ri iria 54 npn subnul A eens ee headwear 54 AAA A E 55 DEDI PE LL 43 MPG addis an 43 pq CcanoniCcaliz enee EA LEE en 42 mHpq Cl sS u c EE o aa 72 mpq_class
76. T RU 36 5 11 Logical and Bit Manipulation Functions 0 0 0 0 0c cece eee eee eee 37 5 12 Input and Output Functions 4 2 4 12 444 Ueber cetera ERU rae wl AC e 38 5 13 Random Number Functions 00 0 ccc ccc nee ence hh 39 5 14 Integer Import and apart 39 5 15 Miscellaneous Functions 0 00 000 ccc cc ehh Al ii GNU MP 4 1 3 6 Rational Number bunctons eee 42 6 1 Initialization and Assignment Functions 0000 c eee eese 42 6 2 Conversion Functions bd ded ii Pada dad dra ata ees 43 6 3 Arithmetic Functions wm ox tss ound e eee aa ls hed boa donde o e aaa eons 43 6 4 Comparison Functions apai sesane ipida pia ai aaa tet ent hh hee 44 6 5 Applying Integer Functions to Rationals 0 cece eee eee eens 44 6 6 Input and Output Functions 2 eere eoe Ae 45 7T Floating point Functions 22999 9 kn Res 46 Gl Initialization PunetiGns 222 cerner A ORC ond ea daa ER 46 1 2 Assignment Functions sire ne dE EEN dl ee he og I Ron E once ere e o o Re 48 7 3 Combined Initialization and Assignment Functions eiie eese 48 TA Conversion Functions Auger id See Eee Und d Roe bd e on ees PL E und 49 To Arithm tic EurctioHs 2 12e ER ALLERLEE as wed asd EORR eee EUR ANER 50 7 6 Comparison Functions 0 000 ehh hh here 50 7 7 Input and Output Functions 00 ec hh heran 51 CS Miscellaneo s Eunctlolls 2222122 9 ia dai ood 51 8 Low level Functions cccccsccvcccscscvescscves 53 g
77. The following three functions are useful for changing the precision during a calculation A typical use would be for adjusting the precision gradually in iterative algorithms like Newton Raphson making the computation precision closely match the actual accurate part of the numbers unsigned long int mpf_get_prec mpf_t op Function Return the current precision of op in bits void mpf set prec mpf_t rop unsigned long int prec Function Set the precision of rop to be at least prec bits The value in rop will be truncated to the new precision This function requires a call to realloc and so should not be used in a tight loop void mpf set prec raw mpf_t rop unsigned long int prec Function Set the precision of rop to be at least prec bits without changing the memory allocated prec must be no more than the allocated precision for rop that being the precision when rop was initialized or in the most recent mpf set prec The value in rop is unchanged and in particular if it had a higher precision than prec it will retain that higher precision New values written to rop will use the new prec Before calling mpf clear or the full mpf set prec another mpf set prec raw call must be made to restore rop to its original allocated precision Failing to do so will have unpredictable results mpf get prec can be used before mpf set prec raw to get the original allocated precision After mpf set prec raw it reflects the p
78. _init_set_ui mpf_t rop unsigned long int op Function void mpf_init_set_si mpf_t rop signed long int op Function void mpf_init_set_d mpf_t rop double op Function Initialize rop and set its value from op The precision of rop will be taken from the active default precision as set by mpf set default prec int mpf init set str mpf_t rop char str int base Function Initialize rop and set its value from the string in str See mpf set str above for details on the assignment operation Note that rop is initialized even if an error occurs Le you have to call mpf clear for it The precision of rop will be taken from the active default precision as set by mpf set default prec 7 4 Conversion Functions double mpf get d mpf_t op Function Convert op to a double double mpf get d 2exp signed long int exp mpf t op Function Find d and exp such that d x 2 with 0 5 d lt 1 is a good approximation to op This is similar to the standard C function frexp long mpf get si mpf_t op Function unsigned long mpf get ui mpf_t op Function Convert op to a long or unsigned long truncating any fraction part If op is too big for the return type the result is undefined See also mpf fits slong p and mpf fits ulong p see Section 7 8 Miscellaneous Float Functions page 51 char mpf get str char str mp_exp_t expptr int base size_t Function n digits mpf_t op Convert op to a string o
79. _t mpn_mul_1 mp_limb_t rp const mp_limb_t s1p mp size t n Function mp_limb_t s2limb Multiply slp n by s2limb and write the n least significant limbs of the product to rp Return the most significant limb of the product slp n and rp n are allowed to overlap provided rp lt slp This is a low level function that is a building block for general multiplication as well as other operations in GMP It is written in assembly for most CPUs Don t call this function if s2limb is a power of 2 use mpn_1shift with a count equal to the logarithm of s2limb instead for optimal speed mp_limb_t mpn addmul 1 mp_limb_t rp const mp_limb_t s1p mp size t Function n mp_limb_t s21imb Multiply slp n and s2limb and add the n least significant limbs of the product to rp n and write the result to rp Return the most significant limb of the product plus carry out from the addition This is a low level function that is a building block for general multiplication as well as other operations in GMP It is written in assembly for most CPUs mp_limb_t mpn submul 1 mp_limb_t rp const mp_limb_t s1p mp size t Function n mp_limb_t s2limb Multiply slp n and s2limb and subtract the n least significant limbs of the product from rp n and write the result to rp Return the most significant limb of the product minus borrow out from the subtraction Chapter 8 Low level Functions 55 This is a low level function that is
80. a building block for general multiplication and division as well as other operations in GMP It is written in assembly for most CPUs mp_limb_t mpn_mul mp_limb_t rp const mp_limb_t s1p mp size t sin Function const mp_limb_t s2p mp_size_t s2n Multiply slp sin and s2p s2n and write the result to rp Return the most significant limb of the result The destination has to have space for sin s2n limbs even if the result might be one limb smaller This function requires that sin is greater than or equal to s2n The destination must be distinct from both input operands void mpn tdiv qr mp_limb_t qp mp limb t rp mp size t qxn const Function mp_limb_t np mp size t nn const mp_limb_t dp mp size t dn Divide np nn by dp dn and put the quotient at qp nn dn 1 and the remainder at rp dn The quotient is rounded towards 0 No overlap is permitted between arguments nn must be greater than or equal to dn The most significant limb of dp must be non zero The qxn operand must be zero mp limb t mpn_divrem mp_limb_t r1p mp size t qxn mp_limb_t rs2p Function mp size t rs2n const mp_limb_t s3p mp size_t s3n This function is obsolete Please call mpn_tdiv_qr instead for best performance Divide rs2p rs2n by s3p s3n and write the quotient at rip with the exception of the most significant limb which is returned The remainder replaces the dividend at rs2p it will be s3n limbs long i e a
81. a previously allocated block ptr of old size bytes to be new size bytes The block may be moved if necessary or if desired and in that case the smaller of old size and new size bytes must be copied to the new location The return value is a pointer to the resized block that being the new location if moved or just ptr if not ptr is never NULL it s always a previously allocated block new_size may be bigger or smaller than old size void deallocate function void ptr size_t size Function De allocate the space pointed to by ptr ptr is never NULL it s always a previously allocated block of size bytes A byte here means the unit used by the sizeof operator The old_size parameters to reallocate function and deallocate function are passed for conve nience but of course can be ignored if not needed The default functions using malloc and friends for instance don t use them No error return is allowed from any of these functions if they return then they must have per formed the specified operation In particular note that allocate_function or reallocate function mustn t return NULL Chapter 14 Custom Allocation 83 Getting a different fatal error action is a good use for custom allocation functions for example giving a graphical dialog rather than the default print to stderr How much is possible when genuinely out of memory is another question though There s currently no defined way for the allocation functions to re
82. al integer usually means a multiple precision integer as defined by the GMP library The C data type for such integers is mpz_t Here are some examples of how to declare such integers mpz_t sum struct foo mpz_t x y mpz_t vec 20 Rational number means a multiple precision fraction The C data type for these fractions is mpq_t For example mpq t quotient Floating point number or Float for short is an arbitrary precision mantissa with a limited precision exponent The C data type for such objects is mpf_t Chapter 3 GMP Basics 17 A limb means the part of a multi precision number that fits in a single machine word We chose this word because a limb of the human body is analogous to a digit only larger and containing several digits Normally a limb is 32 or 64 bits The C data type for a limb is mp_limb_t 3 3 Function Classes There are six classes of functions in the GMP library 1 Functions for signed integer arithmetic with names beginning with mpz_ The associated type is mpz_t There are about 150 functions in this class 2 Functions for rational number arithmetic with names beginning with mpg_ The associated type is mpq_t There are about 40 functions in this class but the integer functions can be used for arithmetic on the numerator and denominator separately 3 Functions for floating point arithmetic with names beginning with mpf_ The associated type is mpf_t There are about 60 functions is this cla
83. algorithm A since it s just as easy and can be done with somewhat less temporary memory 16 4 2 Modular Powering Modular powering is implemented using a 2 ary sliding window algorithm as per Handbook of Applied Cryptography algorithm 14 85 see Appendix B References page 113 k is chosen according to the size of the exponent Larger exponents use larger values of k the choice being made to minimize the average number of multiplications that must supplement the squaring The modular multiplies and squares use either a simple division or the REDC method by Mont gomery see Appendix B References page 113 REDC is a little faster essentially saving N single limb divisions in a fashion similar to an exact remainder see Section 16 2 5 Exact Remainder page 94 The current REDC has some limitations It s only O N so above POWM_THRESHOLD division becomes faster and is used It doesn t attempt to detect small bases but rather always uses a REDC form which is usually a full size operand And lastly it s only applied to odd moduli 16 5 Root Extraction Algorithms 16 5 1 Square Root Square roots are taken using the Karatsuba Square Root algorithm by Paul Zimmermann see Appendix B References page 113 This is expressed in a divide and conquer form but as noted in the paper it can also be viewed as a discrete variant of Newton s method In the Karatsuba multiplication range this is an O M N 2 algorithm wher
84. all mpq t variables when you are done with them void mpq_set mpq t rop mpq t op Function void mpq_set_z mpq t rop mpz t op Function Assign rop from op void mpq set ui mpq t rop unsigned long int op1 unsigned long int op2 Function void mpq set si mpq t rop signed long int op1 unsigned long int op2 Function Set the value of rop to op1 op2 Note that if op1 and op2 have common factors rop has to be passed to mpq_canonicalize before any operations are performed on rop int mpq set str mpq t rop char str int base Function Set rop from a null terminated string str in the given base The string can be an integer like 41 or a fraction like 41 152 The fraction must be in canonical form see Chapter 6 Rational Number Functions page 42 or if not then mpq_canonicalize must be called The numerator and optional denominator are parsed the same as in mpz set str see Sec tion 5 2 Assigning Integers page 29 White space is allowed in the string and is simply ignored The base can vary from 2 to 36 or if base is 0 then the leading characters are used Ox for hex 0 for octal or decimal otherwise Note that this is done separately for the numerator and denominator so for instance OxEF 100 is 239 100 whereas OxEF 0x100 is 239 256 The return value is 0 if the entire string is a valid number or 1 if not Chapter 6 Rational Number Functions void mpq_swap mpq_t rop1 mpq_t rop2 Swap the
85. and plas ceded qa oes 16 3 2 Nomenclature and Types teen ebe gett baie ead aged beeen eet 16 3 9 Function Classes ue x eratac eie SERA EE ep RU dedu qa Rc RR a n e 3 4 Variable Conventions 0 ccc ce enn hh rh Ee 3 5 Parameter Conventions ett ee SE pe eeh as ch e 18 3 6 Memory Management 19 o R entranCy ee Eee Ia epee das re Renee be Greer RES eae dd 19 3 8 Useful Macros and Constant 19 3 9 Compatibility with older veraons 0 0 0 0 c cece eee ete hh 20 3 10 Demonstration pDrogramsg 0 cece cette tent hh hes 20 3 11 Reien an ete Do EF Ire red bak BLA eege eae eee rah ea Pea Reaves 20 3 12 Deler PPP 23 Sr roti xL 25 3 1A4 Autoconf iiis e Hn E er Ee do ddan eau drea Eod der aad ieget 25 Slo e e O O O E DE 26 4 Reporting es gg rancia oes deen E E 27 5 Integer Functions ceres p Rx hr AAA 28 5 1 Initialization Functions 0 000 0 cece tnt tnt tne nee 28 5 2 Assignment Functions 2 52420408 ease RH RR E RENE Pet baa PUR Rugs 29 5 3 Combined Initialization and Assignment Functions 0000s eee eee 29 5 4 Conversion Functions 000 seen 30 5 5 Arithmetic Functions erene iip ce ehh aa 31 5 6 Divisi n noen great demas packed dad Felge e REG E paces 32 5 7 Exponentiation Functonsg 0 0 cece cece rr 34 5 8 Root Extraction Functions ooooooooorrrrrrr tect ene t ha 34 5 9 Number Theoretic Functions 00 0 c ccc cn cence ehh hn 34 5 10 Comparison Functions gh gis ee ee beg ERE ec Gaede ae bx RR
86. antecedents and delete instructions at the end whose results are unwanted Sizes not a multiple of the unrolling can then be handled as desired 106 GNU MP 4 1 3 17 Internals This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters 17 1 Integer Internals mpz_t variables represent integers using sign and magnitude in space dynamically allocated and reallocated The fields are as follows _mp_size The number of limbs or the negative of that when representing a negative integer Zero is represented by _mp_size set to zero in which case the _mp_d data is unused _mp_d A pointer to an array of limbs which is the magnitude These are stored little endian as per the mpn functions so mp d 0 is the least significant limb and _mp_ d ABS mp size 1 is the most significant Whenever mp size is non zero the most significant limb is non zero Currently there s always at least one limb allocated so for instance mpz set ui never needs to reallocate and mpz get ui can fetch mp d 0 unconditionally though its value is then only wanted if mp size is non zero mp alloc mp alloc is the number of limbs currently allocated at mp d and naturally _mp_ alloc gt ABS mp size When an mpz routine is about to or might be abo
87. aracters of the string if base is 0 This is so that numbers like 0 23 are not interpreted as octal White space is allowed in the string and is simply ignored This is not really true white space is ignored in the beginning of the string and within the mantissa but not in other places such as after a minus sign or in the exponent We are considering changing the definition of this function making it fail when there is any white space in the input since that makes a lot of sense Please tell us your opinion about this change Do you really want it to accept 3 14 as meaning 314 as it does now This function returns 0 if the entire string is a valid number in base base Otherwise it returns 1 void mpf_swap mpf_t rop1 mpf_t rop2 Function Swap ropl and rop2 efficiently Both the values and the precisions of the two variables are swapped 7 3 Combined Initialization and Assignment Functions For convenience GMP provides a parallel series of initialize and set functions which initialize the output and then store the value there These functions names have the form mpf_init_set Once the float has been initialized by any of the mpf_init_set functions it can be used as the source or destination operand for the ordinary float functions Don t use an initialize and set function on a variable already initialized void mpf_init_set mpf_t rop mpf_t op Function Chapter 7 Floating point Functions 49 void mpf
88. are certain short passages of text that are listed as Front Cover Texts or Back Cover Texts in the notice that says that the Document is released under this License A Front Cover Text may be at most 5 words and a Back Cover Text may be at most 25 words 116 GNU MP 4 1 3 A Transparent copy of the Document means a machine readable copy represented in a format whose specification is available to the general public that is suitable for revising the document straightforwardly with generic text editors or for images composed of pixels generic paint programs or for drawings some widely available drawing editor and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters A copy made in an otherwise Transparent file format whose markup or absence of markup has been arranged to thwart or discourage subsequent modification by readers is not Transparent An image format is not Transparent if used for any substantial amount of text A copy that is not Transparent is called Opaque Examples of suitable formats for Transparent copies include plain ASCII without markup Texinfo input format LaT X input format SGML or XML using a publicly available DTD and standard conforming simple HTML PostScript or PDF designed for human modifica tion Examples of transparent image formats include PNG XCF and JPG Opaque formats include proprietary formats that
89. array integer conversions mpz_t integer conversions N oz t 6 The conversions accepted are as follows a and A are always supported for mpf_t but depend on the C library for standard C float types m and p depend on the C library aA hex floats C99 style c character d decimal integer eE scientific format float f fixed point float i same as d gG fixed or scientific float m strerror string GLIBC style n store characters written so far o octal integer p pointer S string u unsigned integer xX hex integer ta o x and X are unsigned for the standard C types but for types Z Q and N they are signed u is not meaningful for Z Q and N n can be used with any type even the GMP types Other types or conversions that might be accepted by the C library printf cannot be used through gmp_printf this includes for instance extensions registered with GLIBC register_ printf_function Also currently there s no support for POSIX style numbered arguments perhaps this will be added in the future The precision field has it s usual meaning for integer Z and float F types but is currently undefined for Q and should not be used with that mpf_t conversions only ever generate as many digits as can be accurately represented by the operand the same as mpf_get_str does Zeros will be used if necessary to pad to the reque
90. as well as a re mainder are generally a little slower than the remainder only functions like mpz_ tdiv ui If the quotient is only rarely wanted then it s probably best to just take a remainder and then go back and calculate the quotient if and when it s wanted mpz_divexact_ui can be used if the remainder is zero Rational Arithmetic The mpq functions operate on mpq t values with no common factors in the numerator and denominator Common factors are checked for and cast out as necessary In general cancelling factors every time is the best approach since it minimizes the sizes for subsequent operations However applications that know something about the factorization of the values they re working with might be able to avoid some of the GCDs used for canonical ization or swap them for divisions For example when multiplying by a prime it s enough to check for factors of it in the denominator instead of doing a full GCD Or when forming a big product it might be known that very little cancellation will be possible and so canonicalization can be left to the end The mpq numref and mpq denref macros give access to the numerator and denom inator to do things outside the scope of the supplied mpq functions See Section 6 5 Applying Integer Functions page 44 The canonical form for rationals allows mixed type mpq t and integer additions or subtractions to be done directly with multiples of the denominator This will be somewhat faster than
91. ast cleared between each initialization After a variable has been initialized it may be assigned to any number of times For efficiency reasons avoid excessive initializing and clearing In general initialize near the start of a function and clear near the end For example void foo void 1 mpz t n int i mpz_init n for i 1 i lt 100 i 18 GNU MP 4 1 3 mpz_mul n mpz fdiv oq n 22223 d mpz_clear n 3 5 Parameter Conventions When a GMP variable is used as a function parameter it s effectively a call by reference meaning if the function stores a value there it will change the original in the caller Parameters which are input only can be designated const to provoke a compiler error or warning on attempting to modify them When a function is going to return a GMP result it should designate a parameter that it sets like the library functions do More than one value can be returned by having more than one output parameter again like the library functions A return of an mpz_t etc doesn t return the object only a pointer and this is almost certainly not what s wanted Here s an example accepting an mpz_t parameter doing a calculation and storing the result to the indicated parameter void foo mpz_t result const mpz_t param unsigned long n 1 unsigned long i mpz mul ui result param n for i 1 i lt n i mpz_add_ui result result i 7 int main void mpz_t r
92. atically convert any pointer to a bool so if GMP accepted bool it Chapter 12 C Class Interface 73 would make all sorts of invalid class and pointer combinations compile but almost certainly not do anything sensible Conversions back from the classes to standard C types aren t done automatically instead member functions like get_si are provided see the following sections for details Also there are no automatic conversions from the classes to the corresponding GMP C types instead a reference to the underlying C object can be obtained with the following functions mpz_t mpz_class get_mpz_t Function mpq_t mpq_class get_mpq_t Function mpf_t mpf_class get_mpf_t Function These can be used to call a C function which doesn t have a C class interface For example to set a to the GCD of b and c mpz_class a b c mpz_gcd a get_mpz_t b get_mpz_t c get_mpz_t In the other direction a class can be initialized from the corresponding GMP C type or assigned to if an explicit constructor is used In both cases this makes a copy of the value it doesn t create any sort of association For example mpz t z init and calculate z mpz class x z mpz class y y mpz class z There are no namespace setups in gmpxx h all types and functions are simply put into the global namespace This is what gmp h has done in the past and continues to do for compat ibility The extras pr
93. ats page 75 12 6 C Interface Random Numbers gmp_randclass Class The C class interface to the GMP random number functions uses gmp_randclass to hold an algorithm selection and current state as per gmp_randstate_t gmp_randclass gmp_randclass void randinit gmp randstate t Function Construct a gmp_randclass using a call to the given randinit function see Section 9 1 Random State Initialization page 61 The arguments expected are the same as randinit but with mpz_class instead of mpz_t For example gmp_randclass ri gmp_randinit_default gmp_randclass r2 gmp_randinit_lc_2exp_size 32 gmp_randclass r3 gmp_randinit_lc_2exp a c m2exp gmp_randinit_lc_2exp_size can fail if the size requested is too big the behaviour of gmp_ randclass gmp randclass is undefined in this case perhaps this will change in the future 78 GNU MP 4 1 3 gmp_randclass gmp_randclass gmp randalg t alg Function Construct a emp randclass using the same parameters as emp randinit see Section 9 1 Random State Initialization page 61 This function is obsolete and the above randinit style should be preferred void gmp randclass seed unsigned long int s Function void gmp randclass seed mpz class s Function Seed a random number generator See see Chapter 9 Random Number Functions page 61 for how to choose a good seed mpz class emp randclass get z bits unsigned long bits Function mpz class em
94. b 1 as for example with 3 in mpn_divexact_ by3 Other such factors include 5 17 and 257 but no particular use has been found for this 16 2 6 Small Quotient Division An NxM division where the number of quotient limbs Q N M is small can be optimized somewhat An ordinary basecase division normalizes the divisor by shifting it to make the high bit set shifting the dividend accordingly and shifting the remainder back down at the end of the calculation This is wasteful if only a few quotient limbs are to be formed Instead a division of just the top 2Q limbs of the dividend by the top Q limbs of the divisor can be used to form a trial quotient This requires only those limbs normalized not the whole of the divisor and dividend A multiply and subtract then applies the trial quotient to the M Q unused limbs of the divisor and N Q dividend limbs which includes Q limbs remaining from the trial quotient division The starting trial quotient can be 1 or 2 too big but all cases of 2 too big and most cases of 1 too big are detected by first comparing the most significant limbs that will arise from the subtraction An addback is done if the quotient still turns out to be 1 too big This whole procedure is essentially the same as one step of the basecase algorithm done in a Q limb base though with the trial quotient test done only with the high limbs not an entire Q limb digit product The correctness of this weaker test can be
95. braries All declarations needed to use GMP are collected in the include file gmp h It is designed to work with both C and C compilers include lt gmp h gt Note however that prototypes for GMP functions with FILE parameters are only provided if lt stdio h gt is included too include lt stdio h gt include lt gmp h gt Likewise lt stdarg h gt or lt varargs h gt is required for prototypes with va_list parameters such as gmp_vprintf And lt obstack h gt for prototypes with struct obstack parameters such as gmp_obstack_printf when available All programs using GMP must link against the libgmp library On a typical Unix like system this can be done with lgmp for example gcc myprogram c lgmp GMP C functions are in a separate libgmpxx library This is built and installed if C support has been enabled see Section 2 1 Build Options page 4 For example g mycxxprog cc lgmpxx lgmp GMP is built using Libtool and an application can use that to link if desired see section In troduction in GNU Libtool If GMP has been installed to a non standard location then it may be necessary to use I and L compiler options to point to the right directories and some sort of run time path for a shared library Consult your compiler documentation for instance section Introduction in Using and Porting the GNU Compiler Collection 3 2 Nomenclature and Types In this manu
96. can be read and edited only by proprietary word proces sors SGML or XML for which the DTD and or processing tools are not generally available and the machine generated HTML PostScript or PDF produced by some word processors for output purposes only The Title Page means for a printed book the title page itself plus such following pages as are needed to hold legibly the material this License requires to appear in the title page For works in formats which do not have any title page as such Title Page means the text near the most prominent appearance of the work s title preceding the beginning of the body of the text A section Entitled XYZ means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language Here XYZ stands for a specific section name mentioned below such as Acknowledgements Dedications Endorsements or History To Preserve the Title of such a section when you modify the Document means that it remains a section Entitled XYZ according to this definition The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document These Warranty Disclaimers are considered to be included by reference in this License but only as regards disclaiming warranties any other implication that these Warranty Disclaimers may have is void and has no
97. can be selected For example configure host mips64 sgi irix6 ABI n32 See Section 2 2 ABI and ISA page 9 for the available choices on relevant CPUs and what applications need to do CC CFLAGS By default the C compiler used is chosen from among some likely candidates with gcc normally preferred if it s present The usual CC whatever can be passed to configure to choose something different For some systems default compiler flags are set based on the CPU and compiler The usual CFLAGS whatever can be passed to configure to use something different or to set good flags for systems GMP doesn t otherwise know The CC and CFLAGS used are printed during configure and can be found in each generated Makefile This is the easiest way to check the defaults when considering changing or adding something Note that when CC and CFLAGS are specified on a system supporting multiple ABIs it s important to give an explicit ABI whatever since GMP can t determine the ABI just from the flags and won t be able to select the correct assembler code Chapter 2 Installing GMP 7 CPPFLAGS If just CC is selected then normal default CFLAGS for that compiler will be used if GMP recognises it For example CC gcc can be used to force the use of GCC with default flags and default ABI Any flags like D defines or I include
98. cc DD64 ABI 2 0n The 2 0n ABI means the 32 bit HPPA 1 0 ABI but with a 64 bit limb using long long This is available on HP UX 10 or up when using cc No gcc support is planned for this Applications must be compiled with cc DA2 0 e ABI 1 0 10 GNU MP 4 1 3 HPPA 2 0 CPUs can run all HPPA 1 0 and 1 1 code in the 32 bit HPPA 1 0 ABI No special compiler options are needed for applications All three ABIs are available for CPUs hppa2 0w and hppa2 0 but for CPU hppa2 0n only 2 0n or 1 0 are allowed MIPS under IRIX 6 mips irix 6789 IRIX 6 supports the n32 and 64 ABIs and always has a 64 bit MIPS 3 or better CPU In both these ABIs GMP uses a 64 bit limb A new enough gcc is required 2 95 for instance ABT n32 The n32 ABI is 32 bit pointers and integers but with a 64 bit limb using a long long Applications must be compiled with gcc mabi n32 cc n32 ABI 64 The 64 bit ABI is 64 bit pointers and integers Applications must be compiled with gcc mabi 64 cc 64 Note that MIPS GNU Linux as of kernel version 2 2 doesn t have the necessary support for n32 or 64 and so only gets a 32 bit limb and the MIPS 2 code PowerPC 64 powerpc64 powerpc620 powerpc630 ABT aix64 The AIX 64 ABI uses 64 bit limbs and pointers and is available on systems aix Applications must be compiled and linked with gcc maix64 xlc q64 ABI 32 This
99. char fmt Function int gmp vscanf const char fmt va_list ap Function Read from the standard input stdin int gmp_fscanf FILE fp const char fmt Function int gmp vfscanf FILE fp const char fmt va list ap Function Read from the stream fp int gmp_sscanf const char Ze const char fmt Function int gmp_vsscanf const char s const char fmt va list ap Function Read from a null terminated string s The return value from each of these functions is the same as the standard C99 scanf namely the number of fields successfully parsed and stored n fields and fields read but suppressed by don t count towards the return value If end of file or file error or end of string is reached when a match is required and when no previous non suppressed fields have matched then the return value is EOF instead of 0 A match is required for a literal character in the format string or a field other than n Whitespace in the format string is only an optional match and won t induce an EOF in this fashion Leading whitespace read and discarded for a field doesn t count as a match 11 3 C Formatted Input The following functions are provided in libgmpxx which is built only if C support is enabled see Section 2 1 Build Options page 4 Prototypes are available from lt gmp h gt istream amp operator gt gt istream amp stream mpz_t rop Function Read rop from stream using its ios format
100. coefficients will be roughly b each and the final W b will be an addition like high low wi The w coefficients could be formed by a simple set of cross products like w4 Tratt W3 X391 219 W2 L2Yo0 21Y1 Loy2 etc but this would need all nine z y for 7 7 0 1 2 and would be equivalent merely to a basecase multiply Instead the following approach is used X t and Y t are evaluated and multiplied at 5 points giving values of W t at those points The points used can be chosen in various ways but in GMP the following are used Point Value t 0 ZoYo Which gives wy immediately i 2 4x2 2x1 zo 4y2 2y1 Yo Chapter 16 Algorithms 89 t 1 22 21 Zo Ya 3i Yo t 3 2 224 Axo ya 2y1 Ayo t 00 xz2Yo2 Which gives w4 immediately At t 1 the value calculated is actually 16X 3 Y 3 giving a value for 16W 2 and this is always an integer At t oo the value is actually lim _o AA but it s much easier to think of as simply att giving w4 immediately much like royo at t 0 gives wo immediately Now each of the points substituted into W t wat wo gives a linear combination of the w coefficients and the value of those combinations has just been calculated W 0 Wo 16W 3 w 2w 4w 8w 16ug WD we Ws w w w W 2 16w Bus 4w 20 w W oo wa This is a set of five equations in five unknowns and some
101. cover from an error such as out of memory they must terminate program execution A longjmp or throwing a C exception will have undefined results This may change in the future GMP may use allocated blocks to hold pointers to other allocated blocks This will limit the assumptions a conservative garbage collection scheme can make Since the default GMP allocation uses malloc and friends those functions will be linked in even if the first thing a program does is an mp_set_memory_functions It s necessary to change the GMP sources if this is a problem 84 GNU MP 4 1 3 15 Language Bindings The following packages and projects offer access to GMP from languages other than C though perhaps with varying levels of functionality and efficiency C Fortran Haskell Java Lisp M4 ML Objective Caml Oz GMP C class interface see Chapter 12 C Class Interface page 72 Straightforward interface expression templates to eliminate temporaries ALP http www inria fr saga logiciels ALP Linear algebra and polynomials using templates Arithmos http win www uia ac be u cant arithmos Rationals with infinities and square roots CLN http www ginac de CLN High level classes for arithmetic LiDIA http www informatik tu darmstadt de TI LiDIA A C library for computational number theory Linbox http www linalg org Sparse vectors and matrices NTL http www shoup net ntl A C number theory library O
102. cross compiling cc Note that the target option is not appropriate for GMP It s for use when building compiler tools with host being where they will run and target what they ll produce code for Ordinary programs or libraries like GMP are only interested in the host part being where they ll run Some past versions of GMP used target incorrectly CPU types In general if you want a library that runs as fast as possible you should configure GMP for the exact CPU type your system uses However this may mean the binaries won t run on older members of the family and might run slower on other members older or newer The best idea is always to build GMP for the exact machine type you intend to run it on GNU MP 4 1 3 The following CPUs have specific support See configure in for details of what code and compiler options they select e Alpha alpha alphaev5 alphaev56 alphapca56 alphapca57 alphaev6 alphaev67 alphaev68 e Cray c90 j90 t90 svt e HPPA hppa1 0 hppal 1 hppa2 0 hppa2 0n hppa2 0w e MIPS mips mips3 mips64 e Motorola m68k m68000 m68010 m68020 m68030 m68040 m68060 m68302 m68360 m88k m88110 e POWER power power1 power2 power2sc e PowerPC powerpc po
103. ction starting bit Scan op starting from bit starting bit towards more significant bits until the first 0 or 1 bit respectively is found Return the index of the found bit If the bit at starting bit is already what s sought then starting bit is returned If there s no bit found then ULONG MAX is returned This will happen in mpz_scan0 past the end of a positive number or mpz scan1 past the end of a negative void mpz setbit mpz t rop unsigned long int bit index Function Set bit bit index in rop 38 GNU MP 4 1 3 void mpz_clrbit mpz_t rop unsigned long int bit_index Function Clear bit bit_index in rop int mpz_tstbit mpz_t op unsigned long int bit_index Function Test bit bit index in op and return 0 or 1 accordingly 5 12 Input and Output Functions Functions that perform input from a stdio stream and functions that output to a stdio stream Passing a NULL pointer for a stream argument to any of these functions will make them read from stdin and write to stdout respectively When using any of these functions it is a good idea to include stdio h before gmp h since that will allow gmp h to define prototypes for these functions size t mpz out str FILE stream int base mpz t op Function Output op on stdio stream stream as a string of digits in base base The base may vary from 2 to 36 Return the number of bytes written or if an error occurred return 0 size_t mpz inp s
104. cur too rarely to make it worth checking for them 16 3 2 Accelerated GCD For sizes above GCD_ACCEL_THRESHOLD GMP uses the Accelerated GCD algorithm described independently by Weber and Jebelean the latter as the Generalized Binary algorithm see Appendix B References page 113 This algorithm is still O N but is much faster than the binary algorithm since it does fewer multi precision operations It consists of alternating the k ary reduction by Sorenson and a dmod exact remainder reduction For operands u and v the k ary reduction replaces u with nv du where n and d are single limb values chosen to give two trailing zero limbs on that value which can be stripped n and d are calculated using an algorithm similar to half of a two limb GCD see find_a in mpn generic gcd c When u and v differ in size by more than a certain number of bits a dmod is performed to zero out bits at the low end of the larger It consists of an exact remainder style division applied to an appropriate number of bits see Section 16 2 4 Exact Division page 93 and see Section 16 2 5 Exact Remainder page 94 This is faster than a k ary reduction but useful only when the operands differ in size There s a dmod after each k ary reduction and if the dmod leaves the operands still differing in size then it s repeated The k ary reduction step can introduce spurious factors into the GCD calculated and these are eliminated at the end
105. d independently It should be noted that when a double limb calculation is used it s used for the whole of that GCD it doesn t fall back to single limb part way through This is because as the algorithm proceeds the inputs a and b are reduced but the cofactors x and y grow so the multipliers at each step are applied to a roughly constant total number of limbs 16 3 4 Jacobi Symbol mpz_jacobi and mpz_kronecker are currently implemented with a simple binary algorithm similar to that described for the GCDs see Section 16 3 1 Binary GCD page 95 They re not very fast when both inputs are large Lehmer s multi step improvement or a binary based multi step algorithm is likely to be better When one operand fits a single limb and that includes mpz_kronecker_ui and friends an initial reduction is done with either mpn_mod_1 or mpn_modexact_1_odd followed by the binary algorithm on a single limb The binary algorithm is well suited to a single limb and the whole calculation in this case is quite efficient In all the routines sign changes for the result are accumulated using some bit twiddling avoiding table lookups or conditional jumps 16 4 Powering Algorithms 16 4 1 Normal Powering Normal mpz or mpf powering uses a simple binary algorithm successively squaring and then multiplying by the base when a 1 bit is seen in the exponent as per Knuth section 4 6 3 The left to right variant described there is used rather than
106. de Sparc App Regs The GMP assembler code for both 32 bit and 64 bit Sparc clobbers the application registers g2 g3 and g4 the same way that the GCC default mapp regs does see section SPARC Options in Using the GNU Compiler Collection GCC This makes that code unsuitable for use with the special V9 mcmodel embmedany which uses g4 as a data segment pointer and for applications wanting to use those registers for special purposes In these cases the only suggestion currently is to build GMP with CPU none to avoid the assembler code SunOS 4 6 usr bin m4 lacks various features needed to process asm files and instead configure will automatically use usr 5bin m4 which we believe is always available if not then use GNU m4 x86 CPU Types 1386 selects generic code which will run reasonably well on all x86 chips 1586 pentium or pentiummmx code is good for the intended P5 Pentium chips but quite slow when run on Intel P6 class chips PPro P II P III i386 is a better choice when making binaries that must run on both pentium4 and an SSE2 capable assembler are important for best results on Pentium 4 The specific code is for instance roughly a 2x to 3x speedup over the generic i386 code x86 MMX and SSE2 Code If the CPU selected has MMX code but the assembler doesn t support it a warning is given and non MMX code is used instead This wi
107. default_prec unsigned long int prec Function Set the default precision to be at least prec bits All subsequent calls to mpf_init will use this precision but previously initialized variables are unaffected unsigned long int mpf_get_default_prec void Function Return the default default precision actually used An mpf_t object must be initialized before storing the first value in it The functions mpf_init and mpf_init2 are used for that purpose Chapter 7 Floating point Functions 4T void mpf init mpf_t x Function Initialize x to 0 Normally a variable should be initialized once only or at least be cleared using mpf clear between initializations The precision of x is undefined unless a default precision has already been established by a call to mpf set default prec void mpf_init2 mpf_t x unsigned long int prec Function Initialize x to 0 and set its precision to be at least prec bits Normally a variable should be initialized once only or at least be cleared using mpf clear between initializations void mpf clear mpf_t x Function Free the space occupied by x Make sure to call this function for all mpf_t variables when you are done with them Here is an example on how to initialize floating point variables mpf_t x y mpf_init x use default precision mpf_init2 y 256 precision at least 256 bits Unless the program is about to exit do mpf_clear x mpf_clear y
108. e represented in less This means low precision values or small integers stored in a high precision mpf_t can still be operated on efficiently mp Size can also be greater than mp prec Firstly a value is allowed to use all of the mp prec 1 limbs available at mp d and secondly when mpf set prec raw 108 GNU MP 4 1 3 lowers mp prec it leaves mp size unchanged and so the size can be arbitrarily bigger than mp prec Rounding All rounding is done on limb boundaries Calculating mp prec limbs with the high Bit Shifts non zero will ensure the application requested minimum precision is obtained The use of simple trunc rounding towards zero is efficient since there s no need to examine extra limbs and increment or decrement Since the exponent is in limbs there are no bit shifts in basic operations like mpf_ add and mpf_mul When differing exponents are encountered all that s needed is to adjust pointers to line up the relevant limbs Of course mpf_mul_2exp and mpf_div_2exp will require bit shifts but the choice is between an exponent in limbs which requires shifts there or one in bits which requires them almost everywhere else Use of _mp_prec 1 Limbs The extra limb on _mp_d _mp_prect1 rather than just _mp_prec helps when an mpf routine might get a carry from its operation mpf_add for instance will do an mpn_add of mp prec limbs If there s no carry then that s the result but if there is a carry then it s stored
109. e 70 If the rop read might not be in canonical form then mpq_class canonicalize must be called 76 12 4 C Interface Floats GNU MP 4 1 3 When an expression requires the use of temporary intermediate mpf_class values like f g h x y those temporaries will have the same precision as the destination f Explicit con structors can be used if this doesn t suit mpf class mpf class type op Function mpf class mpf class type op unsigned long prec Function Construct an mpf class Any standard C type can be used except long long and long double and any of the GMP C classes can be used If prec is given the initial precision is that value in bits If prec is not given then the initial precision is determined by the type of op given An mpz class mpq class string or C builtin type will give the default mpf precision see Section 7 1 Initializing Floats page 46 An mpf class or expression will give the precision of that value The precision of a binary expression is the higher of the two operands mpf class f 1 5 default precision mpf class f 1 5 500 500 bits at least mpf class f x precision of x mpf class f abs x precision of x mpf class f g 1000 1000 bits at least mpf class f x y greater of precisions of x and y mpf class amp mpf class operator type op Function Convert and store the given op value to an mpf_class object The same types are accepted
110. e Chapter 14 Custom Allocation page 82 An additional choice enable alloca debug is available to help when debugging memory related problems see Section 3 12 Debugging page 23 FFT Multiplication disable fft By default multiplications are done using Karatsuba 3 way Toom Cook and Fermat FFT The FFT is only used on large to very large operands and can be disabled to save code size if desired Berkeley MP enable mpbsd The Berkeley MP compatibility library libmp and header file mp h are built and installed only if enable mpbsd is used See Chapter 13 BSD Compatible Functions page 80 MPER enable mpfr The optional MPFR functions are built and installed only if enable mpfr is used These are in a separate library libmpfr a and are documented separately too see section Introduction to MPFR in MPFR Assertion Checking enable assert This option enables some consistency checking within the library This can be of use while debugging see Section 3 12 Debugging page 23 Execution Profiling enable profiling prof gprof Profiling support can be enabled either for prof or gprof This adds p or pg respectively to CFLAGS and for some systems adds corresponding mcount calls to the assembler code See Section 3 13 Profiling page 25 D MPN_PATH Various assembler versions of each mpn subroutines are provided Fo
111. e M n is the time to multiply two numbers of n limbs In the FFT multiplication range this grows to a 98 GNU MP 4 1 3 bound of O 6M N 2 In practice a factor of about 1 5 to 1 8 is found in the Karatsuba and Toom 3 ranges growing to 2 or 3 in the FFT range The algorithm does all its calculations in integers and the resulting mpn_sqrtrem is used for both mpz sqrt and mpf_sqrt The extended precision given by mpf_sqrt_ui is obtained by padding with zero limbs 16 5 2 Nth Root Integer Nth roots are taken using Newton s method with the following iteration where A is the input and m is the root to be taken 1 A D 1 i 1 TL i Aj 1 at a The initial approximation a is generated bitwise by successively powering a trial root with or without new 1 bits aiming to be just above the true root The iteration converges quadratically when started from a good approximation When n is large more initial bits are needed to get good convergence The current implementation is not particularly well optimized 16 5 3 Perfect Square mpz_perfect_square_p is able to quickly exclude most non squares by checking whether the input is a quadratic residue modulo some small integers The first test is modulo 256 which means simply examining the least significant byte Only 44 different values occur as the low byte of a square so 82 8 of non squares can be immediately excluded Similar tests modulo primes from 3 to 29 exclude 99 5 of those rema
112. e low digit has become zero The procedure is repeated at the second digit with the next quotient digit 7 1x7 mod 10 subtracting 7x543 3801 leaving 325800 And finally at the third digit with quotient digit 6 8x7 mod 10 subtracting 6x 543 3258 leaving 0 So the quotient is 678 Notice however that the multiplies and subtractions don t need to extend past the low three digits of the dividend since that s enough to determine the three quotient digits For the last quotient digit no subtraction is needed at all On a 2NxN division like this one only about half the work of a normal basecase division is necessary For an NxM exact division producing Q N M quotient limbs the saving over a normal basecase division is in two parts Firstly each of the Q quotient limbs needs only one multiply not a 2x1 divide and multiply Secondly the crossproducts are reduced when Q gt M to QM M M 1 2 or when Q M to Q Q 1 2 Notice the savings are complementary If Q is big then many divisions are saved or if Q is small then the crossproducts reduce to a small number The modular inverse used is calculated efficiently by modlimb_invert in gmp impl h This does four multiplies for a 32 bit limb or six for a 64 bit limb tune modlinv c has some alternate implementations that might suit processors better at bit twiddling than multiplying The sub quadratic exact division described by Jebelean in Exact Division with Karatsuba
113. e mpn functions so mp d 0 is the least significant limb and mp d ABS mp size 1 the most significant The most significant limb is always non zero but there are no other restrictions on its value in particular the highest 1 bit can be anywhere within the limb _mp_prec 1 limbs are allocated to mp d the extra limb being for convenience see below There are no reallocations during a calculation only in a change of precision with mpf set prec The exponent in limbs determining the location of the implied radix point Zero means the radix point is just above the most significant limb Positive values mean a radix point offset towards the lower limbs and hence a value gt 1 as for example in the diagram above Negative exponents mean a radix point further above the highest limb Naturally the exponent can be any value it doesn t have to fall within the limbs as the diagram shows it can be a long way above or a long way below Limbs other than those included in the mp d mp size data are treated as zero The following various points should be noted Low Zeros The least significant limbs mp d 0 etc can be zero though such low zeros can always be ignored Routines likely to produce low zeros check and avoid them to save time in subsequent calculations but for most routines they re quite unlikely and aren t checked Mantissa Size Range The mp size count of limbs in use can be less than mp prec if the value can b
114. e same two multiplies worth of latency that one limb at a time gives This extends to 3 or 4 limbs at a time though the extra work to apply the inverse will almost certainly soon reach the limits of multiplier throughput Chapter 16 Algorithms 93 A similar approach in reverse can be taken to process just half a limb at a time if the divisor is only a half limb In this case the 1x1 multiply for the inverse effectively becomes two i x 1 for each limb which can be a saving on CPUs with a fast half limb multiply or in fact if the only multiply is a half limb and especially if it s not pipelined 16 2 2 Basecase Division Basecase NxM division is like long division done by hand but in base 2 P bits per_limb_ See Knuth section 4 3 1 algorithm D and mpn generic sb_divrem_mn c Briefly stated while the dividend remains larger than the divisor a high quotient limb is formed and the Nx1 product qd subtracted at the top end of the dividend With a normalized divisor most significant bit set each quotient limb can be formed with a 2x1 division and a 1x1 multiplication plus some subtractions The 2x1 division is by the high limb of the divisor and is done either with a hardware divide or a multiply by inverse the same as in Section 16 2 1 Single Limb Division page 92 whichever is faster Such a quotient is sometimes one too big requiring an addback of the divisor but that happens rarely With Q N M being the number of quotien
115. e using one of the following three methods which can be selected with for instance enable alloca malloc reentrant e alloca C library or compiler builtin e malloc reentrant the heap in a re entrant fashion e malloc notreentrant the heap with global variables 8 GNU MP 4 1 3 6 For convenience the following choices are also available disable alloca is the same as enable alloca no e yes a synonym for alloca e no a synonym for malloc reentrant e reentrant alloca if available otherwise malloc reentrant This is the default e notreentrant alloca if available otherwise malloc notreentrant alloca is reentrant and fast and is recommended but when working with large numbers it can overflow the available stack space in which case one of the two malloc methods will need to be used Alternately it might be possible to increase available stack with limit ulimit or setrlimit or under DJGPP with stubedit or _stklen Note that depending on the system the only indication of stack overflow might be a segmentation violation malloc reentrant is as the name suggests reentrant and thread safe but malloc notreentrant is faster and should be used if reentrancy is not required The two malloc methods in fact use the memory allocation functions selected by mp_ set memory functions these being malloc and friends by default Se
116. effect on the meaning of this License VERBATIM COPYING You may copy and distribute the Document in any medium either commercially or noncom mercially provided that this License the copyright notices and the license notice saying this License applies to the Document are reproduced in all copies and that you add no other conditions whatsoever to those of this License You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute However you may accept compensation in exchange for copies If you distribute a large enough number of copies you must also follow the conditions in section 3 You may also lend copies under the same conditions stated above and you may publicly display copies COPYING IN QUANTITY If you publish printed copies or copies in media that commonly have printed covers of the Document numbering more than 100 and the Document s license notice requires Cover Texts you must enclose the copies in covers that carry clearly and legibly all these Cover Texts Front Cover Texts on the front cover and Back Cover Texts on the back cover Both covers must also clearly and legibly identify you as the publisher of these copies The front cover must present the full title with all words of the title equally prominent and visible You may add other material on the covers in addition Copying with changes limited to Appendix C GNU Free Documentation License 117
117. ematics of Computation volume 44 number 170 April 1985 Tudor Jebelean An algorithm for exact division Journal of Symbolic Computation vol ume 15 1993 pp 169 180 Research report version available ftp ftp risc uni linz ac at pub techreports 1992 92 35 ps gz Tudor Jebelean Exact Division with Karatsuba Complexity Extended Abstract RISC Linz technical report 96 31 ftp ftp risc uni linz ac at pub techreports 1996 96 31 ps gz Tudor Jebelean Practical Integer Division with Karatsuba Complexity ISSAC 97 pp 339 341 Technical report available ftp ftp risc uni linz ac at pub techreports 1996 96 29 ps gz Tudor Jebelean A Generalization of the Binary GCD Algorithm ISSAC 93 pp 111 116 Technical report version available ftp ftp risc uni linz ac at pub techreports 1993 93 01 ps gz Tudor Jebelean A Double Digit Lehmer Euclid Algorithm for Finding the GCD of Long Integers Journal of Symbolic Computation volume 19 1995 pp 145 157 Technical report version also available ftp ftp risc uni linz ac at pub techreports 1992 92 69 ps gz 114 GNU MP 4 1 3 Werner Krandick and Tudor Jebelean Bidirectional Exact Integer Division Journal of Symbolic Computation volume 21 1996 pp 441 455 Early technical report version also available ftp ftp risc uni linz ac at pub techreports 1994 94 50 ps gz R Moenck and A Borodin Fast Modular Transforms via Division Proceedings of the 13th Annual
118. ems providing localeconv Hex and octal are supported unlike the standard operator lt lt on double The mantissa will be in hex or octal the exponent will be in decimal For hex the exponent delimiter is an This is as per mpf_out_str ios showbase is supported and will put a base on the mantissa for example hex 0x1 8 or 0x0 8 or octal 01 4 or 00 4 This last form is slightly strange but at least differentiates itself from decimal These operators mean that GMP types can be printed in the usual C way for example mpz t zi int ny cout lt lt iteration lt lt n lt lt value lt lt z lt lt n But note that ostream output and istream input see Section 11 3 C Formatted Input page 70 is the only overloading available for the GMP types and that for instance using with Chapter 10 Formatted Output 67 an mpz_t will have unpredictable results For classes with overloading see Chapter 12 C Class Interface page 72 68 GNU MP 4 1 3 11 Formatted Input 11 1 Formatted Input Strings gmp_scanf and friends accept format strings similar to the standard C scanf see section For matted Input in The GNU C Library Reference Manual A format specification is of the form Lflags width typel conv GMP adds types Z Q and F for mpz t mpq t and mpf_t respectively Z and Q behave like integers Q will read a and a denominator
119. ense conditions 1 ET EE 16 WWM S128 d aei duse Ae geeiert 19 linking ng ides eae ek aa e Rr 9 16 Logical functions en dg be deed Eeer 37 Low level function 53 Lucas number functions oooooooooooo o 36 M Mailing listos rie iris eibar 2 Memory allocation pin 82 Memory Management 19 Miscellaneous float functions 51 Miscellaneous integer functions 41 Modular inverse functions ssssssesurerreen 35 Most significant biti e EEN Rer 41 pio M EP 80 MPER eb ph PIERDE S gana ER nae a p e Gee gad 8 GNU MP 4 1 3 a A buie Hesse ipe testis dosi tides dt E Met TEE 19 Multiplication algorithms o n annaaunsunnun 86 EE 59 Nomenclature kde aes A B6 eau 16 Number theoretic functions n a anaana anaa 34 Numerator and denominator 44 O ostream OULPUE si sols sae came ater es 66 Output functi nS sses icss ascesa tgatesriss 38 45 51 P Packaged builds e iie EELER 11 PalmOS iue osveTEPH oes Py Vade EE 14 Parameter conventions eese eese 18 Particular Systems 2x de bete ri 11 EE 20 Powering algorithms A 9 ANEN de RR aes 97 Powering functons eese 34 50 Precision of floats Lore etr rre Ree 46 Precision of hardware floating point 12 lui PE tie tad Gat ene sos ose eae od voce 4 Prime testing functions 34 printf formatted output s sciscrccsisiasii eses 63 IS eet 25 R Radix convers
120. ependent parts like the documentation but separate the dependent parts Note however that gmp h and mp h are architecture dependent since they encode certain as Chapter 2 Installing GMP 5 pects of libgmp so it will be necessary to ensure both prefix include and exec prefix include are available to the compiler disable shared disable static By default both shared and static libraries are built where possible but one or other can be disabled Shared libraries result in smaller executables and permit code sharing between separate running processes but on some CPUs are slightly slower having a small cost on each function call Native Compilation build CPU VENDOR OS For normal native compilation the system can be specified with build By default configure uses the output from running config guess On some systems config guess can determine the exact CPU type on others it will be necessary to give it explicitly For example configure build ultrasparc sun solaris2 7 In all cases the OS part is important since it controls how libtool generates shared libraries Running config guess is the simplest way to see what it should be if you don t know already Cross Compilation host CPU VENDOR OS When cross compiling the system used for compiling is given by build and the system where the library will run is given by host
121. er split and speedup would be possible but no such root exists for plain integers Going to complex integers with i y 1 doesn t help essentially because in cartesian form it takes three real multiplies to do a complex multiply The existence of 2 th roots of unity in a suitable ring or field lets the fast fourier transform keep splitting and get to O N log r Floating point FFTs use complex numbers approximating Nth roots of unity Some processors have special support for such FFTs But these are not used in GMP since it s very difficult to guarantee an exact result to some number of bits An occasional difference of 1 in the last bit might not matter to a typical signal processing algorithm but is of course of vital importance to GMP 16 2 Division Algorithms 16 2 1 Single Limb Division Nx1 division is implemented using repeated 2x1 divisions from high to low either with a hardware divide instruction or a multiplication by inverse whichever is best on a given CPU The multiply by inverse follows section 8 of Division by Invariant Integers using Multiplication by Granlund and Montgomery see Appendix B References page 113 and is implemented as udiv_qrnnd_preinv in gmp impl h The idea is to have a fixed point approximation to 1 d see invert_limb and then multiply by the high limb plus one bit of the dividend to get a quotient q With d normalized high bit set q is no more than 1 too small Subtracting qd from t
122. eran heed oad cia edad ede 90 16 1 5 Other Multiplication uices E eb ER ed ie 91 16 2 Division Algorithms Lue exa eredi s ARR ha le C Ros E RR RR RU RR cen 92 16 2 1 Single Limb Dtvision 0 ccc cc eect eee hh 92 10 2 2 Basec se Division oso dE ERA de eked eA ee ete aed daa 93 16 2 3 Divide and Conquer Division 93 16 2 4 Exact DivislO0nz ze pen o gia ea E hie Magee Sege ec 93 16 2 5 Exact Remainder berri edad Miva tae ed edad ai eed be do RU End eee 94 16 2 6 Small Quotient Divisions eg BEI be eee rw RR ERE bees 95 16 3 Greatest Common Divisor sssi sue sictiami egia ai akiagana hh hh 95 16 3 Binary GOD i3 ker ted t bee ch wk HS oett der eodd uve ci dodo 95 16 3 2 Accelerated GOD uiii ee wie prenait n i eg a ca o RD Heino tte gn e 96 16 3 3 Extended GOD er dct Le Er at etr ensi dena ORA RU A 96 16 3 4 Jacobi Symbol steep dA HERO ERE ES 97 16 4 Powering Algoritht s us sete eer tee ced d roe ee eda ego es cod d 97 1641 Normal POWerllBg iiu ee e d o e n Y EGRE RO RR c RR ESO te 97 16 4 3 Modular Powering 0 0 nent hh hh 97 160 5 Root Extraction Algorithms 2 on erac dae Medd eed ead ean RA ears 97 16 5 1 Square Rotes cashes ee e qtti Ux genero dog Det t e ee gee edhe e ea ees 97 10 5 2 NRO ier ret Aere AE annette 98 16 03 Perfect Square od eG dad wed nadia ERU n Ro ee ER POS elt Biden d Ped eode 98 16 5 4 Perfect POWGeE cateii disis n eR a aceite SR RG ake a 98 16 6 Radix Conversion i ous eco it
123. es For GCC this means version 2 91 or later Everything described in this chapter is to be considered preliminary and might be subject to incompatible changes if some unforeseen difficulty reveals itself 12 1 C Interface General All the C classes and functions are available with include lt gmpxx h gt Programs should be linked with the libgmpxx and libgmp libraries For example g mycxxprog cc lgmpxx lgmp The classes defined are mpz_class Class mpq_class Class mpf_class Class The standard operators and various standard functions are overloaded to allow arithmetic with these classes For example int main void mpz_class a b c a 1234 b 5678 c atb cout lt lt sum is lt lt c lt lt An cout lt lt absolute value is lt lt abs c lt lt n return 0 An important feature of the implementation is that an expression like a b c results in a single call to the corresponding mpz_add without using a temporary for the b c part Expressions which by their nature imply intermediate values like a b c d e still use temporaries though The classes can be freely intermixed in expressions as can the classes and the standard types long unsigned long and double Smaller types like int or float can also be intermixed since C will promote them Note that bool is not accepted directly but must be explicitly cast to an int first This is because C will autom
124. esn t do any see section Options for Code Generation in Using the GNU Compiler Collection GCC These options must be added to the CFLAGS used in the GMP build see Section 2 1 Build Options page 4 adding them just to an application will have no effect Note also they re a slowdown adding overhead to each function call and each stack allocation Heap Problems The most likely cause of application problems with GMP is heap corruption Fail ing to init GMP variables will have unpredictable effects and corruption arising elsewhere in a program may well affect GMP Initializing GMP variables more than once or failing to clear them will cause memory leaks In all such cases a malloc debugger is recommended On a GNU or BSD system the standard C library malloc has some diagnostic facilities see section Allocation Debugging in The GNU C Library Reference Manual or man 3 malloc Other possibilities in no particular order include http www inf ethz ch personal biere projects ccmalloc http dmalloc com http www perens com FreeSoftware electric fence http packages debian org fda http www gnupdate org components leakbug http people redhat com otaylor memprof http www cbmamiga demon co uk mpatrol The GMP default allocation routines in memory c also have a simple sentinel scheme which can be enabled with define DEBUG in that file This is mainly de signed for detecting buffer overruns during GM
125. established by following the argument of Knuth section 4 3 1 exercise 20 but with the pap gt bf us condition appropriately relaxed 16 3 Greatest Common Divisor 16 3 1 Binary GCD At small sizes GMP uses an O N binary style GCD This is described in many textbooks for example Knuth section 4 5 2 algorithm B It simply consists of successively reducing operands a and b using gcd a b gcd min a b abs a b and also that if a and b are first made odd then abs a b is even and factors of two can be discarded 96 GNU MP 4 1 3 Variants like letting a b become negative and doing a different next step are of interest only as far as they suit particular CPUs since on small operands it s machine dependent factors that determine performance The Euclidean GCD algorithm as per Knuth algorithms E and A reduces using a mod b but this has so far been found to be slower everywhere One reason the binary method does well is that the implied quotient at each step is usually small so often only one or two subtractions are needed to get the same effect as a division Quotients 1 2 and 3 for example occur 67 7 of the time see Knuth section 4 5 3 Theorem E When the implied quotient is large meaning b is much smaller than a then a division is worth while This is the basis for the initial a mod b reductions in mpn gcd and mpn gcd 1 the latter for both Nx1 and 1x1 cases But after that initial reduction big quotients oc
126. esults will be unpredictable For the ui variants the return value is the remainder and in fact returning the remainder is all the div ui functions do For tdiv and cdiv the remainder can be negative so for those the return value is the absolute value of the remainder The 2exp functions are right shifts and bit masks but of course rounding the same as the other functions For positive n both mpz fdiv q 2exp and mpz tdiv q 2exp are simple bitwise right shifts For negative n mpz fdiv q 2exp is effectively an arithmetic right shift treating n as twos complement the same as the bitwise logical functions do whereas mpz_ tdiv_q_2exp effectively treats n as sign and magnitude void mpz mod mpz_t r mpz t n mpz t d Function unsigned long int mpz mod ui mpz_t r mpz_t n unsigned long int d Function Set r to n mod d The sign of the divisor is ignored the result is always non negative mpz mod ui is identical to mpz fdiv r ui above returning the remainder as well as setting r See mpz fdiv ui above if only the return value is wanted void mpz divexact mpz_t q mpz t n mpz t d Function void mpz divexact ui mpz t q mpz t n unsigned long d Function Set q to n d These functions produce correct results only when it is known in advance that d divides n These routines are much faster than the other division functions and are the best choice when exact division is known to occur for example reducing a rational to lowest
127. f digits in base base base can be 2 to 36 Up to n digits digits will be generated Trailing zeros are not returned No more digits than can be accurately represented by op are ever generated If n digits is 0 then that accurate maximum number of digits are generated If str is NULL the result string is allocated using the current allocation function see Chap ter 14 Custom Allocation page 82 The block will be strlen str 1 bytes that being exactly enough for the string and null terminator If str is not NULL it should point to a block of n digits 2 bytes that being enough for the mantissa a possible minus sign and a null terminator When n digits is O to get all significant digits an application won t be able to know the space required and str should be NULL in that case The generated string is a fraction with an implicit radix point immediately to the left of the first digit The applicable exponent is written through the expptr pointer For example the number 3 1416 would be returned as string 31416 and exponent 1 When op is zero an empty string is produced and the exponent returned is 0 50 GNU MP 4 1 3 A pointer to the result string is returned being either the allocated block or the given str 7 5 Arithmetic Functions void mpf_add mpf_t rop mpf_t op1 mpf_t op2 Function void mpf_add_ui mpf_t rop mpf_t op1 unsigned long int op2 Function Set rop to opl op2 void mpf sub mpf t rop mpf t opi mpf t
128. f mp_bits_per_limb Note that the extra limb added here for the high only being non zero is in addition to the extra limb allocated to _mp_d For example with a 32 bit limb an application request for 250 bits will be rounded up to 8 limbs then an extra added for the high being only non zero giving an _mp_prec of 9 _mp_d then gets 10 limbs allocated Reading back with mpf_get_prec will take _mp_prec subtract 1 limb and multiply by 32 giving 256 bits Strictly speaking the fact the high limb has at least one bit means that a float with say 3 limbs of 32 bits each will be holding at least 65 bits but for the purposes of mpf_t it s considered simply to be 64 bits a nice multiple of the limb size 17 4 Raw Output Internals mpz_out_raw uses the following format Chapter 17 Internals 109 size data bytes The size is 4 bytes written most significant byte first being the number of subsequent data bytes or the twos complement negative of that when a negative integer is represented The data bytes are the absolute value of the integer written most significant byte first The most significant data byte is always non zero so the output is the same on all systems irrespective of limb size In GMP 1 leading zero bytes were written to pad the data bytes to a multiple of the limb size mpz_inp_raw will still accept this for compatibility The use of big endian for both the size and data fields is deliberate it ma
129. functions including the modular powering functions mpz_powm and mpz_powm_ui will cause an intentional division by zero This lets a program handle arithmetic exceptions in these functions the same way as for normal C int arithmetic void mpz_cdiv_q mpz_t q mpz t n mpz t d Function void mpz cdiv r mpz_t r mpz t n mpz t d Function void mpz cdiv qr mpz_t q mpz t r mpz t n mpz t d Function unsigned long int mpz cdiv q ui mpz t q mpz t n Function unsigned long int d unsigned long int mpz cdiv r ui mpz t r mpz t n Function unsigned long int d unsigned long int mpz cdiv qr ui mpz t q mpz t r mpz tn Function unsigned long int d unsigned long int mpz cdiv ui mpz n unsigned long int d Function void mpz_cdiv_q_2exp mpz t q mpz t n unsigned long int b Function void mpz cdiv r 2exp mpz_t r mpz t n unsigned long int b Function void mpz fdiv q mpz t q mpz t n mpz t d Function void mpz fdiv r mpz t r mpz t n mpz t d Function void mpz fdiv qr mpz_t q mpz t r mpz t n mpz t d Function unsigned long int mpz fdiv q ui mpz t q mpz t n Function unsigned long int d unsigned long int mpz fdiv r ui mpz_t r mpz_t n Function unsigned long int d unsigned long int mpz fdiv qr ui mpz_t q mpz r mpz tn Function unsigned long int d unsigned long int mpz fdiv ui mpz_t n unsigned long int d Function void mpz fdiv o 2exp mpz_t q mpz t n unsigned long int b Function void mpz fdiv r 2exp
130. gmp h since that will allow gmp h to define prototypes for these functions size_t mpf out str FILE stream int base size t n digits mpf t op Function Print op to stream as a string of digits Return the number of bytes written or if an error occurred return 0 The mantissa is prefixed with an 0 and is in the given base which may vary from 2 to 36 An exponent then printed separated by an e or if base is greater than 10 then by an e The exponent is always in decimal The decimal point follows the current locale on systems providing localeconv Up to n digits will be printed from the mantissa except that no more digits than are accu rately representable by op will be printed n digits can be 0 to select that accurate maximum size_t mpf inp str mpf_t rop FILE stream int base Function Read a string in base base from stream and put the read float in rop The string is of the form M N or if the base is 10 or less alternatively MeN M is the mantissa and N is the exponent The mantissa is always in the specified base The exponent is either in the specified base or if base is negative in decimal The decimal point expected is taken from the current locale on systems providing localeconv The argument base may be in the ranges 2 to 36 or 36 to 2 Negative values are used to specify that the exponent is in decimal Unlike the corresponding mpz function the ba
131. gure requires various Unix like tools On an MS DOS system DJGPP can be used and on MS Windows Cygwin or MINGW can be used http www cygwin com http www delorie com djgpp http www mingw org Microsoft also publishes an Interix Services for Unix which can be used to build GMP on Windows with a normal configure but it s not free software The macos directory contains an unsupported port to MacOS 9 on Power Mac intosh see macos README Note that MacOS X Darwin should use the normal Unix style configure It might be possible to build without the help of configure certainly all the code is there but unfortunately you ll be on your own Build Directory To compile in a separate build directory cd to that directory and prefix the configure command with the path to the GMP source directory For example cd my build dir my sources gmp 4 1 3 configure Not all make programs have the necessary features VPATH to support this In particular SunOS and Slowaris make have bugs that make them unable to build in a separate directory Use GNU make instead prefix and exec prefix The prefix option can be used in the normal way to direct GMP to install under a particular tree The default is usr local exec prefix can be used to direct architecture dependent files like 1ibgmp a to a different location This can be used to share architecture ind
132. hat the multiplications for each x are big blocks allowing Karatsuba and higher algorithms to be used But the cost of calculating the powers b must be overcome SET STR THRESHOLD usually ends up quite big around 5000 digits and on some processors much bigger still SET STR THRESHOLD is based on the input digits and tuned for decimal though it might be better based on a limb count so as to be independent of the base But that sort of count isn t used by the base case and so would need some sort of initial calculation or estimate The main reason SET STR THRESHOLD is so much bigger than the corresponding GET STR PRECOMPUTE THRESHOLD is that mpn mul 1 is much faster than mpn_divrem_1 often by a factor of 10 or more 16 7 Other Algorithms 16 7 1 Factorial Factorials n are calculated by a simple product from 1 to n but arranged into certain sub products First as many factors as fit in a limb are accumulated then two of those multiplied to give a 2 limb product When two 2 limb products are ready they re multiplied to a 4 limb product and when two 4 limbs are ready they re multiplied to an 8 limb product etc A stack of outstanding products is built up with two of the same size multiplied together when ready Arranging for multiplications to have operands the same or nearly the same size means the Karatsuba and higher multiplication algorithms can be used And even on sizes below the Karatsuba threshold an NxN multiply w
133. have fixed this in release 3 3 of their system POWER and PowerPC Bugs in GCC 2 7 2 and 2 6 3 mean it can t be used to compile GMP on POWER or PowerPC If you want to use GCC for these machines get GCC 2 7 2 1 or later Sequent Symmetry Use the GNU assembler instead of the system assembler since the latter has serious bugs Solaris 2 6 The system sed prints an error Output line too long when libtool builds libgmp la This doesn t seem to cause any obvious ill effects but GNU sed is recommended to avoid any doubt Sparc Solaris 2 7 with gcc 2 95 2 in ABI 32 A shared library build of GMP seems to fail in this combination it builds but then fails the tests apparently due to some incorrect data relocations within gmp_ randinit_lc_2exp_size The exact cause is unknown disable shared is rec ommended Windows DLL test programs When creating a DLL version of libgmp libtool creates wrapper scripts like t mu1 for programs that would normally be t mul exe in order to setup the right library paths etc This works fine but the absence of t mul exe etc causes make to think they need recompiling every time which is an annoyance when re running a make check 16 GNU MP 4 1 3 3 GMP Basics Using functions macros data types etc not documented in this manual is strongly discouraged If you do so your application is guaranteed to be incompatible with future versions of GMP 3 1 Headers and Li
134. he dividend gives a remainder and reveals whether q or q 1 is correct The result is a division done with two multiplications and four or five arithmetic operations On CPUs with low latency multipliers this can be much faster than a hardware divide though the cost of calculating the inverse at the start may mean it s only better on inputs bigger than say 4 or 5 limbs When a divisor must be normalized either for the generic C __udiv_qrnnd_c or the multiply by inverse the division performed is actually a2 by d2 where a is the dividend and k is the power necessary to have the high bit of d2 set The bit shifts for the dividend are usually accomplished on the fly meaning by extracting the appropriate bits at each step Done this way the quotient limbs come out aligned ready to store When only the remainder is wanted an alternative is to take the dividend limbs unshifted and calculate r a mod d2 followed by an extra final step r2 mod d2 This can help on CPUs with poor bit shifts or few registers The multiply by inverse can be done two limbs at a time The calculation is basically the same but the inverse is two limbs and the divisor treated as if padded with a low zero limb This means more work since the inverse will need a 2x2 multiply but the four 1x1s to do that are independent and can therefore be done partly or wholly in parallel Likewise for a 2x1 calculating qd l he net effect is to process two limbs with roughly th
135. he functions can return 1 if the C library printf variant in use returns 1 but this shouldn t normally occur int gmp_printf const char fmt Function int gmp_vprintf const char fmt va_list ap Function Print to the standard output stdout Return the number of characters written or 1 if an error occurred int gmp_fprintf FILE fp const char fmt Function int emp vfprintf FILE fp const char fmt va list ap Function Print to the stream fp Return the number of characters written or 1 if an error occurred int emp sprintf char buf const char fmt Function int gmp_vsprintf char buf const char fmt va_list ap Function Form a null terminated string in buf Return the number of characters written excluding the terminating null No overlap is permitted between the space at buf and the string fmt These functions are not recommended since there s no protection against exceeding the space available at buf int gmp_snprintf char buf size t size const char fmt Function int gmp_vsnprintf char buf size_t size const char fmt va list ap Function Form a null terminated string in buf No more than size bytes will be written To get the full output size must be enough for the string and null terminator The return value is the total number of characters which ought to have been produced excluding the terminating null If retval gt size then the actual output has been
136. ie Latdindds cede obieted 6 ees Ae 26 Exact division function 33 Example programme 20 Exec EE 4 Exponentiation functions 34 50 Exportador Ne 40 Extended GOD 4 512 oce dee 35 F Factorial functions ue NEEN dE RR 36 FDL GNU Free Documentation License 115 Fibonacci sequence functions ssssuerccccuu 36 Float arithmetic function 50 Float assignment functions 48 Float comparison function 50 Float conversion functions oocooococooocoo 49 Eloat futctiong o4ecunctalcie fee Gee que dera 46 Float init and assign functions 48 Float initialization functions 46 Float input and output functions 51 Float miscellaneous functions 51 Float sigu testi ose per RR sheen E 51 Floating point mode 12 Floating point functions 0 00 46 Floating point number ssssssrersrrrsrrrr 16 Formatted inputs 00040 med ee eR 68 Formatted output vicio pesar rr 63 FTP of latest version 2 Function classes ae VEER ENEE a renga dd bee e G GMP version number eee eeee 20 jou PETI TEE 16 EE T2 GNU Free Documentation License 115 Greatest common divisor algorithms 95 Greatest common divisor functions 35 H Hardware floating point mode 12 STEE 16 Home EE 2 122 I I O functions dE A dE catered 38 45 51 TMP OLE gen ie dio bet
137. ieces is convenient With some care though six 21x32 53 bit products can be used if one of the lower two 21 bit pieces also uses the sign bit For the mpn mul 1 family of functions on a 64 bit machine the invariant single limb is split at the start into 3 or 4 pieces Inside the loop the bignum operand is split into 32 bit pieces Fast conversion of these unsigned 32 bit pieces to floating point is highly machine dependent In 104 GNU MP 4 1 3 some cases reading the data into the integer unit zero extending to 64 bits then transferring to the floating point unit back via memory is the only option Converting partial products back to 64 bit limbs is usually best done as a signed conversion Since all values are smaller than 253 signed and unsigned are the same but most processors lack unsigned conversions Here is a diagram showing 16x32 bit products for an mpn mul 1 or mpn addmul 1 with a 64 bit limb The single limb operand V is split into four 16 bit parts The multi limb operand U is split in the loop into two 32 bit parts v48 v32 v16 v00 V Operand x u32 u00 U Operand one limb u00 x v00 p00 48 bit products u00 x v16 p16 u00 x v32 p32 u00 x v48 p48 u32 x v00 r32 u32 x v16 r48 u32 x v32 r64 u32 x v48 r80 p32 and r32 can be summed using floating point addition and likewise p48 and r48 p00 and p16 can
138. if present F behaves like a float GMP variables don t require an amp when passed to gmp_scanf since they re already call by reference For example to read say a 5 1234 int n mpz t z emp scanf a 4d WZdNn amp n z mpq t qi q2 emp sscanf 0377 0x10 0x11 Qi 4Qi qi q2 to read say topleft 1 55 2 66 mpf t x y char buf 32 gmp_scanf 431s AFL Ff buf x y All the standard C scanf types behave the same as in the C library scanf and can be freely intermixed with the GMP extensions In the current implementation the standard parts of the format string are simply handed to scanf and only the GMP extensions handled directly The flags accepted are as follows a and 77 will depend on support from the C library and cannot be used with GMP types read but don t store a allocate a buffer string conversions d group digits GLIBC style not GMP types 6 The standard types accepted are as follows h and 1 are portable the rest will depend on the compiler or include files for the type and the C library for the input h short hh char j intmax_t or uintmax_t 1 long int double or wchar_t 11 long long L long double q quad_t or u_quad_t t ptrdiff_t E size_t The GMP types are F mpf_t float conversions Chapter 11 Formatted Input 69 Q mpq_t integer conversions Z mpz_t integer conversions The conversions accepted
139. ill give a basecase multiply more to work on An obvious improvement not currently implemented would be to strip factors of 2 from the products and apply them at the end with a bit shift Another possibility would be to determine the prime factorization of the result which can be done easily and use a powering method at each stage squaring then multiplying in those primes with a 1 in their exponent at that point The advantage would be some multiplies turned into squares 16 7 2 Binomial Coefficients k sary and then evaluating the following product simply from i 2 to i k N T E 1 2 Binomial coefficients 7 are calculated by first arranging k n 2 using 7 i if neces n k It s easy to show that each denominator 7 will divide the product so far so the exact division algorithm is used see Section 16 2 4 Exact Division page 93 The numerators n k i and denominators i are first accumulated into as many fit a limb to save multi precision operations though for mpz bin ui this applies only to the divisors since n is an mpz t and n k i in general won t fit in a limb at all Chapter 16 Algorithms 101 An obvious improvement would be to strip factors of 2 from each multiplier and divisor and count them separately to be applied with a bit shift at the end Factors of 3 and perhaps 5 could even be handled similarly Another possibility if n is not too big would be to determine the prime factorizat
140. imental feature whereby a few bits are left unused at the top of each mp limb t T his can significantly improve carry handling on some processors All the mpn functions accepting limb data will expect the nail bits to be zero on entry and will return data with the nails similarly all zero This applies both to limb vectors and to single limb arguments Nails can be enabled by configuring with enable nails By default the number of bits will be chosen according to what suits the host processor but a particular number can be selected with enable nails N At the mpn level a nail build is neither source nor binary compatible with a non nail build strictly speaking But programs acting on limbs only through the mpn functions are likely to work equally well with either build and judicious use of the definitions below should make any program compatible with either build at the source level For the higher level routines meaning mpz etc a nail build should be fully source and binary compatible with a non nail build GMP NAIL BITS Macro GMP NUMB BITS Macro GMP LIMB BITS Macro GMP NAIL BITS is the number of nail bits or 0 when nails are not in use GMP_NUMB_BITS is the number of data bits in a limb GMP LIMB BITS is the total number of bits in an mp limb t In all cases GMP LIMB BITS GHP NAIL BITS GMP NUMB BITS GMP_NATL_MASK Macro GMP_NUMB_MASK Macro Bit masks for the nail and number parts
141. ining or if a limb is 64 bits then primes up to 53 are used excluding 99 99 A single Nx1 remainder using PP from gmp impl h quickly gives all these remainders A square root must still be taken for any value that passes the residue tests to verify it s really a square and not one of the 0 086 or 0 000156 for 64 bits non squares that get through See Section 16 5 1 Square Root Algorithm page 97 16 5 4 Perfect Power Detecting perfect powers is required by some factorization algorithms Currently mpz_perfect_ power_p is implemented using repeated Nth root extractions though naturally only prime roots need to be considered See Section 16 5 2 Nth Root Algorithm page 98 If a prime divisor p with multiplicity e can be found then only roots which are divisors of e need to be considered much reducing the work necessary To this end divisibility by a set of small primes is checked 16 6 Radix Conversion Radix conversions are less important than other algorithms A program dominated by conver sions should probably use a different data representation 16 6 1 Binary to Radix Conversions from binary to a power of 2 radix use a simple and fast O N bit extraction algo rithm Conversions from binary to other radices use one of two algorithms Sizes below GET STR PRECOMPUTE THRESHOLD use a basic O N method Repeated divisions by b are made where b is the radix and m is the biggest power that fits in a limb But ins
142. integer and set its value to 0 void mpz_init2 mpz_t integer unsigned long n Function Initialize integer with space for n bits and set its value to 0 n is only the initial space integer will grow automatically in the normal way if necessary for subsequent values stored mpz_init2 makes it possible to avoid such reallocations if a maximum size is known in advance void mpz clear mpz t integer Function Free the space occupied by integer Call this function for all mpz t variables when you are done with them void mpz realloc2 mpz t integer unsigned long n Function Change the space allocated for integer to n bits The value in integer is preserved if it fits or is set to 0 if not This function can be used to increase the space for a variable in order to avoid repeated automatic reallocations or to decrease it to give memory back to the heap void mpz array init mpz_t integer array size_t array size Function mp size_t fixed num bits This is a special type of initialization Fixed space of fixed num bits bits is allocated to each of the array size integers in integer array The space will not be automatically increased unlike the normal mpz init but instead an application must ensure it s sufficient for any value stored The following space requirements apply to various functions Chapter 5 Integer Functions 29 e mpz_abs mpz_neg mpz_set mpz_set_si and mpz_set_ui need room for the value they store
143. ion void mpz set si mpz t rop signed long int op Function void mpz set d mpz t rop double op Function void mpz set o mpz t rop mpq t op Function void mpz set f mpz t rop mpf t op Function Set the value of rop from op mpz set d mpz set q and mpz set f truncate op to make it an integer int mpz set str mpz t rop char str int base Function Set the value of rop from str a null terminated C string in base base White space is allowed in the string and is simply ignored The base may vary from 2 to 36 If base is 0 the actual base is determined from the leading characters if the first two characters are Ox or OX hexadecimal is assumed otherwise if the first character is 0 octal is assumed otherwise decimal is assumed This function returns 0 if the entire string is a valid number in base base Otherwise it returns void mpz swap mpz_t rop1 mpz t rop2 Function Swap the values rop and rop2 efficiently 5 3 Combined Initialization and Assignment Functions For convenience GMP provides a parallel series of initialize and set functions which initialize the output and then store the value there These functions names have the form mpz init set 30 GNU MP 4 1 3 Here is an example of using one mpz_t pie mpz_init_set_str pie 3141592653589793238462643383279502884 10 mpz sub pie 2 mpz clear pie F Once the integer has been initialized by any of the m
144. ion algorithms 98 Random number functions 39 61 Random number seeding icsuerueurcrereu 62 Random number State 0 cee ee eee eee 61 Rational arithmetic functions 43 Rational comparison functions 44 Rational conversion functions 43 Rational init and assign 0 42 Rational input and output functions 45 Rational number 46er Pm REPRE RET 16 Rational number functions s s s uasaarsssersue 42 Rational numerator and denominator 44 Ttational Sign tests aere oU ve Bess EE 44 Reentrancy3oosu eR E LE REM ee id 19 References codes AN 113 Reporting DUES 2 ee ee d Ee ai 27 Root extraction algorithms OF Root extraction functions 34 50 S Sample POTS 229 2ce cres tee eee TASSE 20 scanf formatted input o 68 Shared library versioning 11 Digi tests senden ee RET EE dos 37 44 51 Size in E EE 41 n RPM 13 Stack overflow segfaulte T Stripped libraries 0o ooooooooooommo 14 EEN 11 Appendix C Concept Index T Re EE 19 TEE 16 U Upward compatibility 0oooooooo o o o 20 Useful macros and constants o0ooooccooooo o 19 User defined precision 00 eee ee eee 46 V 123 Valgrinds cgit EI AREE bin dos iex e a ded 24 Variable conventions 0000 cee cece een IT Version number
145. ion of the result based on the factorials involved and power up those primes appropriately This would help most when k is near n 2 16 7 3 Fibonacci Numbers The Fibonacci functions mpz fib ui and mpz_fib2_ui are designed for calculating isolated Fi or bb values efficiently For small n a table of single limb values in emp fib table is used On a 32 bit limb this goes up to bis or on a 64 bit limb up to Fo3 For convenience the table starts at F4 Beyond the table values are generated with a binary powering algorithm calculating a pair Fn and F working from high to low across the bits of n The formulas used are Paz AF Fa 2 1 Por Fp Fha For ban bat At each step k is the high b bits of n If the next bit of n is 0 then P5 P5 is used or if it s a 1 then ban bat is used and the process repeated until all bits of n are incorporated Notice these formulas require just two squares per bit of n It d be possible to handle the first few n above the single limb table with simple additions using the defining Fibonacci recurrence bn Fy Fk 1 but this is not done since it usually turns out to be faster for only about 10 or 20 values of n and including a block of code for just those doesn t seem worthwhile If they really mattered it d be better to extend the data table Using a table avoids lots of calculations on small numbers and makes small n go fast A bigger table would make more small n g
146. is the basic 32 bit PowerPC ABI No special compiler options are needed for applications Sparc V9 sparcv9 and ultrasparc ABI 64 The 64 bit V9 ABI is available on Solaris 2 7 and up and GNU Linux GCC 2 95 or up or Sun cc is required Applications must be compiled with gcc m64 mptr64 Wa xarch v9 mcpu v9 cc xarch v9 ABI 32 On Solaris 2 6 and earlier and on Solaris 2 7 with the kernel in 32 bit mode only the plain V8 32 bit ABI can be used since the kernel doesn t save all registers GMP still uses as much of the V9 ISA as it can in these circumstances No special compiler options are required for applications though using something like the following requesting V9 code within the V8 ABI is recommended Chapter 2 Installing GMP 11 gcc mv8plus cc xarch v8plus gcc 2 8 and earlier only supports mv8 though Don t be confused by the names of these sparc m and x options they re called arch but they effectively control the ABI On Solaris 2 7 with the kernel in 32 bit mode a normal native build will reject ABI 64 because the resulting executables won t run ABI 64 can still be built if desired by making it look like a cross compile for example configure build none host sparcv9 sun solaris2 7 ABI 64 2 3 Notes for Package Builds GMP should present no great difficulties for packaging in a binary distribution Libtool is used to build the library and
147. it This just uses the default AC CHECK LIB actions for found or not found but an application that must have GMP would want to generate an error if not found For example AC CHECK LIB gmp gmpz init AC MSG ERROR GNU MP not found see http swox com gmp If functions added in some particular version of GMP are required then one of those can be used when checking For example mpz mul si was added in GMP 3 1 AC CHECK LIB gmp gmpz mul si AC MSG ERROR GNU MP not found or not 3 1 or up see http swox com gmp An alternative would be to test the version number in gmp h using say AC EGREP CPP That would make it possible to test the exact version if some particular sub minor release is known to be necessary 26 GNU MP 4 1 3 An application that can use either GMP 2 or 3 will need to test for gmpz init GMP 3 and up or mpz init GMP 2 and it s also worth checking for libgmp2 since Debian GNU Linux systems used that name in the past For example AC CHECK LIB gmp gmpz init AC CHECK LIB gmp mpz init AC CHECK LIB gmp2 mpz_init In general it s suggested that applications should simply demand a new enough GMP rather than trying to provide supplements for features not available in past versions Occasionally an application will need or want to know the size of a type at configuration or preprocessing time not just with sizeof in the code This can be done in the normal way
148. it2 or mpz_realloc2 is too small since all functions will do a further reallocation if necessary Badly overestimating memory required will waste space though 2exp functions It s up to an application to call functions like mpz_mul_2exp when appropriate General purpose functions like mpz mul make no attempt to identify powers of two or other special forms because such inputs will usually be very rare and testing every time would be wasteful ui and si functions The ui functions and the small number of si functions exist for convenience and should be used where applicable But if for example an mpz t contains a value that fits in an unsigned long there s no need extract it and call a ui function just use the regular mpz function In Place Operations mpz abs mpq abs mpf abs mpz neg mpq neg and mpf_neg are fast when used for in place operations like mpz abs x x since in the current implementation only a single field of x needs changing On suitable compilers GCC for instance this is inlined too mpz add ui mpz sub ui mpf add ui and mpf sub ui benefit from an in place operation like mpz add ui x x y since usually only one or two limbs of x will need to be changed The same applies to the full precision mpz add etc if y is small If y is big then cache locality may be helped but that s all mpz mul is currently the opposite a separate destination is slightly better A call like mpz mul x x y will unless y is only
149. izations of __gmp_expr Next we define assignment of __gmp_expr to mpf_class template lt class T gt mpf_class amp mpf_class operator const __gmp_expr lt T gt amp expr d expr eval this gt get_mpf_t this gt precision return this template lt class Op gt 110 GNU MP 4 1 3 void __gmp_expr lt __gmp_binary_expr lt mpf_class mpf_class Op gt gt eval mpf_t f unsigned long int precision Op eval f expr vali get mpf tO expr val2 get_mpf_t where expr vali and expr val2 are references to the expression s operands here expr is the __gmp_binary_expr stored within the __gmp_expr This way the expression is actually evaluated only at the time of assignment when the required precision that of is known Furthermore the target mpf_t is now available thus we can call mpf add directly with f as the output argument Compound expressions are handled by defining operators taking subexpressions as their argu ments like this template class T class U BEmnp expr gmp binary expr gmp expr T __gmp_expr lt U gt gmp binary plus gt operator const __gmp_expr lt T gt amp expri const gmp expr U amp expr2 1 return gmp expr lt __gmp_binary_expr lt __gmp_expr lt T gt __gmp_expr lt U gt gmp binary plus gt expri expr2 J And the corresponding specializations of __gmp_expr eval template lt class T class U class Op gt void __gmp_expr
150. kes the data easy to read in a hex dump of a file Unfortunately it also means that the limb data must be reversed when reading or writing so neither a big endian nor little endian system can just read and write _mp_d 17 5 C Interface Internals A system of expression templates is used to ensure something like a b c turns into a simple call to mpz_add etc For mpf_class and mpfr_class the scheme also ensures the precision of the final destination is used for any temporaries within a statement like f w x y z These are important features which a naive implementation cannot provide A simplified description of the scheme follows The true scheme is complicated by the fact that expressions have different return types For detailed information refer to the source code To perform an operation say addition we first define a function object evaluating it struct __gmp_binary_plus d static void eval mpf_t f mpf_t g mpf_t h mpf_add f g h F And an additive expression object __gmp_expr lt __gmp_binary_expr lt mpf_class mpf_class __gmp_binary_plus gt gt operator const mpf class amp f const mpf class amp g d return __gmp_expr lt __gmp_binary_expr lt mpf_class mpf_class __gmp_binary_plus gt gt f g The seemingly redundant __gmp_expr lt __gmp_binary_expr lt gt gt is used to encapsulate any possible kind of expression into a single template type In fact even mpf_class etc are typedef special
151. l u2 u3 u4 u d ul d u2 d u3 d u4 d In practice squaring isn t a full 2x faster than multiplying it s usually around 1 5x Less than 1 5x probably indicates mpn_sqr_basecase wants improving on that CPU Chapter 16 Algorithms 87 On some CPUs mpn_mul_basecase can be faster than the generic C mpn_sqr_basecase SQR_ BASECASE_THRESHOLD is the size at which to use mpn_sqr_basecase this will be zero if that routine should be used always 16 1 2 Karatsuba Multiplication The Karatsuba multiplication algorithm is described in Knuth section 4 3 3 part A and various other textbooks A brief description is given here The inputs x and y are treated as each split into two parts of equal length or the most significant part one limb shorter if N is odd high low T To Yi Yo Let b be the power of 2 where the split occurs ie if zo is k limbs yo the same then b gk mp_bits_per_limb With that y x b zy and y y b yo and the following holds zy b b oiyi b z1 zo Yi Yo b 1 zoyo This formula means doing only three multiplies of N 2 x N 2 limbs whereas a basecase multiply of NxN limbs is equivalent to four multiplies of N 2 x N 2 The factors b b etc represent the positions where the three products must be added high low Tiu Toyo T1Y1 SE Toyo E zi zo yi Yo The term x
152. lass op Function int cmp mpq class op1 type op2 Function int cmp type opi mpq class op2 Function double mpq class get d void Function int sgn mpq class op Function These functions provide a C class interface to the corresponding GMP C routines cmp can be used with any of the classes or the standard C types except long long and long double mpz class amp mpq class get num Function mpz class amp mpq class get den Function Get a reference to an mpz class which is the numerator or denominator of an mpq class This can be used both for read and write access If the object returned is modified it modifies the original mpq_class If direct manipulation might produce a non canonical value then mpq class canonicalize must be called before further operations mpz t mpq class get num mpz tU Function mpz t mpq class get den mpz t Function Get a reference to the underlying mpz_t numerator or denominator of an mpq class This can be passed to C functions expecting an mpz_t Any modifications made to the mpz t will modify the original mpq class If direct manipulation might produce a non canonical value then mpq class canonicalize must be called before further operations istream amp operator istream amp stream mpq class amp rop Function Read rop from stream using its ios formatting settings the same as mpq t operator see Section 11 3 C Formatted Input pag
153. le There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 4 Please see the GMP 2 manual for details The Berkeley MP compatibility library see Chapter 13 BSD Compatible Functions page 80 is source and binary compatible with the standard libmp 3 10 Demonstration programs The demos subdirectory has some sample programs using GMP These aren t built or installed but there s a Makefile with rules for them For instance make pexpr pexpr 68 975 10 The following programs are provided e pexpr is an expression evaluator the program used on the GMP web page e The calc subdirectory has a similar but simpler evaluator using lex and yacc e The expr subdirectory is yet another expression evaluator a library designed for ease of use within a C program See demos expr README for more information e factorize is a Pollard Rho factorization program e isprime is a command line interface to the mpz_probab_prime_p function e primes counts or lists primes in an interval using a sieve e qcn is an example use of mpz_kronecker_ui to estimate quadratic class numbers e The perl subdirectory is a comprehensive perl interface to GMP See demos perl INSTALL for more information Documentation is in POD format in demos per1 GMP pm 3 11 Efficiency Small operands On small
154. library see Section 2 1 Build Options page 4 These are likely to be of limited value to most applications Assertion failures are just as likely to indicate memory corruption as a library or compiler bug Applications using the low level mpn functions however will benefit from enable assert since it adds checks on the parameters of most such functions many of which have subtle restrictions on their usage Note however that only the generic C code has checks not the assembler code so CPU none should be used for maximum checking Temporary Memory Checking The build option enable alloca debug arranges that each block of temporary memory in GMP is allocated with a separate call to malloc or the allocation function set with mp_set_memory_functions This can help a malloc debugger detect accesses outside the intended bounds or detect memory not released In a normal build on the other hand temporary memory is allocated in blocks which GMP divides up for its own use or may be allocated with a compiler builtin alloca which will go nowhere near any malloc debugger hooks Maximum Debuggability Checker Valgrind To summarize the above a GMP build for maximum debuggability would be configure disable shared enable assert enable alloca debug host none CFLAGS g For C add enable cxx CXXFLAGS g The checker program http savannah gnu org projects checker can be used with GMP It co
155. ll be an inferior build since the MMX code that s present is there because it s faster than the corresponding plain integer code The same applies to SSE2 Old versions of gas don t support MMX instructions in particular version 1 92 3 that comes with FreeBSD 2 2 8 doesn t and unfortunately there s no newer assem bler for that system Solaris 2 6 and 2 7 as generate incorrect object code for register to register movq instructions and so can t be used for MMX code Install a recent gas if MMX code is wanted on these systems 2 5 Known Build Problems You might find more up to date information at http swox com gmp Compiler link options The version of libtool currently in use rather aggressively strips compiler options when linking a shared library T his will hopefully be relaxed in the future but for now if this is a problem the suggestion is to create a little script to hide them and for instance configure with configure CC gcc with my options 14 DJGPP GNU MP 4 1 3 The DJGPP port of bash 2 03 is unable to run the configure script it exits silently having died writing a preamble to config log Use bash 2 04 or higher make all was found to run out of memory during the final 1ibgmp 1a link on one system tested despite having 64Mb available A separate make libgmp 1a helped perhaps recursing into the various subdirectories uses up memory DESTDIR and shared libgmpxx
156. llaneous Functions int mpz fits ulong p mpz t op Function int mpz fits slong p mpz t op Function int mpz fits uint p mpz t op Function int mpz fits sint p mpz t op Function int mpz fits ushort p mpz t op Function int mpz fits sshort p mpz t op Function Return non zero iff the value of op fits in an unsigned long int signed long int unsigned int signed int unsigned short int or signed short int respectively Otherwise re turn zero int mpz odd p mpz t op Macro int mpz even p mpz _t op Macro Determine whether op is odd or even respectively Return non zero if yes zero if no These macros evaluate their argument more than once size t mpz size mpz t op Function Return the size of op measured in number of limbs If op is zero the returned value will be Zero size_t mpz_sizeinbase mpz t op int base Function Return the size of op measured in number of digits in the given base base can vary from 2 to 36 The sign of op is ignored just the absolute value is used The result will be either exact or 1 too big If base is a power of 2 the result is always exact If op is zero the return value is always 1 This function can be used to determine the space required when converting op to a string The right amount of allocation is normally two more than the value returned by mpz_sizeinbase one extra for a minus sign and one for the null terminator It will be noted that mpz_sizeinbase op 2 can be
157. loat Functions page 51 The older style random number functions don t accept a gmp_randstate_t parameter but in stead share a global variable of that type They use a default algorithm and are currently not seeded though perhaps that will change in the future The new functions accepting a gmp_randstate_t are recommended for applications that care about randomness 9 1 Random State Initialization void gmp_randinit_default gmp_randstate_t state Function Initialize state with a default algorithm This will be a compromise between speed and randomness and is recommended for applications with no special requirements void gmp_randinit_lc_2exp gmp randstate_t state mpz_t a Function unsigned long c unsigned long m2exp Initialize state with a linear congruential algorithm X aX c mod Amien The low bits of X in this algorithm are not very random The least significant bit will have a period no more than 2 and the second bit no more than 4 etc For this reason only the high half of each X is actually used When a random number of more than m2exp 2 bits is to be generated multiple iterations of the recurrence are used and the results concatenated int emp randinit lc 2exp size gmp randstate t state unsigned long Function size Initialize state for a linear congruential algorithm as per emp randinit lc 2exp a c and m2exp are selected from a table chosen so that size bits or more of each X will be used ie m2exp 2
158. m mpz rop mpz t opi mpz t op2 Function void mpz lcm ui mpz_t rop mpz t opi unsigned long op2 Function Set rop to the least common multiple of op and op2 rop is always positive irrespective of the signs of op1 and op2 rop will be zero if either op1 or op2 is zero int mpz invert mpz t rop mpz t opi mpz t op2 Function Compute the inverse of op1 modulo op2 and put the result in rop If the inverse exists the return value is non zero and rop will satisfy 0 rop op2 If an inverse doesn t exist the return value is zero and rop is undefined int mpz jacobi mpz t a mpz t b Function Calculate the Jacobi symbol 7 This is defined only for b odd int mpz legendre mpz t a mpz t p Function Calculate the Legendre symbol 2 This is defined only for p an odd positive prime and for such p it s identical to the Jacobi symbol int mpz kronecker mpz t a mpz_t b Function int mpz kronecker si mpz t a long b Function int mpz kronecker ui mpz t a unsigned long b Function int mpz si kronecker long a mpz t b Function int mpz ui kronecker unsigned long a mpz t b Function Calculate the Jacobi symbol 2 with the Kronecker extension 2 2 when a odd or 2 0 when a even 36 GNU MP 4 1 3 When b is odd the Jacobi symbol and Kronecker symbol are identical so mpz_kronecker_ui etc can be used for mixed precision Jacobi symbols too For more information see Henri Cohen section 1 4
159. mni F77 http phase hpcc jp Omni home html Arbitrary precision floats Glasgow Haskell Compiler http www haskell org ghc Kaffe http www kaffe org Kissme http kissme sourceforge net GNU Common Lisp http www gnu org software gcl gcl html In the process of switching to GMP for bignums Librep http librep sourceforge net XEmacs 21 5 18 beta and up http www xemacs org Optional big integers rationals and floats using GMP GNU m4 betas http www seindal dk rene gnu Optionally provides an arbitrary precision mpeval MLton compiler http www mlton org MLGMP nhttp www di ens fr monniaux programmes html en Numerix http pauillac inria fr quercia Optionally using GMP Chapter 15 Language Bindings 85 Pascal Perl Pike Prolog Python Scheme Smalltalk Other Mozart http www mozart oz org GNU Pascal Compiler http www gnu pascal de GMP unit Numerix http pauillac inria fr quercia For Free Pascal optionally using GMP GMP module see demos perl in the GMP sources Math GMP http www cpan org Compatible with Math BigInt but not as many functions as the GMP module above Math BigInt GMP http www cpan org Plug Math GMP into normal Math BigInt operations mpz module in the standard distribution http pike ida liu se SWI Prolog http www swi psy uva nl projects SWI Prolog Arbitrary precision floats mpz module in the standard distribution http www
160. n mpz_init r mpz_init_set_str n 123456 0 foo r n 20L gmp_printf Zd n r return 0 foo works even if the mainline passes the same variable for param and result just like the library functions But sometimes it s tricky to make that work and an application might not want to bother supporting that sort of thing For interest the GMP types mpz_t etc are implemented as one element arrays of certain struc tures This is why declaring a variable creates an object with the fields GMP needs but then using it as a parameter passes a pointer to the object Note that the actual fields in each mpz_t etc are for internal use only and should not be accessed directly by code that expects to be compatible with future GMP releases Chapter 3 GMP Basics 19 3 6 Memory Management The GMP types like mpz_t are small containing only a couple of sizes and pointers to allocated data Once a variable is initialized GMP takes care of all space allocation Additional space is allocated whenever a variable doesn t have enough mpz_t and mpq_t variables never reduce their allocated space Normally this is the best policy since it avoids frequent reallocation Applications that need to return memory to the heap at some particular point can use mpz_realloc2 or clear variables no longer needed mpf_t variables in the current implementation use a fixed amount of space determined by the chosen precision and allocated at initializa
161. n and s2p s2n and write the sin least significant limbs of the result to rp Return carry either 0 or 1 This function requires that sin is greater than or equal to s2n 54 GNU MP 4 1 3 mp limb t mpn sub n mp_limb_t rp const mp_limb_t s1p const Function mp_limb_t s2p mp_size_t n Subtract s2p n from slp n and write the n least significant limbs of the result to rp Return borrow either 0 or 1 This is the lowest level function for subtraction It is the preferred function for subtraction since it is written in assembly for most CPUs mp limb t mpn sub 1 mp limb t rp const mp_limb_t s1p mp size t n Function mp_limb_t s21imb Subtract s2limb from slp n and write the n least significant limbs of the result to rp Return borrow either 0 or 1 mp limb t mpn sub mp_limb_t rp const mp_limb_t s1p mp size_t sin Function const mp_limb_t s2p mp size_t s2n Subtract s2p s2n from slp sln and write the sin least significant limbs of the result to rp Return borrow either 0 or 1 This function requires that sin is greater than or equal to s2n void mpn_mul_n mp_limb_t rp const mp_limb_t s1p const mp_limb_t Function s2p mp _size_t n Multiply slp n and s2p n and write the 2 n limb result to rp The destination has to have space for 2 n limbs even if the product s most significant limb is zero No overlap is permitted between the destination and either source mp_limb
162. n that everyone finds out that there is no warranty for the GNU MP library If it is modified by someone else and passed on we want their recipients to know that what they have is not what we distributed so that any problems introduced by others will not reflect on our reputation The precise conditions of the license for the GNU MP library are found in the Lesser General Public License version 2 1 that accompanies the source code see COPYING LIB Certain demon stration programs are provided under the terms of the plain General Public License version 2 see COPYING 2 GNU MP 4 1 3 1 Introduction to GNU MP GNU MP is a portable library written in C for arbitrary precision arithmetic on integers rational numbers and floating point numbers It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types Many applications use just a few hundred bits of precision but some applications may need thousands or even millions of bits GMP is designed to give good performance for both by choosing algorithms based on the sizes of the operands and by carefully keeping the overhead at a minimum The speed of GMP is achieved by using fullwords as the basic arithmetic type by using sophis ticated algorithms by including carefully optimized assembly code for the most common inner loops for many different CPUs and by a general emphasis on speed as opposed t
163. ne gece y ed 39 Initialization and assignment functions 29 42 48 Initialization functions 28 46 Input functions AN SEN EENS dee es 38 45 51 Install prefix gu Dk dee ri Eredi A Installing GMP iii oid stig settee be eos Bed e rati 4 Instruction Set Architecture 9 ang 16 Integer arithmetic functionS 0ooooooooo 31 Integer assignment functions 29 Integer bit manipulation functions 37 Integer comparison functions oooooooooooo 36 Integer conversion functions 30 Integer division functions 32 Integer exponentiation functions 34 Integer export es eed x ted REFER ERE seas 40 Integer f nctions 9 Ie e o EUER PES 28 Integer Importo clio dae 39 Integer init and assign eseussserrrerrrerss 29 Integer initialization functions 28 Integer input and output functions 38 Integer miscellaneous Dunctons ss 41 Integer random number functions 39 Integer root functions 0005 34 Integer sign tests EE 37 Introduction ae peiora nannan ea E 2 EE 9 istream Ehe GEET 70 J Jacobi symbol functions 35 K Kronecker symbol functions 00oooooooooo 35 L Latest version of OM 2 Least common multiple functions 35 ITT 16 Libtool versioning 0 cess ee eee ee ee til Lic
164. nt p void Function bool mpf class fits ulong p void Function bool mpf class fits ushort p void Function int sgn mpf_class op Function mpf_class sqrt mpf class op Function mpf_class trunc mpf_class op Function These functions provide a C class interface to the corresponding GMP C routines cmp can be used with any of the classes or the standard C types except long long and long double The accuracy provided by hypot is not currently guaranteed unsigned long int mpf_class get_prec Function void mpf_class set_prec unsigned long prec Function void mpf class set prec raw unsigned long prec Function Get or set the current precision of an mpf_class The restrictions described for mpf set prec raw see Section 7 1 Initializing Floats page 46 apply to mpf_class set prec raw Note in particular that the mpf class must be restored to it s allocated precision before being destroyed T his must be done by applica tion code there s no automatic mechanism for it 12 5 C Interface MPFR The C class interface to MPFR is provided if MPFR is enabled see Section 2 1 Build Options page 4 This interface must be regarded as preliminary and possibly subject to incompatible changes in the future since MPFR itself is preliminary All definitions can be obtained with include lt mpfrxx h gt This defines mpfr_class Class which behaves similarly to mpf_class see Section 12 4 C Interface Flo
165. ntains a stub library which means GMP applications compiled with checker can use a normal GMP build A build of GMP with checking within GMP itself can be made This will run very very slowly Configure with configure host none pc linux gnu CC checkergcc host none must be used since the GMP assembler code doesn t support the checking scheme The GMP C features cannot be used since current versions of checker 0 9 9 1 don t yet support the standard C library The valgrind program http valgrind kde org is a memory checker for x86s It translates and emulates machine instructions to do strong checks for uninitialized data at the level of individual bits memory accesses through bad pointers and memory leaks Chapter 3 GMP Basics 25 Recent versions of Valgrind are getting support for MMX and SSE SSE2 instruc tions for past versions GMP will need to be configured not to use those ie for an x86 without them for instance plain i486 Other Problems Any suspected bug in GMP itself should be isolated to make sure it s not an appli cation problem see Chapter 4 Reporting Bugs page 27 3 13 Profiling Running a program under a profiler is a good way to find where it s spending most time and where improvements can be best sought Depending on the system it may be possible to get a flat profile meaning simple timer sampling of the program counter with no special GMP build options just a p
166. o fast it s just a question of balancing size against desired speed For GMP the code is kept compact with the emphasis primarily on a good powering algorithm mpz fib2 ui returns both F and PF but mpz_fib_ui is only interested in Fj In this case the last step of the algorithm can become one multiply instead of two squares One of the following two formulas is used according as n is odd or even Fy FF 2F4 1 Bau 2Fp Fx 43 2F amp Fy 1 2 1 Fox here is the same as above just rearranged to be a multiply For interest the 2 1 term both here and above can be applied just to the low limb of the calculation without a carry or borrow into further limbs which saves some code size See comments with mpz fib ui and the internal mpn_fib2_ui for how this is done 16 7 4 Lucas Numbers mpz lucnum2 ui derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas Ly Fx 2Fx 1 Lys 2F Pra mpz lucnum ui is only interested in L and some work can be saved Trailing zero bits on n can be handled with a single square each La L 2 1 102 GNU MP 4 1 3 And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers similar to what mpz_fib_ui does Doxy1 5Fx 1 2Fx Fu 4 1 16 8 Assembler Coding The assembler subroutines in GMP are the most significant source of speed at small to moderate sizes At larger sizes algorithm selec
167. o simplicity or elegance There is carefully optimized assembly code for these CPUs ARM DEC Alpha 21064 21164 and 21264 AMD 29000 AMD K6 K6 2 and Athlon Hitachi SuperH and SH 2 HPPA 1 0 1 1 and 2 0 Intel Pentium Pentium Pro II III Pentium 4 generic x86 Intel IA 64 1960 Motorola MC68000 MC68020 MC88100 and MC88110 Motorola IBM PowerPC 32 and 64 National NS32000 IBM POWER MIPS R3000 R4000 SPARCv7 SuperSPARC generic SPARCv8 UltraSPARC DEC VAX and Zilog Z8000 Some optimizations also for Cray vector systems Clipper IBM ROMP RT and Pyramid AP XP There are two public mailing lists of interest One for general questions and discussions about usage of the GMP library and one for discussions about development of GMP There s more information about the mailing lists at http swox com mailman listinfo These lists are not for bug reports The proper place for bug reports is bug gmp gnu org See Chapter 4 Reporting Bugs page 27 for info about reporting bugs For up to date information on GMP please see the GMP web pages at http swox com gmp The latest version of the library is available at ftp ftp gnu org gnu gmp Many sites around the world mirror ftp gnu org please use a mirror near you see http www gnu org order ftp html for a full list There are three public mailing lists of interest One for release announcements one for gen eral questions and discussions about usage of the GMP libra
168. oe eo Rocha ae es 36 npz Cdiv Qiii e PE Y dd xo ERR 32 mpz cdiv Q QED EE 32 npz cdiv Giens ge as eR ienna pin anni es 32 MPZ Cdiv AT oie disks ner rueda adage ade 32 npz cdiv dr Ui eM EE 32 npz 6diVv E seme eb ia 32 npz cdiv r 26XD ives Hiei ia b pads 32 Dpz cdiy r ul 26a e a Rente Ye 32 mpz cdiv it bpd nea oes oe ias 32 mpz CLASS re Sieten E perdere REP H 72 npz class fits sint po ie ses sr ooreen 74 mpz elass fits slong pi esse 1088 hes 74 npz class fits sshort p ee enn 74 npz class fits int D e s 74 mpz class fits ulong ebe rea 74 Mp2_class ifits short p 4 ecrit 74 Mp2 Class iget lia a a de antes 74 ue EE Ta mpz class igat 81 22 etd da 74 Mp4 class get EE 74 npz cl ss impz cl sS o Ree pee T3 Mpz ti E r E S E EEE 28 npz CUE EE 38 jino Ae o Ai E N E A E E EET 36 npz cemp od snl a Hee Ae a eas 36 MPZ EENS dose 36 mp2 GMP ii ie 36 npz cmpabs e Me wt Eana Ee E ES 37 npz cmUpabs d er E epe prre He HS 34 npz empabs ui re ER E ROSE ae SS 3T IDDZ COl ocneseeLeueereE a pd x apps 3T mpz congruent 2exp p e I ee e e SES 3o mpz congr ent Pei ee esee EU ES 33 mpz congr ent ul p i leneesue vee ends 3 npz divexaCti i e rara e be 33 nhpz divexact D EE 33 npz divisible 2exp p oooc e temp ag 33 npz divisible D 4 542 eerta Rc etae qs 33 mpz divisible usd Poi io eet eges 33 MPZ even A ae 4 5 8 EES EREN 41 Npz exporioctcef E Gerd ad E ERE Id E PG 40 npz fac et EES EEN BEE a ech rte
169. of a limb GMP_NAIL_MASK is 0 when nails are not in use GMP_NAIL_MASK is not often needed since the nail part can be obtained with x gt gt GMP_NUMB_ BITS and that means one less large constant which can help various RISC chips GMP_NUMB_MAX Macro The maximum value that can be stored in the number part of a limb This is the same as GMP_NUMB_MASK but can be used for clarity when doing comparisons rather than bit wise operations 60 GNU MP 4 1 3 The term nails comes from finger or toe nails which are at the ends of a limb arm or leg numb is short for number but is also how the developers felt after trying for a long time to come up with sensible names for these things In the future the distant future most likely a non zero nail might be permitted giving non unique representations for numbers in a limb vector This would help vector processors since carries would only ever need to propagate one or two limbs Chapter 9 Random Number Functions 61 9 Random Number Functions Sequences of pseudo random numbers in GMP are generated using a variable of type gmp_ randstate_t which holds an algorithm selection and a current state Such a variable must be initialized by a call to one of the gmp_randinit functions and can be seeded with one of the gmp_randseed functions The functions actually generating random numbers are described in Section 5 13 Integer Ran dom Numbers page 39 and Section 7 8 Miscellaneous F
170. of op Otherwise return the least significant part of op with the same sign as op If op is too big to fit in a signed long int the returned result is probably not very useful To find out if the value will fit use the function mpz fits slong p double mpz get d mpz t op Function Convert op to a double double mpz get d 2exp signed long int exp mpz_t op Function Find d and exp such that d x 2 with 0 5 d lt 1 is a good approximation to op char mpz get str char str int base mpz t op Function Convert op to a string of digits in base base The base may vary from 2 to 36 Chapter 5 Integer Functions 31 If str is NULL the result string is allocated using the current allocation function see Chap ter 14 Custom Allocation page 82 The block will be strlen str 1 bytes that being exactly enough for the string and null terminator If str is not NULL it should point to a block of storage large enough for the result that being mpz_sizeinbase op base 2 The two extra bytes are for a possible minus sign and the null terminator A pointer to the result string is returned being either the allocated block or the given str mp_limb_t mpz_getlimbn mpz_t op mp size t n Function Return limb number n from op The sign of op is ignored just the absolute value is used The least significant limb is number 0 mpz_size can be used to find how many limbs make up op mpz_getlimbn returns zero if n
171. of the gmp randinit functions Sec tion 9 1 Random State Initialization page 61 before invoking this function void mpz random mpz_t rop mp size t max size Function Generate a random integer of at most max size limbs The generated random number doesn t satisfy any particular requirements of randomness Negative random numbers are generated when max size is negative This function is obsolete Use mpz_urandomb or mpz_urandomm instead void mpz random2 mpz_t rop mp size t max size Function Generate a random integer of at most max size limbs with long strings of zeros and ones in the binary representation Useful for testing functions and algorithms since this kind of random numbers have proven to be more likely to trigger corner case bugs Negative random numbers are generated when max size is negative This function is obsolete Use mpz_rrandomb instead 5 14 Integer Import and Export mpz t variables can be converted to and from arbitrary words of binary data with the following functions void mpz import mpz t rop size_t count int order int size int Function endian size_t nails const void op Set rop from an array of word data at op 40 GNU MP 4 1 3 The parameters specify the format of the data count many words are read each size bytes order can be 1 for most significant word first or 1 for least significant first Within each word endian can be 1 for most significant byte first 1 for least significan
172. oid fun T f T g 1 fun2 f T f g Good j 80 GNU MP 4 1 3 13 Berkeley MP Compatible Functions These functions are intended to be fully compatible with the Berkeley MP library which is available on many BSD derived U ix systems The enable mpbsd option must be used when building GNU MP to make these available see Chapter 2 Installing GMP page 4 The original Berkeley MP library has a usage restriction you cannot use the same variable as both source and destination in a single function call The compatible functions in GNU MP do not share this restriction inputs and outputs may overlap It is not recommended that new programs are written using these functions Apart from the incomplete set of functions the interface for initializing MINT objects is more error prone and the pow function collides with pow in libm a Include the header mp h to get the definition of the necessary types and functions If you are on a BSD derived system make sure to include GNU mp h if you are going to link the GNU libmp a to your program This means that you probably need to give the I lt dir gt option to the compiler where lt dir gt is the directory where you have GNU mp h MINT itom signed short int initial_value Function Allocate an integer consisting of a MINT object and dynamic limb space Initialize the integer to initial_ value Return a pointer to the MINT object MINT xtom
173. on titles M Delete any section Entitled Endorsements Such a section may not be included in the Modified Version N Do not retitle any existing section to be Entitled Endorsements or to conflict in title with any Invariant Section O Preserve any Warranty Disclaimers If the Modified Version includes new front matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document you may at your option designate some or all of these sections as invariant To do this add their titles to the list of Invariant Sections in the Modified Version s license notice These titles must be distinct from any other section titles You may add a section Entitled Endorsements provided it contains nothing but endorse ments of your Modified Version by various parties for example statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard You may add a passage of up to five words as a Front Cover Text and a passage of up to 25 words as a Back Cover Text to the end of the list of Cover Texts in the Modified Version Only one passage of Front Cover Text and one of Back Cover Text may be added by or through arrangements made by any one entity If the Document already includes a cover text for the same cover previously added by you or by arrangement made by the same entity you are acting on behalf of you may not add anothe
174. one half of the entire aggregate the Document s Cover Texts may be placed on covers that bracket the Document within the aggregate or the electronic equivalent of covers if the Document is in electronic form Otherwise they must appear on printed covers that bracket the whole aggregate TRANSLATION Translation is considered a kind of modification so you may distribute translations of the Document under the terms of section 4 Replacing Invariant Sections with translations requires special permission from their copyright holders but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections You may include a translation of this License and all the license notices in the Document and any Warranty Disclaimers provided that you also include the original English version of this License and the original versions of those notices and disclaimers In case of a disagreement between the translation and the original version of this License or a notice or disclaimer the original version will prevail If a section in the Document is Entitled Acknowledgements Dedications or History the requirement section 4 to Preserve its Title section 1 will typically require changing the actual title TERMINATION You may not copy modify sublicense or distribute the Document except as expressly provided for under this License Any other attempt to copy modify sublicense o
175. one limb make a temporary copy of x before forming the result Normally that copying will only be a tiny fraction of the time for the multiply so this is not a particularly important consideration mpz set mpq set mpq set num mpf set etc make no attempt to recognise a copy of something to itself so a call like mpz_set x x will be wasteful Naturally that would never be written deliberately but if it might arise from two pointers to the same object then a test to avoid it might be desirable if x y mpz_set x y 22 GNU MP 4 1 3 Note that it s never worth introducing extra mpz_set calls just to get in place op erations If a result should go to a particular variable then just direct it there and let GMP take care of data movement Divisibility Testing Small Integers mpz divisible ui p and mpz_congruent_ui_p are the best functions for testing whether an mpz_t is divisible by an individual small integer They use an algorithm which is faster than mpz tdiv ui but which gives no useful information about the actual remainder only whether it s zero or a particular value However when testing divisibility by several small integers it s best to take a re mainder modulo their product to save multi precision operations For instance to test whether a number is divisible by any of 23 29 or 31 take a remainder modulo 23 x 29 x 31 20677 and then test that The division functions like mpz_tdiv_q_ui which give a quotient
176. ons is that each source area needs at least one limb No size argument may be zero Unless otherwise stated in place operations are allowed where source and destination are the same but not where they only partly overlap The mpn functions are the base for the implementation of the mpz_ mpf_ and mpq_ functions This example adds the number beginning at slp and the number beginning at s2p and writes the sum at destp All areas have n limbs cy mpn_add_n destp sip s2p n In the notation used here a source operand is identified by the pointer to the least significant limb and the limb count in braces For example slp sin mp_limb_t mpn_add_n mp_limb_t rp const mp_limb_t s1p const Function mp_limb_t s2p mp_size_t n Add slp n and s2p n and write the n least significant limbs of the result to rp Return carry either 0 or 1 This is the lowest level function for addition It is the preferred function for addition since it is written in assembly for most CPUs For addition of a variable to itself De slp equals s2p use mpn_1shift with a count of 1 for optimal speed mp limb t mpn add 1 mp limb t rp const mp_limb_t s1p mp size t n Function mp_limb_t s21imb Add slp n and s2limb and write the n least significant limbs of the result to rp Return carry either O or 1 mp limb t mpn add mp_limb_t rp const mp_limb_t s1p mp size_t sin Function const mp_limb_t s2p mp size_t s2n Add slp si
177. operands the time for function call overheads and memory allocation can be significant in comparison to actual calculation This is unavoidable in a general purpose variable precision library although GMP attempts to be as efficient as it can on both large and small operands Chapter 3 GMP Basics 21 Static Linking On some CPUs in particular the x86s the static libgmp a should be used for maximum speed since the PIC code in the shared libgmp so will have a small overhead on each function call and global data address For many programs this will be insignificant but for long calculations there s a gain to be had Initializing and clearing Avoid excessive initializing and clearing of variables since this can be quite time consuming especially in comparison to otherwise fast operations like addition A language interpreter might want to keep a free list or stack of initialized variables ready for use It should be possible to integrate something like that with a garbage collector too Reallocations An mpz_t or mpq_t variable used to hold successively increasing values will have its memory repeatedly realloced which could be quite slow or could fragment memory depending on the C library If an application can estimate the final size then mpz_init2 or mpz_realloc2 can be called to allocate the necessary space from the beginning see Section 5 1 Initializing Integers page 28 It doesn t matter if a size set with mpz_in
178. ovided by gmpxx h follow GMP naming conventions and are unlikely to clash with anything 12 2 C Interface Integers void mpz class mpz class type n Function Construct an mpz class All the standard C types may be used except long long and long double and all the GMP C classes can be used Any necessary conversion follows the corresponding C function for example double follows mpz set d see Section 5 2 Assigning Integers page 29 void mpz class mpz class mpz_t zZ Function Construct an mpz class from an mpz t The value in z is copied into the new mpz class there won t be any permanent association between it and z void mpz class mpz class const char s Function void mpz class mpz class const char s int base Function void mpz class mpz class const string amp s Function void mpz class mpz class const string amp s int base Function Construct an mpz class converted from a string using mpz set str see Section 5 2 As signing Integers page 29 If the base is not given then 0 is used mpz class operator mpz class a mpz class d Function 74 GNU MP 4 1 3 mpz class operator mpz class a mpz_class d Function Divisions involving mpz class round towards zero as per the mpz tdiv q and mpz tdiv r functions see Section 5 6 Integer Division page 32 This is the same as the C99 and operators The mpz fdiv or mpz cdiv functions can always be called directly if de
179. p randclass get z bits mpz class bits Function Generate a random integer with a specified number of bits mpz class emp randclass get z range mpz class n Function Generate a random integer in the range 0 to n 1 inclusive mpf class emp randclass get f Function mpf class gmp_randclass get_f unsigned long prec Function Generate a random float f in the range 0 lt f 1 f will be to prec bits precision or if prec is not given then to the precision of the destination For example gmp randclass r mpf class f 0 512 512 bits precision f r get fO random number 512 bits 12 7 C Interface Limitations mpq_class and Templated Reading A generic piece of template code probably won t know that mpq_class requires a canonicalize call if inputs read with operator gt gt might be non canonical This can lead to incorrect results operator gt gt behaves as it does for reasons of efficiency A canonicalize can be quite time consuming on large operands and is best avoided if it s not necessary But this potential difficulty reduces the usefulness of mpq c1ass Perhaps a mech anism to tell operator what to do will be adopted in the future maybe a pre processor define a global flag or an ios flag pressed into service Or maybe at the risk of inconsistency the mpq class operator could canonicalize and leave mpq_t operator not doing so for use on those occasions when that s acceptable
180. p2 Function int mpf emp d mpf_t opi double op2 Function int mpf cmp ui mpf_t opi unsigned long int op2 Function int mpf cmp si mpf_t op1 signed long int op2 Function Compare opl and op2 Return a positive value if op1 gt op2 zero if opl op2 and a negative value if op1 lt op2 Chapter 7 Floating point Functions 51 int mpf_eq mpf_t op1 mpf_t op2 unsigned long int op3 Function Return non zero if the first op3 bits of op1 and op2 are equal zero otherwise I e test of opl and op2 are approximately equal Caution Currently only whole limbs are compared and only in an exact fashion In the future values like 1000 and 0111 may be considered the same to 3 bits on the basis that their difference is that small void mpf reldiff mpf_t rop mpf_t opi mpf t op2 Function Compute the relative difference between op and op and store the result in rop This is op1 op2 op1 int mpf_sgn mpf_t op Macro Return 1 if op gt 0 0 if op 0 and 1 if op lt 0 This function is actually implemented as a macro It evaluates its arguments multiple times 7 7 Input and Output Functions Functions that perform input from a stdio stream and functions that output to a stdio stream Passing a NULL pointer for a stream argument to any of these functions will make them read from stdin and write to stdout respectively When using any of these functions it is a good idea to include stdio h before
181. page 88 generalizes to split into an arbitrary number of pieces as per Knuth section 4 3 3 algorithm C This is not currently used though it s possible a Toom 4 might fit in between Toom 3 and the FFTs The notes here are merely for interest In general a split into r 1 pieces is made and evaluations and pointwise multiplications done at 2r 1 points A 4 way split does 7 pointwise multiplies 5 way does 9 etc Asymptotically an r 1 way algorithm is O N 9 e9 D Only the pointwise multiplications count towards big O complexity but the time spent in the evaluate and interpolate stages grows with r and has a significant practical impact with the asymptotic advantage of each r realized only at bigger and bigger sizes The overheads grow as O Nr whereas in an r 2 FFT they grow only as O N logr 92 GNU MP 4 1 3 Knuth algorithm C evaluates at points 0 1 2 2r but exercise 4 uses r 0 r and the latter saves some small multiplies in the evaluate stage or rather trades them for additions and has a further saving of nearly half the interpolate steps The idea is to separate odd and even final coefficients and then perform algorithm C steps C7 and C8 on them separately The divisors at step C7 become j and the multipliers at C8 become 2tj j Splitting odd and even parts through positive and negative points can be thought of as using 1 as a square root of unity If a 4th root of unity was available then a furth
182. pter 5 Integer Functions 39 5 13 Random Number Functions The random number functions of GMP come in two groups older function that rely on a global state and newer functions that accept a state parameter that is read and modified Please see the Chapter 9 Random Number Functions page 61 for more information on how to use and not to use random number functions void mpz urandomb mpz_t rop gmp_randstate_t state unsigned long int Function n Generate a uniformly distributed random integer in the range 0 to 2 1 inclusive The variable state must be initialized by calling one of the gmp_randinit functions Sec tion 9 1 Random State Initialization page 61 before invoking this function void mpz urandomm mpz_t rop gmp_randstate_t state mpz t n Function Generate a uniform random integer in the range 0 to n 1 inclusive The variable state must be initialized by calling one of the gmp randinit functions Sec tion 9 1 Random State Initialization page 61 before invoking this function void mpz_rrandomb mpz t rop gmp randstate t state unsigned long int Function n Generate a random integer with long strings of zeros and ones in the binary representation Useful for testing functions and algorithms since this kind of random numbers have proven to be more likely to trigger corner case bugs The random number will be in the range 0 to 2 1 inclusive The variable state must be initialized by calling one
183. pz_init_set functions it can be used as the source or destination operand for the ordinary integer functions Don t use an initialize and set function on a variable already initialized void mpz_init_set mpz t rop mpz t op Function void mpz_init_set_ui mpz t rop unsigned long int op Function void mpz_init_set_si mpz t rop signed long int op Function void mpz_init_set_d mpz t rop double op Function Initialize rop with limb space and set the initial numeric value from op int mpz_init_set_str mpz_t rop char str int base Function Initialize rop and set its value like mpz_set_str see its documentation above for details If the string is a correct base base number the function returns 0 if an error occurs it returns 1 rop is initialized even if an error occurs I e you have to call mpz_clear for it 5 4 Conversion Functions This section describes functions for converting GMP integers to standard C types Functions for converting to GMP integers are described in Section 5 2 Assigning Integers page 29 and Section 5 12 I O of Integers page 38 unsigned long int mpz_get_ui mpz_t op Function Return the value of op as an unsigned long If op is too big to fit an unsigned long then just the least significant bits that do fit are returned The sign of op is ignored only the absolute value is used signed long int mpz get si mpz t op Function If op fits into a signed long int return the value
184. r but you may replace the old one on explicit permission from the previous publisher that added the old one The author s and publisher s of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version COMBINING DOCUMENTS You may combine the Document with other documents released under this License under the terms defined in section 4 above for modified versions provided that you include in the combination all of the Invariant Sections of all of the original documents unmodified and list them all as Invariant Sections of your combined work in its license notice and that you preserve all their Warranty Disclaimers The combined work need only contain one copy of this License and multiple identical Invariant Sections may be replaced with a single copy If there are multiple Invariant Sections with the same name but different contents make the title of each such section unique by adding at the end of it in parentheses the name of the original author or publisher of that section if known or else a unique number Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work In the combination you must combine any sections Entitled History in the various original documents forming one section Entitled History likewise combine any sections Entitled Acknowledgements and any
185. r 16 Algorithms 91 For a mod 2 1 product an FFT k is an O N U algorithm the exponent representing 2 recursed modular multiplies each 1 2 71 the size of the original Each successive k is an asymptotic improvement but overheads mean each is only faster at bigger and bigger sizes In the code MUL FFT TABLE and SQR_FFT_TABLE are the thresholds where each k is used Each new k effectively swaps some multiplying for some shifts adds and overheads A mod 2 1 product can be formed with a normal N x N gt 2N bit multiply plus a subtraction so an FFT and Toom 3 etc can be compared directly A k 4 FFT at O N 3 can be expected to be the first faster than Toom 3 at O N 4 In practice this is what s found with MUL_FFT_ MODF_THRESHOLD and SQR_FFT_MODF_THRESHOLD being between 300 and 1000 limbs depending on the CPU So far it s been found that only very large FFTs recurse into pointwise multiplies above these sizes When an FFT is to give a full product the change of N to 2N doesn t alter the theoretical complexity for a given k but for the purposes of considering where an FFT might be first used it can be assumed that the FFT is recursing into a normal multiply and that on that basis it s doing 2 recursed multiplies each 1 2 the size of the inputs making it O N This would mean k 7 at O N would be the first FFT faster than Toom 3 In practice MUL_ FFT_THRESHOLD and SQR_FFT_THRESHOLD have been found to be in the
186. r a given CPU a search is made though a path to choose a version of each For example sparcv8 has MPN_PATH sparc32 v8 sparc32 generic which means look first for v8 code then plain sparc32 which is v7 and finally fall back on generic C Knowledgeable users with special requirements can specify a different path Normally this is completely unnecessary Chapter 2 Installing GMP 9 Documentation The document you re now reading is gmp texi The usual automake targets are available to make PostScript gmp ps and or DVI gmp dvi HTML can be produced with makeinfo html see section Generating HTML in Texinfo Or alternately texi2html see section About in Texinfo To HTML PDF can be produced with texi2dvi pdf see section PDF Output in Texinfo or with pdftex Some supplementary notes can be found in the doc subdirectory 2 2 ABI and ISA ABI Application Binary Interface refers to the calling conventions between functions meaning what registers are used and what sizes the various C data types are ISA Instruction Set Architecture refers to the instructions and registers a CPU has available Some 64 bit ISA CPUs have both a 64 bit ABI and a 32 bit ABI defined the latter for com patibility with older CPUs in the family GMP supports some CPUS like this in both ABIs In fact within GMP ABI means a combination of chip ABI plus how GMP chooses to use it
187. r distribute the Document is void and will automatically terminate your rights under this License However parties who have received copies or rights from you under this License will not have their licenses terminated so long as such parties remain in full compliance FUTURE REVISIONS OF THIS LICENSE The Free Software Foundation may publish new revised versions of the GNU Free Doc umentation License from time to time Such new versions will be similar in spirit to the present version but may differ in detail to address new problems or concerns See http www gnu org copyleft Each version of the License is given a distinguishing version number If the Document specifies that a particular numbered version of this License or any later version applies to it you have the option of following the terms and conditions either of that specified version or of any later version that has been published not as a draft by the Free Software Foundation If the Document does not specify a version number of this License you may choose any version ever published not as a draft by the Free Software Foundation 120 GNU MP 4 1 3 C 1 ADDENDUM How to use this License for your documents To use this License in a document you have written include a copy of the License in the document and put the following copyright and license notices just after the title page Copyright C year your name Permission is granted to copy distribute and or modify
188. raW 0 00 T npf cl ar iscsvedoc e APA ones ed TERRA Ub 4T Mp CMP eid Mita ee inane ii G 50 A ee Cem Pere es ee wasted 50 npf cmp Sh iii ER 50 A rii ote te PETTE RETE PERNE ITOS 50 mpfi KEE 50 npf div 2eXp ic da ethyl Rena P a dk PE 50 Mp div Ulese serei nae Sea UR md dece AUC os 50 MPP ceci ira dia 51 Appendix C Function and Type Index npf fits Sint D va Rer bx eds 52 mpf fits slong peine SOD 52 npf fits sshort p c rb toes 52 Mpf fits Unt Pics ea de pas rn RUE 52 mpf fits Ulong p e rS 52 npf fits ushort p aee reb 52 npf EEN 51 A EE ne Seed Rae E 49 mpf get d 260XD ei t uret edie eae d 49 mpf get default DEE co oet ree eet EXS 46 Mpf get precio eo 4T MPL E EE 49 NpfZpet SUr 4 92 A E 49 pf got dd dad 49 npf initu ias ped edes 4T npf init S6t bid 48 npf init SSCL sages us sri ae Grounds E 49 Epf init _ SOULS EE 49 Mpf EE E EE 49 npf init Set i ENEE ieee intip mpeg 48 ME EE 47 ppf inp SUE dnde 51 mpf inteBer EE 52 A tiie e naa A EE 50 Mpf m l een ee tea e ae es 50 npf mul EE 50 WE NCP eso sre s eeertu vache eed DRghcbrst ud 50 npf out SUE ie hs EE EE 51 pf POR ui ree kireg eh cepa EE 50 npf random2 ovv RESI HEY REP ds 52 npf reldiff i iuge e Rn rE ER wee ir 51 Mpf Seb iced cide tor eee ernie banded Pees 48 Dpf set Ati a ra 48 mpf set default prec sess esese 46 npf set EE 4T npf set prec Taw EE 4T npf set dq iaa ia 48 let Sil liec hae es eee aed ERR 48 npf set
189. random float of at most max size limbs with long strings of zeros and ones in the binary representation The exponent of the number is in the interval exp to exp in limbs This function is useful for testing functions and algorithms since these kind of random numbers have proven to be more likely to trigger corner case bugs Negative random numbers are generated when max size is negative Chapter 8 Low level Functions 53 8 Low level Functions This chapter describes low level GMP functions used to implement the high level GMP func tions but also intended for time critical user code These functions start with the prefix mpn_ The mpn functions are designed to be as fast as possible not to provide a coherent calling interface The different functions have somewhat similar interfaces but there are variations that make them hard to use These functions do as little as possible apart from the real multiple precision computation so that no time is spent on things that not all callers need A source operand is specified by a pointer to the least significant limb and a limb count A destination operand is specified by just a pointer It is the responsibility of the caller to ensure that the destination has enough space for storing the result With this way of specifying operands it is possible to perform computations on subranges of an argument and store the result into a subrange of a destination A common requirement for all functi
190. rces combines well with loop unrolling since a prefetch can be initiated once per unrolled loop or more than once if the loop covers more than one cache line On CPUs without write allocate caches prefetching destinations will ensure individual stores don t go further down the cache hierarchy limiting bandwidth Of course for calculations which are slow anyway like mpn divrem 1 write throughs might be fine The distance ahead to prefetch will be determined by memory latency versus throughput The aim of course is to have data arriving continuously at peak throughput Some CPUs have limits on the number of fetches or prefetches in progress If a special prefetch instruction doesn t exist then a plain load can be used but in that case care must be taken not to attempt to read past the end of an operand since that might produce a segmentation violation Some CPUs or systems have hardware that detects sequential memory accesses and initiates suitable cache movements automatically making life easy 16 8 5 Floating Point Floating point arithmetic is used in GMP for multiplications on CPUs with poor integer multi pliers It s mostly useful for mpn mul 1 mpn_addmul_1 and mpn submul 1 on 64 bit machines and mpn mul basecase on both 32 bit and 64 bit machines With IEEE 53 bit double precision floats integer multiplications producing up to 53 bits will give exact results Breaking a 64x64 multiplication into eight 16x32 48 bit p
191. rec value set 48 GNU MP 4 1 3 mpf_set_prec_raw is an efficient way to use an mpf_t variable at different precisions during a calculation perhaps to gradually increase precision in an iteration or just to use various different precisions for different purposes during a calculation 7 2 Assignment Functions These functions assign new values to already initialized floats see Section 7 1 Initializing Floats page 46 void mpf_set mpf_t rop mpf_t op Function void mpf_set_ui mpf_t rop unsigned long int op Function void mpf_set_si mpf_t rop signed long int op Function void mpf_set_d mpf_t rop double op Function void mpf_set_z mpf_t rop mpz_t op Function void mpf_set_q mpf_t rop mpq_t op Function Set the value of rop from op int mpf_set_str mpf_t rop char str int base Function Set the value of rop from the string in str The string is of the form MON or if the base is 10 or less alternatively MeN M is the mantissa and N is the exponent The mantissa is always in the specified base The exponent is either in the specified base or if base is negative in decimal The decimal point expected is taken from the current locale on systems providing localeconv The argument base may be in the ranges 2 to 36 or 36 to 2 Negative values are used to specify that the exponent is in decimal Unlike the corresponding mpz function the base will not be determined from the leading ch
192. red whereas GET_ STR_DC_THRESHOLD assumes that s already available which is the case when recursing Since the base case produces digits from least to most significant but they want to be stored from most to least it s necessary to calculate in advance how many digits there will be or at least be sure not to underestimate that For GMP the number of input bits is multiplied by chars_per_bit_exactly from mp_bases rounding up The result is either correct or one too big Examining some of the high bits of the input could increase the chance of getting the exact number of digits but an exact result every time would not be practical since in general the difference between numbers 100 and 99 is only in the last few bits and the work to identify 99 might well be almost as much as a full conversion mpf_get_str doesn t currently use the algorithm described here it multiplies or divides by a power of b to move the radix point to the just above the highest non zero digit or at worst one above that location then multiplies by b to bring out digits This is O N and is certainly not optimal The r b scheme described above for using multiplications to bring out digits might be useful for more than a single limb Some brief experiments with it on the base case when recursing didn t give a noticable improvement but perhaps that was only due to the implementation Something similar would work for the sub quadratic divisions too though
193. riant Sections and required Cover Texts given in the Document s license notice H Include an unaltered copy of this License I Preserve the section Entitled History Preserve its Title and add to it an item stating at least the title year new authors and publisher of the Modified Version as given on the Title Page If there is no section Entitled History in the Document create one stating the title year authors and publisher of the Document as given on its Title Page then add an item describing the Modified Version as stated in the previous sentence J Preserve the network location if any given in the Document for public access to a Transparent copy of the Document and likewise the network locations given in the 118 GNU MP 4 1 3 Document for previous versions it was based on These may be placed in the History section You may omit a network location for a work that was published at least four years before the Document itself or if the original publisher of the version it refers to gives permission K For any section Entitled Acknowledgements or Dedications Preserve the Title of the section and preserve in the section all the substance and tone of each of the contributor acknowledgements and or dedications given therein L Preserve all the Invariant Sections of the Document unaltered in their text and in their titles Section numbers or the equivalent are not considered part of the secti
194. rn value is zero count must be in the range 1 to mp bits per limb 1 The regions sp n and rp n may overlap provided rp lt sp This function is written in assembly for most CPUs int mpn cmp const mp_limb_t s1p const mp_limb_t s2p mp_size_t n Function Compare Jain n and s2p n and return a positive value if s1 gt s2 0 if they are equal or a negative value if s1 s2 mp size t mpn_gcd mp_limb_t rp mp_limb_t s1p mp size t sin Function mp limb t s2p mp size t s2n Set rp retval to the greatest common divisor of slp sin and s2p s2n The result can be up to s2n limbs the return value is the actual number produced Both source operands are destroyed slp sln must have at least as many bits as s2p s2n s2p s2n must be odd Both operands must have non zero most significant limbs No overlap is permitted between s1p sin and s2p s2n mp_limb_t mpn_gcd_1 const mp_limb_t s1p mp size t s n mp_limb_t Function s2limb Return the greatest common divisor of s1p sln and s2limb Both operands must be non Zero mp size t mpn gcdext mp_limb_t r1p mp_limb_t r2p mp_size_t r2n Function mp_limb_t s1p mp size t sin mp_limb_t s2p mp size t s2n Calculate the greatest common divisor of slp s1n and s2p s2n Store the gcd at rlp retval and the first cofactor at r2p r2n with r2n negative if the cofactor is negative rip and r2p should each have room for sin 1
195. ry and one for discussions about development of GMP These lists are not for bug reports For more information see http swox com mailman listinfo The proper place for bug reports is bug gmp gnu org See Chapter 4 Reporting Bugs page 27 for information about reporting bugs 1 1 How to use this Manual Everyone should read Chapter 3 GMP Basics page 16 If you need to install the library yourself then read Chapter 2 Installing GMP page 4 If you have a system with multiple Chapter 1 Introduction to GNU MP 3 ABIs then read Section 2 2 ABI and ISA page 9 for the compiler options that must be used on applications The rest of the manual can be used for later reference although it is probably a good idea to glance through it GNU MP 4 1 3 2 Installing GMP GMP has an autoconf automake libtool based configuration system On a Unix like system a basic build can be done with configure make Some self tests can be run with make check And you can install under usr local by default with make install If you experience problems please report them to bug gmp gnu org See Chapter 4 Reporting Bugs page 27 for information on what to include in useful bug reports 2 1 Build Options All the usual autoconf configure options are available run configure help for a summary The file INSTALL autoconf has some generic installation information too Non Unix Systems confi
196. s int mpq_sgn mpq t op Macro Return 1 if op gt 0 0 if op 0 and 1 if op 0 This function is actually implemented as a macro It evaluates its arguments multiple times int mpq equal mpq t opi mpq t op2 Function Return non zero if op1 and op2 are equal zero if they are non equal Although mpq cmp can be used for the same purpose this function is much faster 6 5 Applying Integer Functions to Rationals The set of mpq functions is quite small In particular there are few functions for either input or output The following functions give direct access to the numerator and denominator of an mpq t Note that if an assignment to the numerator and or denominator could take an mpq t out of the canonical form described at the start of this chapter see Chapter 6 Rational Number Functions page 42 then mpq_canonicalize must be called before any other mpq functions are applied to that mpq t mpz_t mpq numref mpq t op Macro mpz t mpq denref mpq t op Macro Return a reference to the numerator and denominator of op respectively The mpz functions can be used on the result of these macros void mpq get num mpz t numerator mpq rational Function void mpq get den mpz_t denominator mpq t rational Function void mpq set num mpq t rational mpz_t numerator Function void mpq set den mpq t rational mpz t denominator Function Get or set the numerator or denominator of a rational These f
197. s exact i e if op is rop to the nth power void mpz_sqrt mpz t rop mpz t op Function Set rop to op the truncated integer part of the square root of op void mpz sqrtrem mpz_t ropi mpz_t rop2 mpz t op Function Set rop1 to op like mpz sqrt Set rop2 to the remainder op rop1 which will be zero if op is a perfect square If rop1 and rop2 are the same variable the results are undefined int mpz perfect power p mpz t op Function Return non zero if op is a perfect power i e if there exist integers a and b with b gt 1 such that op a Under this definition both 0 and 1 are considered to be perfect powers Negative values of op are accepted but of course can only be odd perfect powers int mpz perfect square p mpz t op Function Return non zero if op is a perfect square i e if the square root of op is an integer Under this definition both 0 and 1 are considered to be perfect squares 5 9 Number Theoretic Functions int mpz probab prime p mpz t n int reps Function Determine whether n is prime Return 2 if n is definitely prime return 1 if n is probably prime without being certain or return 0 if n is definitely composite This function does some trial divisions then some Miller Rabin probabilistic primality tests reps controls how many such tests are done 5 to 10 is a reasonable number more will reduce the chances of a composite being returned as probably prime Chapter
198. s many limbs as the divisor In addition to an integer quotient qxn fraction limbs are developed and stored after the integral limbs For most usages qxn will be zero It is required that rs2n is greater than or equal to s3n It is required that the most significant bit of the divisor is set If the quotient is not needed pass rs2p s3n as rlp Aside from that special case no overlap between arguments is permitted Return the most significant limb of the quotient either 0 or 1 The area at rip needs to be rs2n s3n qxn limbs large mp limb t mpn_divrem_1 mp_limb_t rip mp size t qxn Function mp_limb_t s2p mp size t s2n mp_limb_t s3limb mp_limb_t mpn_divmod_1 mp_limb_t r1p mp_limb_t s2p mp _size_t s2n Macro mp_limb_t s3limb Divide s2p s2n by s3limb and write the quotient at rlp Return the remainder The integer quotient is written to rlp qxn s2n and in addition qxn fraction limbs are developed and written to rlp qxn Either or both s2n and qxn can be zero For most usages qxn will be zero mpn_divmod_1 exists for upward source compatibility and is simply a macro calling mpn_ divrem 1 with a qxn of 0 56 GNU MP 4 1 3 The areas at r1p and s2p have to be identical or completely separate not partially overlap ping mp_limb_t mpn_divmod mp_limb_t r1p mp_limb_t rs2p mp size t rs2n Function const mp_limb_t s3p mp _size_t s3n This function is obsolete Please call mpn_tdiv_qr ins
199. s required by the preprocessor should be set in CPPFLAGS rather than CFLAGS Compiling is done with both CPPFLAGS and CFLAGS but preprocessing uses just CPPFLAGS This distinction is because most preprocessors won t accept all the flags the compiler does Preprocessing is done separately in some configure tests and in the ansi2knr support for K amp R compilers C Support enable cxx C support in GMP can be enabled with enable cxx in which case a C compiler will be required As a convenience enable cxx detect can be used to enable C support only if a compiler can be found The C support consists of a library libgmpxx 1la and header file gmpxx h A separate libgmpxx 1la has been adopted rather than having C objects within libgmp la in order to ensure dynamic linked C programs aren t bloated by a dependency on the C standard library and to avoid any chance that the C compiler could be required when linking plain C programs libgmpxx 1la will use certain internals from libgmp 1la and can only be expected to work with libgmp 1la from the same GMP version Future changes to the rele vant internals will be accompanied by renaming so a mismatch will cause unresolved symbols rather than perhaps mysterious misbehaviour In general libgmpxx 1a will be usable only with the C compiler that built it since name mangling and runtime suppor
200. se will not be determined from the leading characters of the string if base is 0 This is so that numbers like 0 23 are not interpreted as octal Return the number of bytes read or if an error occurred return 0 7 8 Miscellaneous Functions void mpf_ceil mpf_t rop mpf_t op Function 52 GNU MP 4 1 3 void mpf_floor mpf_t rop mpf_t op Function void mpf_trunc mpf_t rop mpf_t op Function Set rop to op rounded to an integer mpf_ceil rounds to the next higher integer mpf_floor to the next lower and mpf_trunc to the integer towards zero int mpf_integer_p mpf_t op Function Return non zero if op is an integer int mpf_fits_ulong_p mpf_t op Function int mpf_fits_slong_p mpf_t op Function int mpf_fits_uint_p mpf_t op Function int mpf_fits_sint_p mpf_t op Function int mpf_fits_ushort_p mpf_t op Function int mpf_fits_sshort_p mpf_t op Function Return non zero if op would fit in the respective C data type when truncated to an integer void mpf_urandomb mpf_t rop gmp_randstate_t state unsigned long int Function nbits Generate a uniformly distributed random float in rop such that 0 lt rop lt 1 with nbits significant bits in the mantissa The variable state must be initialized by calling one of the gmp_randinit functions Sec tion 9 1 Random State Initialization page 61 before invoking this function void mpf random2 mpf_t rop mp size t max size mp_exp_t exp Function Generate a
201. sed on them make syntax error On certain versions of SCO OpenServer 5 and IRIX 6 5 the native make is unable to handle the long dependencies list for 1ibgmp la The symptom is a syntax error on the following line of the top level Makefile libgmp la libgmp_la_OBJECTS libgmp 1a DEPENDENCIES Either use GNU Make or as a workaround remove 1ibgmp 1a DEPENDENCIES from that line which will make the initial build work but if any recompiling is done libgmp la might not be rebuilt MacOS X and GCC Libtool currently only knows how to create shared libraries on MacOS X using the native cc which is a modified GCC not a plain GCC A static only build should work though disable shared Also libtool currently cannot build C shared libraries on MacOS X so if enable cxx is desired then disable shared must be used Hopefully this will be fixed in the future Motorola 68k ABI The GMP assembler code has been written for the SVR4 standard ABI GCC option mshort changes the calling conventions and is not currently supported We believe the PalmOS calling conventions are similarly different and are likewise not currently supported Chapter 2 Installing GMP 15 NeXT prior to 3 3 The system compiler on old versions of NeX T was a massacred and old GCC even if it called itself cc This compiler cannot be used to build GMP you need to get a real GCC and install that NeXT may
202. sequences The method for choosing a seed is critical if the generated numbers are to be used for important applications such as generating cryptographic keys Traditionally the system time has been used to seed but care needs to be taken with this If an application seeds often and the resolution of the system clock is low then the same sequence of numbers might be repeated Also the system time is quite easy to guess so if unpredictability is required then it should definitely not be the only source for the seed value On some systems there s a special device dev random which provides random data better suited for use as a seed Chapter 10 Formatted Output 63 10 Formatted Output 10 1 Format Strings gmp_printf and friends accept format strings similar to the standard C printf see section Formatted Output in The GNU C Library Reference Manual A format specification is of the form flags width precision type conv GMP adds types Z Q and F for mpz t mpq t and mpf_t respectively and N for an mp_ limb_t array Z Q and N behave like integers Q will print a and a denominator if needed F behaves like a float For example mpz_t z gmp_printf As is an mpz Zd n here z mpq t q gmp_printf a hex rational 40Qx n q mpf t f int n gmp_printf fixed point mpf Ff with d digits Wm n f n const mp limb t ptr mp
203. sired For example mpz class q a d mpz fdiv q q get mpz tO a get mpz tO d get mpz tO mpz class abs mpz class op1 Function int cmp mpz class op1 type op2 Function int cmp type opi mpz class op2 Function double mpz Claes get d void Function long mpz class get si void Function unsigned long mpz class get ui void Function bool mpz class fits sint p void Function bool mpz class fits slong p void Function bool mpz class fits sshort p void Function bool mpz class fits uint p void Function bool mpz class fits ulong p void Function bool mpz class fits ushort p void Function int sgn mpz class op Function mpz class sqrt mpz_class op Function These functions provide a C class interface to the corresponding GMP C routines cmp can be used with any of the classes or the standard C types except long long and long double Overloaded operators for combinations of mpz class and double are provided for completeness but it should be noted that if the given double is not an integer then the way any rounding is done is currently unspecified The rounding might take place at the start in the middle or at the end of the operation and it might change in the future Conversions between mpz class and double however are defined to follow the corresponding C functions mpz get d and mpz set d And comparisons are always made exactly as per mpz cmp d 12 3 C Interface Rationals In all
204. ss 4 Functions compatible with Berkeley MP such as itom madd and mult The associated type is MINT 5 Fast low level functions that operate on natural numbers These are used by the functions in the preceding groups and you can also call them directly from very time critical user programs These functions names begin with mpn_ The associated type is array of mp_ limb_t There are about 30 hard to use functions in this class 6 Miscellaneous functions Functions for setting up custom allocation and functions for gen erating random numbers 3 4 Variable Conventions GMP functions generally have output arguments before input arguments This notation is by analogy with the assignment operator The BSD MP compatibility functions are exceptions having the output arguments last GMP lets you use the same variable for both input and output in one call For example the main function for integer multiplication mpz mul can be used to square x and put the result back in x with mpz mul x x x Before you can assign to a GMP variable you need to initialize it by calling one of the special initialization functions When you re done with a variable you need to clear it out using one of the functions for that purpose Which function to use depends on the type of variable See the chapters on integer functions rational number functions and floating point functions for details A variable should only be initialized once or at le
205. sted precision This happens even for an f conversion of an mpf_t which is an integer for instance 210 in an mpf t of 128 bits precision will only produce about 40 digits then pad with zeros to the decimal point An empty precision field like Fe or 4 Ff can be used to specifically request just the significant digits The decimal point character or string is taken from the current locale settings on systems which provide localeconv see section Locales and Internationalization in The GNU C Library Reference Manual The C library will normally do the same for standard float output The format string is only interpreted as plain chars multibyte characters are not recognised Perhaps this will change in the future Chapter 10 Formatted Output 65 10 2 Functions Each of the following functions is similar to the corresponding C library function The basic printf forms take a variable argument list The vprintf forms take an argument pointer see section Variadic Functions in The GNU C Library Reference Manual or man 3 va_start It should be emphasised that if a format string is invalid or the arguments don t match what the format specifies then the behaviour of any of these functions will be unpredictable GCC format string checking is not available since it doesn t recognise the GMP extensions The file based functions gmp_printf and gmp_fprintf will return 1 to indicate a write error All t
206. stom Allocation By default GMP uses malloc realloc and free for memory allocation and if they fail GMP prints a message to the standard error output and terminates the program Alternate functions can be specified to allocate memory in a different way or to have a different error action on running out of memory This feature is available in the Berkeley compatibility library see Chapter 13 BSD Compatible Functions page 80 as well as the main GMP library void mp set memory functions Function void alloc func ptr size_t void realloc func ptr void size_t size_t void free func ptr void size t Replace the current allocation functions from the arguments If an argument is NULL the corresponding default function is used These functions will be used for all memory allocation done by GMP apart from temporary space from alloca if that function is available and GMP is configured to use it see Section 2 1 Build Options page 4 Be sure to call mp set memory functions only when there are no active GMP objects allocated using the previous memory functions Usually that means calling it before any other GMP function The functions supplied should fit the following declarations void allocate function size_t alloc size Function Return a pointer to newly allocated space with at least alloc size bytes void reallocate function void ptr size t old size size t Function new size Resize
207. sub optimal in the presence of simple factor izations or little prospect for cancellation but GMP has no way to know when this will occur As per Section 3 11 Efficiency page 20 that s left to applications The mpq_t framework might still suit with mpq numref and mpq_denref for direct access to the numerator and denominator or of course mpz t variables can be used directly Chapter 17 Internals 107 17 3 Float Internals Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this mpf_t floats have a variable precision mantissa and a single machine word signed exponent The mantissa is represented using sign and magnitude most significant limb least significant limb mp exp mp d lt radix point mp size gt The fields are as follows mp size mp prec _mp_d _Mp_exp The number of limbs currently in use or the negative of that when representing a negative value Zero is represented by _mp_size and _mp_exp both set to zero and in that case the _mp_d data is unused In the future _mp_exp might be undefined when representing zero The precision of the mantissa in limbs In any calculation the aim is to produce _mp_prec limbs of result the most significant being non zero A pointer to the array of limbs which is the absolute value of the mantissa These are stored little endian as per th
208. t are usually incompatible between different compilers CXX CXXFLAGS When C support is enabled the C compiler and its flags can be set with vari ables CXX and CXXFLAGS in the usual way The default for CXX is the first compiler that works from a list of likely candidates with g normally preferred when avail able The default for CXXFLAGS is to try CFLAGS CFLAGS without g then for g either g 02 or 02 or for other compilers g or nothing Trying CFLAGS this way is convenient when using gcc and g together since the flags for gcc will usually suit g It s important that the C and C compilers match meaning their startup and runtime support routines are compatible and that they generate code in the same ABI if there s a choice of ABIs on the system configure isn t currently able to check these things very well itself so for that reason disable cxx is the default to avoid a build failure due to a compiler mismatch Perhaps this will change in the future Incidentally it s normally not good enough to set CXX to the same as CC Although gcc for instance recognises foo cc as C code only g will invoke the linker the right way when building an executable or shared library from object files Temporary Memory enable alloca lt choice gt D y y GMP allocates temporary workspac
209. t first or 0 for the native endianness of the host CPU The most significant nails bits of each word are skipped this can be 0 to use the full words There is no sign taken from the data rop will simply be a positive integer An application can handle any sign itself and apply it for instance with mpz neg There are no data alignment restrictions on op any address is allowed Here s an example converting an array of unsigned long data most significant element first and host byte order within each value unsigned long a 20 mpz_t Z mpz import z 20 1 sizeof a 0 O 0 a This example assumes the full sizeof bytes are used for data in the given type which is usually true and certainly true for unsigned long everywhere we know of However on Cray vector systems it may be noted that short and int are always stored in 8 bytes and with sizeof indicating that but use only 32 or 46 bits The nails feature can account for this by passing for instance 8 sizeof int INT_BIT void mpz export void rop size_t countp int order int size int Function endian size t nails mpz t op Fill rop with word data from op The parameters specify the format of the data produced Each word will be size bytes and order can be 1 for most significant word first or 1 for least significant first Within each word endian can be 1 for most significant byte first 1 for least significant first or 0 for the native endianness of the host C
210. t limbs this is an O QM algorithm and will run at a speed similar to a basecase QxM multiplication differing in fact only in the extra multiply and divide for each of the Q quotient limbs 16 2 3 Divide and Conquer Division For divisors larger than DIV_DC_THRESHOLD division is done by dividing Or to be precise by a recursive divide and conquer algorithm based on work by Moenck and Borodin Jebelean and Burnikel and Ziegler see Appendix B References page 113 The algorithm consists essentially of recognising that a 2NxN division can be done with the basecase division algorithm see Section 16 2 2 Basecase Division page 93 but using N 2 limbs as a base not just a single limb This way the multiplications that arise are N 2 x N 2 and can take advantage of Karatsuba and higher multiplication algorithms see Section 16 1 Multiplication Algorithms page 86 The digits of the quotient are formed by recursive Nx N 2 divisions If the N 2 x N 2 multiplies are done with a basecase multiplication then the work is about the same as a basecase division but with more function call overheads and with some subtractions separated from the multiplies These overheads mean that it s only when N 2 is above MUL_ KARATSUBA_THRESHOLD that divide and conquer is of use DIV DC THRESHOLD is based on the divisor size N so it will be somewhere above twice MUL_ KARATSUBA THRESHOLD but how much above depends on the CPU An optimized mpn mul
211. tead for best performance mp_limb_t mpn_divexact_by3 mp_limb_t rp mp_limb_t sp mp_size_t n Macro mp_limb_t mpn_divexact_by3c mp_limb_t rp mp_limb_t sp Function mp size t n mp_limb_t carry Divide sp n by 3 expecting it to divide exactly and writing the result to rp n If 3 divides exactly the return value is zero and the result is the quotient If not the return value is non zero and the result won t be anything useful mpn_divexact_by3c takes an initial carry parameter which can be the return value from a previous call so a large calculation can be done piece by piece from low to high mpn_ divexact_by3 is simply a macro calling mpn_divexact_by3c with a 0 carry parameter These routines use a multiply by inverse and will be faster than mpn divrem 1 on CPUs with fast multiplication but slow division The source a result q size n initial carry i and return value c satisfy cb a i 3q where p Qmp bits_per_limb The return c is always 0 1 or 2 and the initial carry i must also be 0 1 or 2 these are both borrows really When c 0 clearly q a i 3 When c Z 0 the remainder a i mod 3 is given by 3 c because b 1 mod 3 when mp bits per limb is even which is always so currently mp_limb_t mpn mod 1 mp_limb_t s1p mp size t sin mp_limb_t s21imb Function Divide slp s1n by s2limb and return the remainder s1n can be zero mp_limb_t mpn_bdivmod mp limb t rp mp limb t s1p
212. tead of simply using the remainder r from such divisions an extra divide step is done to give a fractional limb representing Chapter 16 Algorithms 99 r b The digits of r can then be extracted using multiplications by b rather than divisions Special case code is provided for decimal allowing multiplications by 10 to optimize to shifts and adds Above GET_STR_PRECOMPUTE_THRESHOLD a sub quadratic algorithm is used For an input f powers b of the radix are calculated until a power between t and y is reached t is then divided by that largest power giving a quotient which is the digits above that power and a remainder which is those below These two parts are in turn divided by the second highest power and so on recursively When a piece has been divided down to less than GET_STR_DC_THRESHOLD limbs the basecase algorithm described above is used The advantage of this algorithm is that big divisions can make use of the sub quadratic divide and conquer division see Section 16 2 3 Divide and Conquer Division page 93 and big divisions tend to have less overheads than lots of separate single limb divisions anyway But in any case the cost of calculating the powers br must first be overcome GET_STR_PRECOMPUTE_THRESHOLD and GET_STR_DC_THRESHOLD represent the same basic thing the point where it becomes worth doing a big division to cut the input in half GET_STR_ PRECOMPUTE_THRESHOLD includes the cost of calculating the radix power requi
213. tees Db REX adele 35 npz lucnum l 2i due a pase els 36 npz lucnuml2 Ui wee Geeks Re oa ee aoe eS 36 npz MO eroe ode y bs Eden Pr P ee Bed anges 33 MPZ MOd Mis it det 33 MZ MMU ic EE al mpz m l 26Xp vc wc e ey ee UR p E 31 mpz m l Si ira rs DERECE rES 31 npz m l UG padres pa eee pad E Peg ees 31 MPZ neg ehr A lee Ee E exp UE rS 31 mpz nextprime 2 eee seek aan EE 35 mpz odd pcssoiieefebRREREXe id 41 npZ o t raW eness era a Pap ie Race e dee dna 38 npz out StE c ek a pde 38 mpz perfect power EEN 34 mpz perfect square p e l9 34 MPZ_POPCOUNt saiek dn Lees deeds de op deve dads 37 npz pOWJul eizgeesiuoaniu ea AUT Re 34 MPZ 25i pe ITE PDDII HM E E 34 npz pow iria ege a ee eed re ey Ries 34 npz probab prime pi ee ee eee om 34 npz randolc si e sme ao ere 39 mpz randonm2 6 eLe6e0rt Ree tx Der ere 39 npz re ll 62 2 nk ebore np be 28 npz remoV6 ieeerkhsm bd er hera ee ewe aces 36 npZ TEE dU ed e apnd aa 34 npz rr ndolb i2 2 903 aake ep ER 39 npz ScanO iereiankta e ERG Eu ds eben 3T mpz SCanl i222 ee y roo e ERRARE e er 3T MPZ Set T 29 Appendix C Function and Type Index hpz Set d na ev er EE 29 npz Set f posre larredi e skeen kee 29 mpz Set Jeetne iia id Had oes eee 29 MPZ Set SI erea ii e a E E 29 MPZ Set SUE so dale g Bab mode peu ego er ed la 29 ipz Set ui cu om ere re Rer a EE UE PERPE 29 npz setbit o l ol4a e essc cb PEE Y P eg 3T DpZ SON ese dig e be b a bes Seas Sr npz D
214. the following constructors if a fraction is given then it should be in canonical form or if not then mpq class canonicalize called void mpq class mpq class type op Function void mpq class mpq class integer num integer den Function Construct an mpq class The initial value can be a single value of any type or a pair of integers mpz class or standard C integer types representing a fraction except that long long and long double are not supported For example mpq class q 99 mpq class q 1 75 mpq class q 1 3 Chapter 12 C Class Interface 75 void mpq_class mpq_class mpq t q Function Construct an mpq class from an mpq_t The value in q is copied into the new mpq class there won t be any permanent association between it and q void mpq class mpq class const char s Function void mpq class mpq class const char e int base Function void mpq class mpq class const string amp s Function void mpq class mpq class const string amp s int base Function Construct an mpq class converted from a string using mpq set str see Section 6 1 Ini tializing Rationals page 42 If the base is not given then 0 is used void mpq_class canonicalize Function Put an mpq_class into canonical form as per Chapter 6 Rational Number Functions page 42 All arithmetic operators require their operands in canonical form and will return results in canonical form mpq class abs mpq c
215. ting settings istream amp operator istream amp stream mpq t rop Function Read rop from stream using its ios formatting settings An integer like 123 will be read or a fraction like 5 9 If the fraction is not in canonical form then mpq canonicalize must be called see Chapter 6 Rational Number Functions page 42 istream amp operator istream amp stream mpf t rop Function Read rop from stream using its ios formatting settings Hex or octal floats are not supported but might be in the future These operators mean that GMP types can be read in the usual C way for example mpz t zZ hose But note that istream input and ostream output see Section 10 3 C Formatted Output page 66 is the only overloading available for the GMP types and that for instance using with Chapter 11 Formatted Input 71 an mpz_t will have unpredictable results For classes with overloading see Chapter 12 C Class Interface page 72 72 GNU MP 4 1 3 12 C Class Interface This chapter describes the C class based interface to GMP All GMP C language types and functions can be used in C programs since gmp h has extern C qualifiers but the class interface offers overloaded functions and operators which may be more convenient Due to the implementation of this interface a reasonably recent C compiler is required one supporting namespaces partial specialization of templates and member templat
216. tion so their size doesn t change All memory is allocated using malloc and friends by default but this can be changed see Chapter 14 Custom Allocation page 82 Temporary memory on the stack is also used via alloca but this can be changed at build time if desired see Section 2 1 Build Options page 4 3 7 Reentrancy GMP is reentrant and thread safe with some exceptions e If configured with enable alloca malloc notreentrant or with enable alloca notreentrant when alloca is not available then naturally GMP is not reentrant e mpf_set_default_prec and mpf_init use a global variable for the selected precision mpf_ init2 can be used instead and in the C interface an explicit precision to the mpf_class constructor e mpz_random and the other old random number functions use a global random state and are hence not reentrant The newer random number functions that accept a gmp_randstate_t parameter can be used instead e gmp_randinit obsolete returns an error indication through a global variable which is not thread safe Applications are advised to use gmp_randinit_1c_2exp instead e mp_set_memory_functions uses global variables to store the selected memory allocation functions e If the memory allocation functions set by a call to mp set memory functions or malloc and friends by default are not reentrant then GMP will not be reentrant either e If the standard I O functions such as fwrite are not reen
217. tion becomes more important but of course speedups in low level routines will still speed up everything proportionally Carry handling and widening multiplies that are important for GMP can t be easily expressed in C GCC asm blocks help a lot and are provided in longlong h but hand coding low level routines invariably offers a speedup over generic C by a factor of anything from 2 to 10 16 8 1 Code Organisation The various mpn subdirectories contain machine dependent code written in C or assembler The mpn generic subdirectory contains default code used when there s no machine specific version of a particular file Each mpn subdirectory is for an ISA family Generally 32 bit and 64 bit variants in a family cannot share code and will have separate directories Within a family further subdirectories may exist for CPU variants 16 8 2 Assembler Basics mpn addmul 1 and mpn_submul_1 are the most important routines for overall GMP performance All multiplications and divisions come down to repeated calls to these mpn_add_n mpn_sub_n mpn_lshift and mpn_rshift are next most important On some CPUs assembler versions of the internal functions mpn_mul_basecase and mpn_sqr_ basecase give significant speedups mainly through avoiding function call overheads They can also potentially make better use of a wide superscalar processor The restrictions on overlaps between sources and destinations see Chapter 8 Low level
218. tional and useful document free in the sense of freedom to assure everyone the effective freedom to copy and redistribute it with or without modifying it either commercially or noncommercially Secondarily this License preserves for the author and publisher a way to get credit for their work while not being considered responsible for modifications made by others This License is a kind of copyleft which means that derivative works of the document must themselves be free in the same sense It complements the GNU General Public License which is a copyleft license designed for free software We have designed this License in order to use it for manuals for free software because free software needs free documentation a free program should come with manuals providing the same freedoms that the software does But this License is not limited to software manuals it can be used for any textual work regardless of subject matter or whether it is published as a printed book We recommend this License principally for works whose purpose is instruction or reference 1 APPLICABILITY AND DEFINITIONS This License applies to any manual or other work in any medium that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License Such a notice grants a world wide royalty free license unlimited in duration to use that work under the conditions stated herein The Document below refers to any s
219. tr mpz t rop FILE stream int base Function Input a possibly white space preceded string in base base from stdio stream stream and put the read integer in rop The base may vary from 2 to 36 If base is 0 the actual base is determined from the leading characters if the first two characters are Ox or 0X hexadecimal is assumed otherwise if the first character is 0 octal is assumed otherwise decimal is assumed Return the number of bytes read or if an error occurred return 0 size t mpz out raw FILE stream mpz t op Function Output op on stdio stream stream in raw binary format The integer is written in a portable format with 4 bytes of size information and that many bytes of limbs Both the size and the limbs are written in decreasing significance order i e in big endian The output can be read with mpz inp raw Return the number of bytes written or if an error occurred return 0 The output of this can not be read by mpz inp raw from GMP 1 because of changes necessary for compatibility between 32 bit and 64 bit machines size_t mpz inp raw mpz t rop FILE stream Function Input from stdio stream stream in the format written by mpz out raw and put the result in rop Return the number of bytes read or if an error occurred return 0 This routine can read the output from mpz out raw also from GMP 1 in spite of changes necessary for compatibility between 32 bit and 64 bit machines Cha
220. trant then the GMP I O functions using them will not be reentrant either e It s safe for two threads to read from the same GMP variable simultaneously but it s not safe for one to read while the another might be writing nor for two threads to write simultaneously It s not safe for two threads to generate a random number from the same gmp_randstate_t simultaneously since this involves an update of that variable 3 8 Useful Macros and Constants const int mp bits per limb Global Constant The number of bits per limb GNU MP VERSION Macro GNU MP VERSION MINOR Macro 20 GNU MP 4 1 3 GNU MP VERSION PATCHLEVEL Macro The major and minor GMP version and patch level respectively as integers For GMP i j these numbers will be i j and 0 respectively For GMP i j k these numbers will be i j and k respectively const char const gmp_version Global Constant 4 19 The GMP version number as a null terminated string in the form ij or i j k This release is 4 1 3 3 9 Compatibility with older versions This version of GMP is upwardly binary compatible with all 4 x and 3 x versions and upwardly compatible at the source level with all 2 x versions with the following exceptions e mpn_gcd had its source arguments swapped as of GMP 3 0 for consistency with other mpn functions e mpf_get_prec counted precision slightly differently in GMP 3 0 and 3 0 1 but in 3 1 re verted to the 2 x sty
221. truncated to the first size 1 characters and a null appended No overlap is permitted between the region buf size and the fmt string Notice the return value is in ISO C99 snprintf style This is so even if the C library vsnprintf is the older GLIBC 2 0 x style int gmp asprintf char pp const char mt Function int gmp vasprintf char pp const char fmt va_list ap Function Form a null terminated string in a block of memory obtained from the current memory allocation function see Chapter 14 Custom Allocation page 82 The block will be the size of the string and null terminator Put the address of the block in pp Return the number of characters produced excluding the null terminator Unlike the C library asprintf gmp_asprintf doesn t return 1 if there s no more memory available it lets the current allocation function handle that 66 GNU MP 4 1 3 int gmp_obstack_printf struct obstack ob const char fmt Function int gmp_obstack_vprintf struct obstack ob const char fmt va_list ap Function Append to the current obstack object in the same style as obstack_printf Return the number of characters written A null terminator is not written fmt cannot be within the current obstack object since the object might move as it grows These functions are available only when the C library provides the obstack feature which probably means only on GNU systems see section Obstacks in The GNU
222. uch manual or work Any member of the public is a licensee and is addressed as you You accept the license if you copy modify or distribute the work in a way requiring permission under copyright law A Modified Version of the Document means any work containing the Document or a portion of it either copied verbatim or with modifications and or translated into another language A Secondary Section is a named appendix or a front matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document s overall subject or to related matters and contains nothing that could fall directly within that overall subject Thus if the Document is in part a textbook of mathematics a Secondary Section may not explain any mathematics The relationship could be a matter of historical connection with the subject or with related matters or of legal commercial philosophical ethical or political position regarding them The Invariant Sections are certain Secondary Sections whose titles are designated as being those of Invariant Sections in the notice that says that the Document is released under this License If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant The Document may contain zero Invariant Sections If the Document does not identify any Invariant Sections then there are none The Cover Texts
223. ultiplier x multiplicand void mpq mul 2exp mpq t rop mpq t op1 unsigned long int op2 Set rop to op1 x 2 void mpq div mpq t quotient mpq t dividend mpq t divisor Set quotient to dividend divisor void mpq_div_2exp mpq t rop mpq t op1 unsigned long int op2 Set rop to op1 2 void mpq_neg mpq t negated_operand mpq t operand Set negated_operand to operand void mpq_abs mpq t rop mpq t op Set rop to the absolute value of op Function Function Function Function Function Function Function Function 44 GNU MP 4 1 3 void mpq_inv mpq t inverted number mpq_t number Function Set inverted number to 1 number If the new denominator is zero this routine will divide by zero 6 4 Comparison Functions int mpq cmp mpq t opi mpq t op2 Function Compare opl and op2 Return a positive value if op1 gt op2 zero if opl op2 and a negative value if op1 lt op2 To determine if two rationals are equal mpq equal is faster than mpq cmp int mpq cmp ui mpq t opi unsigned long int num2 unsigned long int den2 Macro int mpq cmp si mpq_t op1 long int num2 unsigned long int den2 Macro Compare op and num2 den2 Return a positive value if op1 gt num2 den2 zero if op1 num2 den2 and a negative value if op1 lt num2 den2 num2 and den2 are allowed to have common factors These functions are implemented as a macros and evaluate their arguments multiple time
224. unctions are equivalent to calling mpz set with an appropriate mpq numref or mpq denref Direct use of mpq_numref or mpq_denref is recommended instead of these functions Chapter 6 Rational Number Functions 45 6 6 Input and Output Functions When using any of these functions it s a good idea to include stdio h before gmp h since that will allow gmp h to define prototypes for these functions Passing a NULL pointer for a stream argument to any of these functions will make them read from stdin and write to stdout respectively size t mpq out str FILE stream int base mpq_t op Function Output op on stdio stream stream as a string of digits in base base The base may vary from 2 to 36 Output is in the form num den or if the denominator is 1 then just num Return the number of bytes written or if an error occurred return 0 size_t mpq inp str mpq t rop FILE stream int base Function Read a string of digits from stream and convert them to a rational in rop Any initial white space characters are read and discarded Return the number of characters read including white space or 0 if a rational could not be read The input can be a fraction like 17 63 or just an integer like 123 Reading stops at the first character not in this form and white space is not permitted within the string If the input might not be in canonical form then mpq canonicalize must be called see Chapter
225. used to locate the most significant 1 bit in op counting from 1 Unlike the bitwise functions which start from 0 See Section 5 11 Logical and Bit Manipulation Functions page 37 42 GNU MP 4 1 3 6 Rational Number Functions This chapter describes the GMP functions for performing arithmetic on rational numbers These functions start with the prefix mpq_ Rational numbers are stored in objects of type mpq t All rational arithmetic functions assume operands have a canonical form and canonicalize their result The canonical from means that the denominator and the numerator have no common factors and that the denominator is positive Zero has the unique representation 0 1 Pure assignment functions do not canonicalize the assigned variable It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable void mpq_canonicalize mpq t op Function Remove any factors that are common to the numerator and denominator of op and make the denominator positive 6 1 Initialization and Assignment Functions void mpq init mpq t dest rational Function Initialize dest rational and set it to 0 1 Each variable should normally only be initialized once or at least cleared out using the function mpq clear between each initialization void mpq clear mpq t rational number Function Free the space occupied by rational number Make sure to call this function for
226. ut to increase mp size it checks mp alloc to see whether there s enough space and reallocates if not MPZ_REALLOC is generally used for this The various bitwise logical functions like mpz_and behave as if negative values were twos com plement But sign and magnitude is always used internally and necessary adjustments are made during the calculations Sometimes this isn t pretty but sign and magnitude are best for other routines Some internal temporary variables are setup with MPZ TMP INIT and these have mp d space obtained from TMP ALLOC rather than the memory allocation functions Care is taken to ensure that these are big enough that no reallocation is necessary since it would have unpredictable consequences 17 2 Rational Internals mpq t variables represent rationals using an mpz_t numerator and denominator see Section 17 1 Integer Internals page 106 The canonical form adopted is denominator positive and non zero no common factors between numerator and denominator and zero uniquely represented as 0 1 It s believed that casting out common factors at each stage of a calculation is best in general A GCD is an O N operation so it s better to do a few small ones immediately than to delay and have to do a big one later Knowing the numerator and denominator have no common factors can be used for example in mpq_mul to make only two cross GCDs necessary not four This general approach to common factors is badly
227. values rop and rop2 efficiently 6 2 Conversion Functions double mpq get d mpq t op Convert op to a double void mpq set d mpq t rop double op void mpq set f mpq t rop mpf t op Set rop to the value of op without rounding char mpq get str char str int base mpq t op 43 Function Function Function Function Function Convert op to a string of digits in base base The base may vary from 2 to 36 The string will be of the form num den or if the denominator is 1 then just num If str is NULL the result string is allocated using the current allocation function see Chap ter 14 Custom Allocation page 82 The block will be strlen str 1 bytes that being exactly enough for the string and null terminator If str is not NULL it should point to a block of storage large enough for the result that being mpz sizeinbase mpq numref op base mpz sizeinbase mpq denref op base 3 The three extra bytes are for a possible minus sign possible slash and the null terminator A pointer to the result string is returned being either the allocated block or the given str 6 3 Arithmetic Functions void mpq add mpq t sum mpq t addend1 mpq t addend2 Set sum to addend1 addend2 void mpq sub mpq t difference mpq t minuend mpq t subtrahend Set difference to minuend subtrahend void mpq mul mpq t product mpq t multiplier mpq t multiplicand Set product to m
228. version Squaring follows the same procedure as multiplication but there s only one X t and it s eval uated at 5 points and those values squared to give values of W t The interpolation is then identical and in fact the same interpolate3 subroutine is used for both squaring and multi plying Toom 3 is asymptotically O N1465 the exponent being log 5 log 3 representing 5 recursive multiplies of 1 3 the original size This is an improvement over Karatsuba at O N 8 though Toom Cook does more work in the evaluation and interpolation and so it only realizes its ad vantage above a certain size 90 GNU MP 4 1 3 Near the crossover between Toom 3 and Karatsuba there s generally a range of sizes where the difference between the two is small MUL_TOOM3_THRESHOLD is a somewhat arbitrary point in that range and successive runs of the tune program can give different values due to small variations in measuring A graph of time versus size for the two shows the effect see tune README At the fairly small sizes where the Toom 3 thresholds occur it s worth remembering that the asymptotic behaviour for Karatsuba and Toom 3 can t be expected to make accurate predictions due of course to the big influence of all sorts of overheads and the fact that only a few recursions of each are being performed Even at large sizes there s a good chance machine dependent effects like cache architecture will mean actual performance deviates from what
229. version info is set appropriately having started from 3 0 0 in GMP 3 0 The GMP 4 series will be upwardly binary compatible in each release and will be upwardly binary compatible with all of the GMP 8 series Additional function interfaces may be added in each release so on systems where libtool versioning is not fully checked by the loader an auxiliary mechanism may be needed to express that a dynamic linked application depends on a new enough GMP An auxiliary mechanism may also be needed to express that libgmpxx la from enable cxx see Section 2 1 Build Options page 4 requires 1ibgmp la from the same GMP version since this is not done by the libtool versioning nor otherwise A mismatch will result in unresolved symbols from the linker or perhaps the loader Using DESTDIR or a prefix override with make install and a shared libgmpxx may run into a libtool relinking problem see Section 2 5 Known Build Problems page 13 When building a package for a CPU family care should be taken to use host or build to choose the least common denominator among the CPUs which might use the package For example this might necessitate i386 for x86s or plain sparc meaning V7 for SPARCs Users who care about speed will want GMP built for their exact CPU type to make use of the available optimizations Providing a way to suitably rebuild a package may be useful This could be as simple
230. werpc64 powerpc401 powerpc403 powerpc405 powerpc505 powerpc601 powerpc602 powerpc603 powerpc603e powerpc604 powerpc604e powerpc620 powerpc630 powerpc740 powerpc7400 powerpc7450 powerpc750 powerpc801 powerpc821 powerpc823 powerpc860 e SPARC sparc sparcv8 microsparc supersparc sparcv9 ultrasparc ultrasparc2 ultrasparc2i ultrasparc3 sparc64 e 80x86 family 1386 1486 1586 pentium pentiummmx pentiumpro pentium2 pentium3 pentium4 k6 k62 k63 athlon e Other a29k arm clipper i960 ns32k pyramid sh sh2 vax 28k CPUs not listed will use generic C code Generic C Build ABT If some of the assembly code causes problems or if otherwise desired the generic C code can be selected with CPU none For example configure host none unknown freebsd3 5 Note that this will run quite slowly but it should be portable and should at least make it possible to get something running if all else fails On some systems GMP supports multiple ABIs application binary interfaces meaning data type sizes and calling conventions By default GMP chooses the best ABI available but a particular ABI
231. y choices provided by MS C is desired then the suggestion is to use the GMP string functions and confine I O to the application Motorola 68k CPU Types m68k is taken to mean 68000 m68020 or higher will give a performance boost on applicable CPUs m68360 can be used for CPU32 series chips m68302 can be used for Dragonball series chips though this is merely a synonym for m68000 OpenBSD 2 6 m4 in this release of OpenBSD has a bug in eval that makes it unsuitable for asm file processing configure will detect the problem and either abort or choose another m4 in the PATH The bug is fixed in OpenBSD 2 7 so either upgrade or use GNU m4 Power CPU Types In GMP CPU types power and powerpc will each use instructions not available on the other so it s important to choose the right one for the CPU that will be used Currently GMP has no assembler code support for using just the common instruction subset To get executables that run on both the current suggestion is H Chapter 2 Installing GMP 13 to use the generic C code CPU none possibly with appropriate compiler options like mcpu common for gcc CPU rs6000 which is not a CPU but a family of workstations is accepted by config sub but is currently equivalent to none Sparc CPU Types sparcv8 or supersparc on relevant systems will give a significant performance increase over the V7 co

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