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LibTomMath User Manual v0.33

1. SI Single Dicite e Taori sk Reo E Gg 3 1 2 Long 5 3 1 8 Initialize and Setting Constants 3 2 Comparisons 3 2 1 Unsigned 3 2 2 Signed 3 4 3 single Digi ma a ea AR 3 3 Logical 3 3 1 Multiplication by two ss RUSSES IS d 3 3 3 AND OR and Operations Cae taal Se GS as ees Sade ak See eee ee ages eee 9 51 Negation ee ea m ee 3 5 2 Absolute 2 442454 44443445 RE ERES 3 6 Integer Division and Remainder 4 Multiplication and Squaring 4 1 Multiplication 4 2 Squaring 4 3 Tuning Polynomial Basis Routines 9 1 Straight Division ees T T CCP Tv p 6 Exponentiation 6 1 Single Digit 6 2 Modular 6 3 Root Finding 19 19 19 20 21 22 23 24 25 26 26 28 29 29 29 29 29 30 31 31 33 33 35 35 36 37 40 41 Prime Numbers 7 1 rnalbDivisiion IR 2 Fermat uos wo eR ORO SU URS REOR 8 7 3 Miler Rabm 7 51 Required Number of 7 4
2. amp b eb 9 mp div a amp b amp a dc a a b a b c a mod b This allows operands to be re used which can make programming simpler 2 5 Initialization 2 5 1 Single Initialization A single mp int can be initialized with the mp_init function int mp_init mp_int a This function expects a pointer to an mp int structure and will initialize the members of the structure so the mp int represents the default integer which is zero If the functions returns MP OKAY then the mp int is ready to be used by the other LibTomMath functions int main void mp int number int result if result mp_init amp number MP OKAY printf Error initializing the number fs mp_error_to_string result return EXIT_FAILURE use the number return EXIT_SUCCESS 2 5 2 Single Free When you are finished with an mp int it is ideal to return the heap it used back to the system The following function provides this functionality 12 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH void mp_clear mp_int a The function expects a pointer to a previously initialized mp int structure and frees the heap it uses It sets the pointer within the mp_int to NULL which is used to prevent double free situations Is is legal to call mp clear twice on the same mp int in a row int main void mp int number int result if result mp init number MP OKAY printf Error ini
3. display printf number1 number2 Alu mp get int amp numberi free terms and return mp_clear_multi amp number1 amp number2 NULL return EXIT_SUCCESS If this program succeeds it shall output the following numberi number2 262911 4 2 SQUARING 33 4 2 Squaring Since squaring can be performed faster than multiplication it is performed it s own function instead of just using mp mul int mp sqr mp int a mp int b Will square a and store it in b Like the case of multiplication there are four different squaring algorithms all which can be called from mp sqr It is ideal to use mp sqr over mp mul when squaring terms 4 3 Tuning Polynomial Basis Routines Both of the Toom Cook and Karatsuba multiplication algorithms are faster than the traditional O n approach that the Comba and baseline algorithms use At 1 464973 and 1 584962 running times respectfully they require considerably less work For example a 10000 digit multiplication would take roughly 724 000 single precision multiplications with Toom Cook or 100 000 000 single precision multiplications with the standard Comba a factor of 138 So why not always use Karatsuba or Toom Cook simple answer is that they have so much overhead that they re not actually faster than Comba until you hit distinct cutoff points For Karatsuba with the default configuration GCC 3 3 1 and an Athlon XP processor the cutof
4. printf Error getting norm mp_error_to_string result return EXIT_FAILURE get mp value if result mp_montgomery_setup amp c amp mp MP_OKAY 5 3 MONTGOMERY REDUCTION 39 if if if if if printf Error setting up montgomery 478 mp_error_to_string result return EXIT_FAILURE normalize so now a is equal to aR result mp_mulmod amp a amp R amp b amp a MP_OKAY printf Error computing aR 5 mp error to string result return EXIT FAILURE square a to get c a 2R 2 result mp_sqr amp a amp c MP OKAY printf Error squaring 5 mp_error_to_string result return EXIT_FAILURE now reduce back down to a 2R 2 R 1 a 2R result mp montgomery reduce amp c amp b mp MP OKAY printf Error reducing 5 mp error to string result return EXIT FAILURE multiply a to get c a 3R 2 result mp mul a amp c amp c MP OKAY printf Error reducing 4s mp error to string result return EXIT FAILURE now reduce back down to c a 3R 2 R 1 a 3R result mp montgomery reduce amp c amp b mp MP OKAY printf Error reducing 5 mp error to string result return EXIT FAILURE 40 CHAPTER 5 MODULAR REDUCTION now reduce again back down to a 3R R 1 a 3 if result mp_montgom
5. ifdefs which further define symbols All of the symbols technically they re macros represent a given C source file For instance BN MP ADD C represents the file bn mp add c When a define has been enabled the function in the respective file will be compiled and linked into the library Accordingly when the define is absent the file will not be compiled and not contribute any size to the library You will also note that the header tommath_class h is actually recursively included it includes itself twice This is to help resolve as many dependencies as possible In the last pass the symbol LTM LAST will be defined This is useful for trims 1 4 2 Build Tweaks A tweak is an algorithm alternative For example to provide tradeoffs usu ally between size and space They can be enabled at any pass of the configu ration phase Define Purpose BN MP DIV SMALL Enables a slower smaller and equally functional mp div function 1 43 Build Trims A trim is a manner of removing functionality from a function that is not required For instance to perform RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed Build trims are meant to be defined on the last pass of the configuration which means they are to be defined only if LTM LAST has been defined 1 5 PURPOSE OF LIBTOMMATH 5 Moduli Related Restriction Undefine Exponentiation with o
6. CHAPTER 3 BASIC OPERATIONS mp_error_to_string result return EXIT_FAILURE set the number to 5 mp_set number 5 switch mp_cmp_d amp number 7 case MP_GT printf number case MP_EQ printf number case MP_LT printf number A lv NNN 2 2 H done with it mp_clear number return EXIT_SUCCESS If this program functions properly it will print out the following number lt 7 3 3 Logical Operations Logical operations are operations that can be performed either with simple shifts or boolean operators such as AND XOR and OR directly These operations are very quick 3 3 1 Multiplication by two Multiplications and divisions by any power of two can be performed with quick logical shifts either left or right depending on the operation When multiplying or dividing by two a special case routine can be used which are as follows int mp_mul_2 mp_int a mp_int b int mp_div_2 mp_int a mp_int b The former will assign twice a to b while the latter will assign half a to b These functions are fast since the shift counts and maskes are hardcoded into the routines 3 3 LOGICAL OPERATIONS 27 int main void mp_int number int result if result mp_init amp number MP_OKAY printf Error initializing the number js mp_error_to_string result return EXIT_FAILURE set the number to 5 mp set amp number 5
7. Primality Testing M5 Next Prime 4 d GG b d wow A EAR 7 6 Random 1 6 1 Extended 8 Input and Output 8 1 ASCII Conversions 8 1 1 To ASCH 8 1 2 From ASCII 8 2 Binary Conversions 9 Algebraic Functions 9 1 Extended Euclidean Algorithm 9 2 Greatest Common O 9 4 Jacobi Symbol 9 5 Modular Inverse 3 Least Common Multiple 9 6 Single Digit Functions eee eee en List of Figures TT 7 2 1 Return 2222 2 224 4 9 3 1 Comparison Codesfora b o o eee 22 aa 34 48 vii Chapter 1 Introduction 1 1 What is LibTomMath LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating large integer numbers It was written in portable ISO C source code so that it will build on any platform with a conforming compiler In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how to implement bignum math However the resulting code has proven to be very useful It has been used by numerous universities commercial and open source software developers It has been used on a variety of platforms ranging from Linux and Windows based x86 to ARM based Gameboys and PPC bas
8. a int main void mp int number int result if result mp init number MP OKAY printf Error initializing the number fs mp error to string result return EXIT_FAILURE 3 1 SMALL CONSTANTS 21 set the number to 654321 note this is bigger than 127 if result mp_set_int amp number 654321 MP OKAY printf Error setting the value of the number s mp_error_to_string result return EXIT_FAILURE printf number lu mp_get_int amp number we re done with it mp_clear amp number return EXIT_SUCCESS This should output the following if the program succeeds number 654321 3 1 3 Initialize and Setting Constants To both initialize and set small constants the following two functions are avail able int mp_init_set mp_int a mp_digit b int mp_init_set_int mp_int a unsigned long b Both functions work like the previous counterparts except they first mp init before setting the values int main void mp int numberi number2 int result initialize and set a single digit if result mp init set numberi 100 MP OKAY printf Error setting numberi s mp error to string result return EXIT FAILURE 22 CHAPTER 3 BASIC OPERATIONS initialize and set a long if result mp_init_set_int amp number2 1023 MP OKAY printf Error setting number2 s mp error to string result return EX
9. b mu int result initialize a b to desired values mp_init mu c and set to 1 we want to compute a 3 mod b get mu value if result mp_reduce_setup amp mu b MP_OKAY 4 printf Error getting mu 48 mp_error_to_string result return EXIT_FAILURE square a to get a 2 if result mp_sqr amp a amp c MP OKAY printf Error squaring 4s mp error to string result return EXIT FAILURE 5 3 MONTGOMERY REDUCTION 37 now reduce modulo b if result mp_reduce amp c amp b amp mu MP OKAY printf Error reducing 5 mp error to string result return EXIT FAILURE multiply a to get c a 3 if result mp mul a amp c amp c MP OKAY printf Error reducing 45 mp error to string result return EXIT FAILURE now reduce modulo b if result mp_reduce amp c amp b amp mu MP OKAY printf Error reducing 4s mp error to string result return EXIT FAILURE c now equals a 3 mod b return EXIT_SUCCESS This program will calculate a mod b if all the functions succeed 5 3 Montgomery Reduction Montgomery is a specialized reduction algorithm for any odd moduli Like Barrett reduction a pre computation step is required This is accomplished with the following int mp montgomery setup mp int mp digit For the given odd moduli a the precomputat
10. build as a shared library for issue the following make f makefile shared This requires the libtool package common on most Linux BSD systems It will build LibTomMath as both shared and static then install by default into usr lib as well as install the header files in usr include The shared library resource will be called libtommath la while the static library called libtommath a Generally you use libtool to link your application against the shared object There is limited support for making a DLL in windows via the make file cygwin dll makefile It requires Cygwin to work with since it requires the auto export import functionality The resulting DLL and import library libtommath dll a can be used to link LibTomMath dynamically to any Win dows program using Cygwin 1 4 BUILD CONFIGURATION 3 1 3 3 Testing To build the library and the test harness type make test This will build the library test and mtest mtest The test program will accept test vectors and verify the results mtest mtest will generate test vectors using the MPI library by Michael Frombergel Simply pipe mtest into test using mtest mtest test If you do not have a dev urandom style RNG source you will have to write your own PRNG and simply pipe that into mtest For example if your PRNG program is called myprng simply invoke myprng mtest mtest test yprng This will
11. multiply by two if result mp mul 2 amp number amp number MP OKAY printf Error multiplying the number 75 mp_error_to_string result return EXIT_FAILURE switch mp_cmp_d amp number 7 case MP GT printf 2 number case MP EQ printf 2 number case MP LT printf 2 number lt Moo MZ ND NT c H o now divide by two if result mp_div_2 amp number amp number MP OKAY printf Error dividing the number 78 mp_error_to_string result return EXIT_FAILURE switch mp_cmp_d amp number 7 case MP_GT printf 2 number 2 gt 7 case MP EQ printf 2 number 2 7 break case MP LT printf 2 number 2 7 A we re done with it mp_clear number return EXIT_SUCCESS 28 CHAPTER 3 BASIC OPERATIONS If this program is successful it will print out the following text 2 number gt 7 2 number 2 lt 7 Since 10 gt 5 lt 7 To multiply by a power of two the following function can be used int mp_mul_2d mp_int a int b mp_int This will multiply a by 2 and store the result in c If the value of b is less than or equal to zero the function will copy a to c without performing any further actions To divide by a power of two use the following int mp div 2d mp int int b mp int c mp int d Which will divide a by 2 store the quotient in and the remainder in d If b lt 0 th
12. 9 1 Extended Euclidean Algorithm int mp_exteuclid mp_int a mp_int b mp_int U1 mp int U2 mp int U3 This finds the triple U1 U2 U3 using the Extended Euclidean algorithm such that the following equation holds 01 5 02 03 9 1 Any of the U1 U2 U3 paramters can be set to NULL if they are not desired 9 2 Greatest Common Divisor int mp gcd mp int a mp int b mp int c This will compute the greatest common divisor of a and b and store it in c 9 3 Least Common Multiple int mp lcm mp int a mp int b mp int c This will compute the least common multiple of a and b and store it in c 51 52 CHAPTER 9 ALGEBRAIC FUNCTIONS 9 4 Jacobi Symbol int mp jacobi mp int a mp int p int This will compute the Jacobi symbol for a with respect to p If p is prime this essentially computes the Legendre symbol The result is stored in c and can take on one of three values 1 0 1 If p is prime then the result will be 1 when a is not a quadratic residue modulo p The result will be 0 if a divides p and the result will be 1 if a is a quadratic residue modulo p 9 5 Modular Inverse int mp_invmod mp_int a mp_int b mp_int Computes the multiplicative inverse of a modulo b and stores the result in such that 1 mod 6 9 6 Single Digit Functions For those using small numbers snicker snicker there are several helper func tions int mp_add_d mp_int a mp_digit b m
13. Data Types The basic multiple precision integer type is known as the mp int within LibTomMath This data type is used to organize all of the data required to ma nipulate the integer it represents Within LibTomMath it has been prototyped as the following typedef struct int used alloc sign mp_digit dp mp_int Where mp digit is a data type that represents individual digits of the integer By default an mp digit is the ISO C unsigned long data type and each digit is 28 bits long The mp digit type can be configured to suit other platforms by defining the appropriate macros All LTM functions that use the mp int type will expect a pointer to mp int structure You must allocate memory to hold the structure itself by yourself whether off stack or heap it doesn t matter The very first thing that must be done to use an int is that it must be initialized 2 4 Function Organization The arithmetic functions of the library are all organized to have the same style prototype That is source operands are passed on the left and the destination is on the right For instance mp add amp a amp b dc c atb mp mul amp a amp c mp div amp a amp b amp d a b d a mod b 2 5 INITIALIZATION 11 Another feature of the way the functions have been implemented is that source operands can be destination operands as well For instance
14. IT FAILURE display printf Number1 Number2 lu Alu mp get int numberi mp get int amp number2 clear mp_clear_multi amp number1 amp number2 NULL return EXIT SUCCESS If this program succeeds it shall output Numberi Number2 100 1023 3 2 Comparisons Comparisons in LibTomMath are always performed in a left to right fashion There are three possible return codes for any comparison Result Code Meaning MP GT ab MP EQ a b MP LT a lt b Figure 3 1 Comparison Codes for a b In figure 3 1 two integers and b are being compared In this case a is said to be to the left of b 3 2 COMPARISONS 3 2 1 Unsigned comparison An unsigned comparison considers only the digits themselves and not the as sociated sign flag of the mp int structures This is analogous to an absolute comparison The function mp_cmp_mag will compare two mp int variables based on their digits only int mp_cmp mp_int a mp_int b This will compare a to b placing a to the left of b This function cannot fail and will return one of the three compare codes listed in figure int main void mp int numberi number2 int result if result mp init multi amp numberi amp number2 NULL MP OKAY printf Error initializing the numbers s mp error to string result return EXIT FAILURE set the numberi to 5 mp set amp
15. LibTomMath User Manual v0 33 Tom St Denis tomstdenisQiahu ca December 22 2004 This text the library and the accompanying textbook are all hereby placed in the public domain This book has been formatted for B5 176x250 paper using the IXTEX book macro package Open Source Open Academia Open Minds Tom 5 Denis Ontario Canada Contents 1 Introduction 1 1 1 451 41 1 a a a A AR eat 1 1 3 Building LiblomMath ees 2 1 3 1 Static Libraries 2l 2 13 2 Shared Libraries 2 1 33 TESTO uoo eed moneo AR dede RU Be 3 3 1 4 1 Build Depends oc sema mnn 4 14 2 Build Tweaks 4 1 4 3 Build Trini 4 1 5 PurposeofLiblomMath lens 5 2 Getting Started with LibTomMath 9 E MIB UM ATEM 9 2 2 Return 9 2 3 Data TYPES o e uu eee eS Se RARA gts 10 beh A de 10 A A eae 11 Bolas wy ae BO 11 boh Rows MEME EVE 11 ey ty Boe HR 12 2 5 4 Other 13 2 6 Maintenance 15 2 6 1 Reducing Memory Usage 15 T 16 111 3 Basic Operations 3 1 Small
16. ange 0 lt lt 02 Dimminished radix reductions are much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time 5 5 UNRESTRICTED DIMMINSHED RADIX 41 Since the moduli are restricted this algorithm is not particularly useful for something like Rabin RSA or BBS cryptographic purposes This reduction al gorithm is useful for Diffie Hellman and ECC where fixed primes are acceptable Note that unlike Montgomery reduction there is no normalization process The result of this function is equal to the correct residue 5 5 Unrestricted Dimminshed Radix Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the form 2 p for 0 lt p 6 In this sense the unrestricted reductions are more flexible as they can be applied to a wider range of numbers int mp_reduce_2k_setup mp_int mp digit This will compute the required d value for the given moduli a int mp reduce 2k mp int a mp int n mp digit d This will reduce a in place modulo n with the pre computed value d From my experience this routine is slower than mp dr reduce but faster for most moduli sizes than the Montgomery reduction 42 CHAPTER 5 MODULAR REDUCTION Chapter 6 Exponentiation 6 1 Single Digit Exponentiation int mp_expt_d mp_int a mp_digit b mp_int b This computes a using a simple binary left to right algorithm It is faster than
17. bably prime result is set to one Otherwise result is set to zero Note that is suggested that you use the Miller Rabin test instead of the Fermat test since all of the failures of Miller Rabin are a subset of the failures of the Fermat test 7 31 Required Number of Tests Generally to ensure a number is very likely to be prime you have to perform the Miller Rabin with at least half dozen or so unique bases However it has been proven that the probability of failure goes down as the size of the input goes up This is why a simple function has been provided to help out int mp_prime_rabin_miller_trials int size This returns the number of trials required for a 2796 or lower probability of failure for a given size expressed in bits This comes in handy specially since larger numbers are slower to test For example a 512 bit number would require ten tests whereas a 1024 bit number would only require four tests You should always still perform a trial division before a Miller Rabin test though 7 4 Primality Testing int mp_prime_is_prime mp_int a int t int result This will perform a trial division followed by t rounds of Miller Rabin tests on a and store the result in result If a passes all of the tests result is set to one otherwise it is set to zero Note that t is bounded by 1 lt t lt PRIME SIZE where PRIME SIZE is the number of primes in the prime number table by default this is 256 7 5 Next Prime int m
18. d mp int a mp int b mp int c int mp xor mp int a mp int b mp int c Which compute c a b where is one of OR AND or XOR 3 4 Addition and Subtraction compute an addition or subtraction the following two functions can be used int mp add mp int a mp int b mp int c int mp sub mp int a mp int b mp int c Which perform c a b where is one of signed addition or subtraction The operations are fully sign aware 3 5 Sign Manipulation 3 5 1 Negation Simple integer negation can be performed with the following int mp_neg mp_int a mp_int b Which assigns a to b 3 5 2 Absolute Simple integer absolutes can be performed with the following int mp abs mp int a mp int b Which assigns a to b 30 CHAPTER 3 BASIC OPERATIONS 3 6 Integer Division and Remainder To perform a complete and general integer division with remainder use the following function int mp div mp int a mp int b mp int c mp int d This divides a by b and stores the quotient in c and d The signed quotient is computed such that bc d a Note that either of c or d can be set to NULL if their value is not required If b is zero the function returns MP VAL Chapter 4 Multiplication and Squaring 4 1 Multiplication A full signed integer multiplication can be performed with the following int mp_mul mp_int a mp_int b mp_int Which assigns the full sig
19. dd moduli only N S MP EXPTMOD C MP REDUCE C MP REDUCE SETUP C S MP MUL HIGH DIGS C FAST S MP MUL HIGH DIGS C ZZZZ 4 Exponentiation with random odd moduli he above plus the following MP REDUCE 2K C MP REDUCE 2K SETUP C MP REDUCE IS 2K C MP_DR_IS_MODULUS_C MP DR REDUCE C MP DR SETUP C Modular inverse odd moduli only MP INVMOD SLOW C W w w w w yo w w w w Z Z Z Z ZZZ 2 Modular inverse both smaller slower FAST MP INVMOD C Operand Size Related Restriction Undefine Moduli 2560 bits N MP MONTGOMERY REDUCE C N S MP MUL DIGS C N S MP MUL HIGH DIGS C N S MP SQR C N MP KARATSUBA MUL C N MP KARATSUBA SQR C N_MP_TOOM_MUL_C N MP TOOM SQR C Polynomial Schmolynomial Y w 1 5 Purpose of LibTomMath Unlike GNU MP GMP Library LIP OpenSSL or various other commercial kits Miracl LibTomMath was not written with bleeding edge performance in mind First and foremost LibTomMath was written to be entirely open Not only is the source code public domain unlike various other GPL etc licensed code not only is the code freely downloadable but the source code is also acces sible for computer science students attempting to learn BigNum or multiple precision arithmetic techniques 6 CHAPTER 1 INTRODUCTION LibTomMath was written to be an instructive collection of source code This is why there are many comments o
20. ed MacOS machines 1 2 License As of the v0 25 the library source code has been placed in the public domain with every new release As of the v0 28 release the textbook Implementing Multiple Precision Arithmetic has been placed in the public domain with every new release as well This textbook is meant to compliment the project by providing a more solid walkthrough of the development algorithms used in the library Since 1 are in the public domain everyone is entitled to do with them Note that the MPI files under mtest are copyrighted by Michael Fromberger They are not required to use LibTomMath 2 CHAPTER 1 INTRODUCTION as they see fit 1 3 Building LibTomMath LibTomMath is meant to be very GCC friendly as it comes with a makefile well suited for GCC However the library will also build in MSVC Borland out of the box For any other ISO C compiler a makefile will have to be made by the end developer 1 3 1 Static Libraries build as a static library for issue the following make command This will build the library and archive the object files in libtom math a Now you link against that and include tommath h within your programs Alternatively to build with MSVC issue the following nmake f makefile msvc This will build the library and archive the object files in tommath lib This has been tested with MSVC version 6 00 with service pack 5 1 3 2 Shared Libraries
21. en the function simply copies a over to and zeroes d The variable d may be passed as a NULL value to signal that the remainder is not desired 3 3 2 Polynomial Basis Operations Strictly speaking the organization of the integers within the int structures is what is known as a polynomial basis This simply means a field element is stored by divisions of a radix For example if f x ES yix for any vector y then the array of digits in y are said to be the polynomial basis representation of z if f 8 z for a given radix f To multiply by the polynomial g x x all you have todo is shift the digits of the basis left one place The following function provides this operation int mp lshd mp int a int b This will multiply a in place by z which is equivalent to shifting the digits left b places and inserting zeroes in the least significant digits Similarly to divide by a power of x the following function is provided void mp rshd mp int a int b This will divide a in place by z and discard the remainder This function cannot fail as it performs the operations in place and no new digits are required to complete it 3 4 ADDITION AND SUBTRACTION 29 3 33 AND OR Operations While AND OR and XOR operations are not typical bignum functions they can be useful in several instances The three functions are prototyped as follows int mp or mp int a mp int b mp int int mp an
22. ere are reasons not to While LibTomMath is slower than libraries such as GnuMP it is not normally significantly slower On x86 machines the difference is normally a factor of two when performing modular exponentiations Essentially the only time you wouldn t use LibTomMath is when blazing speed is the primary concern CHAPTER 1 INTRODUCTION Chapter 2 Getting Started with LibTomMath 2 1 Building Programs In order to use LibTomMath you must include tommath h and link against the appropriate library file typically libtommath a There is no library initial ization required and the entire library is thread safe 2 2 Return Codes There are three possible return codes a function may return Code Meaning MP_OKAY The function succeeded MP_VAL The function input was invalid MP_MEM Heap memory exhausted MP_YES Response is yes MP_NO Response is no Figure 2 1 Return Codes The last two codes listed are not actually return ed by a function They 9 10 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH are placed in an integer the caller must provide the address of an integer it can store to which the caller can access To convert one of the three return codes to a string use the following function char mp_error_to_string int code This will return a pointer to a string which describes the given error code It will not work for the return codes MP_YES and MP_NO 2 3
23. ery_reduce amp c amp b mp MP_OKAY printf Error reducing s mp_error_to_string result return EXIT_FAILURE c now equals a 3 mod b return EXIT SUCCESS This particular example does not look too efficient but it demonstrates the point of the algorithm By normalizing the inputs the reduced results are always of the form aR for some variable a This allows a single final reduction to correct for the normalization and the fast reduction used within the algorithm For more details consider examining the file bn mp exptmod fast c 5 4 Restricted Dimminished Radix Dimminished Radix reduction refers to reduction with respect to moduli that are ameniable to simple digit shifting and small multiplications In this case the restricted variant refers to moduli of the form 3 p for some k gt 0 and 0 lt p lt B where 8 is the radix default to 2 5 As in the case of Montgomery reduction there is a pre computation phase required for a given modulus void mp dr setup mp int mp digit This computes the value required for the modulus a and stores it in d This function cannot fail and does not return any error codes After the computation a reduction can be performed with the following int mp dr reduce mp int mp int mp digit mp This reduces a in place modulo b with the pre computed value mp b must be of a restricted dimminished radix form and a must be in the r
24. f point is roughly 110 digits about 70 for the Intel P4 That is at 110 digits Karatsuba and Comba multiplications just about break even and for 1104 digits Karatsuba is faster Toom Cook has incredible overhead and is probably only useful for very large inputs So far no known cutoff points exist and for the most part I just set the cutoff points very high to make sure they re not called A demo program in the etc directory of the project called tune c can be used to find the cutoff points This can be built with GCC as follows make XXX Where is one of the following entries from the table 4 1 When the program is running it will output a series of measurements for different cutoff points It will first find good Karatsuba squaring and multi plication points Then it proceeds to find Toom Cook points Note that the Toom Cook tuning takes a very long time as the cutoff points are likely to be very high 34 CHAPTER 4 MULTIPLICATION AND SQUARING Value of XXX Meaning tune Builds portable tuning application tune86 Builds x86 pentium and up program for COFF tune86c Builds x86 program for Cygwin tune86l Builds x86 program for Linux ELF format Figure 4 1 Build Names for Tuning Programs 5 Modular Reduction Modular reduction is process of taking the remainder of one quantity divided by another Expressed as 5 1 the modular reduction is equivalent to the
25. igits are significant that is contribute to the value of the mp int The alloc parameter dictates how many digits are currently available in the array If you need to perform an operation that requires more digits you will have to mp grow the mp int to your desired size int mp grow mp int a int size This will grow the array of digits of a to size If the alloc parameter is already bigger than size the function will not do anything int main void mp int number int result if result mp init number MP OKAY printf Error initializing the number 4s mp error to string result return EXIT_FAILURE use the number We need to add 20 digits to the number if result mp grow amp number number alloc 20 MP OKAY printf Error growing the number 5 mp error to string result return EXIT FAILURE use the number 2 6 MAINTENANCE FUNCTIONS we re done with it mp_clear amp number return EXIT_SUCCESS 17 18 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH Chapter 3 Basic Operations 3 1 Small Constants Setting mp ints to small constants is a relatively common operation To acco modate these instances there are two small constant assignment functions The first function is used to set a single digit constant while the second sets an ISO C style unsigned long constant The reason for both functions is efficiency Setting a single digit is
26. ion value is placed in mp The reduction is computed with the following int mp montgomery reduce mp int mp int mp digit mp 38 CHAPTER 5 MODULAR REDUCTION This reduces a in place modulo m with the pre computed value mp a must be in the range 0 lt a lt 02 Montgomery reduction is faster than Barrett reduction for moduli smaller than the comba limit With the default setup for instance the limit is 127 digits 3556 bits Note that this function is not limited to 127 digits just that it falls back to a baseline algorithm after that point An important observation is that this reduction does not return a mod m but aR mod m where R 8 n is n number of digits in m and is radix used default is 228 quickly calculate R the following function was provided int mp_montgomery_calc_normalization mp_int a mp_int b Which calculates a R for the odd moduli b without using multiplication or division The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system For example to calculate a mod b using Montgomery reduction the value of a can be normalized by multiplying it by R Consider the following code snippet int main void mp_int a b c R mp_digit mp int result initialize a b to desired values mp init c and set c to 1 get normalization if result mp_montgomery_calc_normalization amp R b MP_OKAY
27. mp_set amp number1 5 set the number2 to 6 mp_set amp number2 6 if result mp_neg amp number2 amp number2 MP_OKAY printf Error negating number2 s 1 function uses the mp neg function which is discussed in section 3 2 COMPARISONS 25 mp_error_to_string result return EXIT_FAILURE switch mp_cmp amp number1 amp number2 case MP GT printf number1 gt number2 break case MP EQ printf numberi number2 break case MP LT printf numberi lt number2 break done with it mp clear multi numberi amp number2 NULL return EXIT SUCCESS If this completes successfully it should print the following numberi gt number2 3 2 3 Single Digit To compare a single digit against an mp int the following function has been provided int mp_cmp_d mp_int a mp_digit b This will compare a to the left of b using a signed comparison Note that it will always treat b as positive This function is rather handy when you have to compare against small values such as 1 which often comes up in cryptography The function cannot fail and will return one of the tree compare condition codes listed in figure 3 1 int main void mp_int number int result if result mp_init amp number MP_OKAY printf Error initializing the number 5 2This function uses the mp_neg function which is discussed in section 26
28. nction has been provided int mp init copy mp int a mp int b This function will initialize a and make it a copy of b if all goes well int main void mp_int numi num2 int result initialize and do work on numi We want copy of numi in num2 now if result mp init copy amp num2 amp num1 MP OKAY printf Error initializing the copy 5 mp error to string result return EXIT FAILURE 14 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH now num2 is ready and contains copy of numi done with them mp clear multi amp numi amp num2 NULL return EXIT SUCCESS Another less common initializer is mp_init_size which allows the user to ini tialize an mp int with a given default number of digits By default all initializers allocate MP PREC digits This function lets you override this behaviour int mp init size mp int a int size The size parameter must be greater than zero If the function succeeds the mp int a will be initialized to have size digits which are all initially zero int main void mp_int number int result we need a 60 digit number if result mp_init_size amp number 60 MP_OKAY printf Error initializing the number 4s mp error to string result return EXIT_FAILURE use the number return EXIT_SUCCESS 2 6 MAINTENANCE FUNCTIONS 15 2 6 Maintenance Function
29. ned product ab to c This function actually breaks into one of four cases which are specific multiplication routines optimized for given parameters First there are the Toom Cook multiplications which should only be used with very large inputs This is followed by the Karatsuba multiplications which are for moderate sized inputs Then followed by the Comba and baseline multipliers Fortunately for the developer you don t really need to know this unless you really want to fine tune the system mp_mul will determine on its what routine to use automatically when it is called int main void mp_int numberi number2 int result Initialize the numbers if result mp_init_multi amp number1 Some tweaking may be required 31 32 CHAPTER 4 MULTIPLICATION AND SQUARING amp number2 NULL MP_OKAY 1 printf Error initializing the numbers s mp_error_to_string result return EXIT_FAILURE set the terms if result mp_set_int amp number 257 MP_OKAY printf Error setting numberl 5 mp error to string result return EXIT FAILURE if result mp set int amp number2 1023 MP OKAY printf Error setting number2 5 mp error to string result return EXIT FAILURE multiply them if result mp mul amp numberi amp number2 amp numberi MP OKAY printf Error multiplying terms 475 mp error to string result return EXIT FAILURE
30. nly one function per source file and often use middle road approach where don t cut corners for an extra 2 speed increase Source code alone cannot really teach how the algorithms work which is why I also wrote a textbook that accompanies the library beat that So you may be thinking should I use LibTomMath and the answer is a definite maybe Let me tabulate what I think are the pros and cons of LibTom Math by comparing it to the math routines from GnuP GP 3GnuPG v1 2 3 versus LibTomMath v0 28 1 5 PURPOSE OF LIBTOMMATH 7 Criteria Pro Con Notes Few lines of code per file X GnuPG 300 9 LibTomMath 76 04 Commented function prototypes X GnuPG function names are cryptic Speed X LibTomMath is slower Totally free X GPL has unfavourable restrictions Large function base X GnuPG is barebones Four modular reduction algorithms X Faster modular exponentiation Portable X GnuPG requires configuration to build Figure 1 1 LibTomMath Valuation It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only small portion of the entire application However LibTom Math was written with cryptography in mind It provides essentially all of the functions a cryptosystem would require when working with large integers So it may feel tempting to just rip the math code out of GnuPG or GnuMP where it was taken from originally in your own application but I think th
31. numberi 5 set the number2 to 6 mp_set amp number2 6 if result mp neg number2 amp number2 MP OKAY printf Error negating number2 s mp error to string result return EXIT FAILURE switch mp_cmp_mag amp number1 amp number2 case MP GT printf numberi gt number2 break case MP EQ printf number1i number2 break case MP LT printf numberi lt number2 break we re done with it mp clear multi numberi amp number2 NULL 24 CHAPTER 3 BASIC OPERATIONS return EXIT_SUCCESS If this progrand completes successfully it should print the following numberi lt number2 This is because 6 6 and obviously 5 6 3 2 2 Signed comparison To compare two mp int variables based on their signed value the mp_cmp function is provided int mp cmp mp int a mp int b This will compare a to the left of b It will first compare the signs of the two mp int variables If they differ it will return immediately based on their signs If the signs are equal then it will compare the digits individually This function will return one of the compare conditions codes listed in figure int main void mp int numberi number2 int result if result mp init multi amp numberi amp number2 NULL MP OKAY printf Error initializing the numbers 5 mp error to string result return EXIT_FAILURE set the numberi to 5
32. output a row of numbers that are increasing Each column is a different test such as addition multiplication etc that is being performed The numbers represent how many times the test was invoked If an error is detected the program will exit with a dump of the relevent numbers it was working with 1 4 Build Configuration LibTomMath can configured at build time in three phases we shall call de pends tweaks and trims Each phase changes how the library is built and they are applied one after another respectively To make the system more powerful you can tweak the build process Classes are defined in the file tommath_superclass h By default the symbol LTM_ALL shall be defined which simply instructs the system to build all of the functions This is how LibTomMath used to be packaged This will give you access to every function LibTomMath offers However there are cases where such a build is not optional For instance you want to perform RSA operations You don t need the vast majority of the library to perform these operations Aside from LTM_ALL there is another pre defined class SC_RSA_1 which works in conjunction with the RSA from LibTomCrypt Additional classes can be defined base on the need of the user 2 copy of MPI is included in the package 4 CHAPTER 1 INTRODUCTION 1 4 1 Build Depends In the file tommath_class h you will see a large list of defines followed by a series of
33. p prime next prime mp int int t int bbs style 7 6 RANDOM PRIMES 47 This finds the next prime after a that passes mp_prime_is_prime with t tests Set bbs_style to one if you want only the next prime congruent to 3 mod 4 otherwise set it to zero to find any next prime 7 6 Random Primes int mp_prime_random mp_int a int t int size int bbs ltm_prime_callback cb void dat This will find a prime greater than 2565126 which can be bbs_style or not depending on bbs and must pass t rounds of tests The Itm prime callback is a typedef for typedef int ltm prime callback unsigned char dst int len void dat Which is a function that must read len bytes and return the amount stored into dst The dat variable is simply copied from the original input It can be used to pass RNG context data to the callback The function mp_prime_random is more suitable for generating primes which must be secret as in the case of RSA since there is no skew on the least significant bits Note As of v0 30 of the LibTomMath library this function has been depre cated It is still available but users are encouraged to use the new mp_prime_random_ex function instead 7 6 1 Extended Generation int mp_prime_random_ex mp_int a int t int size int flags ltm_prime_callback cb void dat This will generate a prime in a using t tests of the primality testing algorithms The variable size specifies the bit length of the prime desi
34. p_int int mp sub d mp int mp digit b mp int int mp mul d mp int mp digit b mp int int mp div d mp int mp digit b mp int mp digit int mp mod d mp int mp digit b mp digit These work like the full mp int capable variants except the second parameter bis a mp digit These functions fairly handy if you have to work with relatively small numbers since you will not have to allocate an entire mp int to store a number like 1 or 2 Index mp add mp add d mp and mp clear mp clear multi mp cemp 24 bcm Dm mp cmp mag 23 mp div 30 mp div d mp dr reduce mp dr setup EQ mp error_to_string mp expt d mp exptmod mp exteuclid mp gcd mp get int mp grow MP GT mp init mp init copy mp init multi mp init set mp init set int mp init size mp int 53 mp invmod mp jacobi mp lcm 15 MP _ 9 mp_montgomery _calc_normalization mp montgomery reduce mp montgomery setup 37 mp mul mp mul 2 mp_mul_2d mp_mul_d mp n root 44 mp neg 29 MP NO 9 MP_OKAY 9 mp or _ fermat mp prime is divisible mp prime is prime mp prime miller rabin mp prime next prime mp prime rabin miller trials mp prime random mp prime random ex mp radix size 54 mp read radix mp read unsigned bin mp reduce mp red
35. quick but the domain of a digit can change it s always at least 0 127 3 1 1 Single Digit Setting a single digit can be accomplished with the following function void mp set mp int a mp digit b This will zero the contents of a and make it represent an integer equal to the value of b Note that this function has a return type of void It cannot cause an error so it is safe to assume the function succeeded int main void mp_int number int result if result mp_init amp number MP_OKAY 19 20 CHAPTER 3 BASIC OPERATIONS printf Error initializing the number 4s mp error to string result return EXIT_FAILURE set the number to 5 mp_set number 5 we re done with it mp_clear amp number return EXIT_SUCCESS 3 1 2 Long Constants To set a constant that is the size of an ISO C unsigned long and larger than a single digit the following function can be used int mp_set_int mp_int a unsigned long b This will assign the value of the 32 bit variable b to the mp int a Unlike mp set this function will always accept a 32 bit input regardless of the size of a single digit However since the value may span several digits this function can fail if it runs out of heap memory To get the unsigned long copy of an mp int the following function can be used unsigned long mp get int mp int a This will return the 32 least significant bits of the int
36. red The variable flags specifies one of several options available see fig which can be OR ed together The callback parameters are used as in mp_prime_random 48 CHAPTER 7 PRIME NUMBERS Flag Meaning LTM_PRIME_BBS Make the prime congruent to 3 modulo 4 LTM PRIME SAFE Make a prime p such that p 1 2 is also prime This option implies LTM PRIME BBS as well LTM PRIME 2MSB OFF Makes sure that the bit adjacent to the most significant bit Is forced to zero LTM_PRIME_2MSB_ON Makes sure that the bit adjacent to the most significant bit Is forced to one Figure 7 1 Primality Generation Options 8 Input and Output 8 1 ASCII Conversions 8 1 1 To ASCII int mp_toradix mp_int a char str int radix This still store a in str as a base radix string of ASCII chars This function appends a NUL character to terminate the string Valid values of radix line in the range 2 64 To determine the size exact required by the conversion before storing any data use the following function int mp_radix_size mp_int a int radix int size This stores in size the number of characters including space for the NUL terminator required Upon error this function returns an error code and size will be zero 8 1 2 From ASCII int mp_read_radix mp_int a char str int radix This will read the base radix NUL terminated st
37. remainder of b divided by c a b mod 5 1 Of particular interest to cryptography are reductions where b is limited to the range 0 lt b lt since particularly fast reduction algorithms can be written for the limited range Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation algorithm mp_exptmod when an appro priate modulus is detected 5 1 Straight Division In order to effect an arbitrary modular reduction the following algorithm is provided int mp mod mp int mp int b mp int This reduces a modulo b and stores the result in c The sign of c shall agree with the sign of b This algorithm accepts an input a of any range and is not limited by 0 lt a lt b 35 36 CHAPTER 5 MODULAR REDUCTION 5 2 Barrett Reduction Barrett reduction is a generic optimized reduction algorithm that requires computation to achieve a decent speedup over straight division First a mu value must be precomputed with the following function int mp_reduce_setup mp_int a mp_int b Given a modulus in b this produces the required mu value in a For any given modulus this only has to be computed once Modular reduction can now be performed with the following int mp_reduce mp_int mp_int b mp_int This will reduce a in place modulo b with the precomputed mu value in c a must be in the range 0 lt a lt 02 int main void mp_int a
38. repeated multiplications by a for all values of b greater than three 6 2 Modular Exponentiation int mp_exptmod mp_int mp_int X mp_int mp_int Y This computes Y mod using a variable width sliding window algo rithm This function will automatically detect the fastest modular reduction technique to use during the operation For negative values of X the operation is performed as Y G mod mod P provided that gcd G 1 This function is actually a shell around the two internal exponentiation functions This routine will automatically detect when Barrett Montgomery Restricted and Unrestricted Dimminished Radix based exponentiation can be used Generally moduli of the a restricted dimminished radix form lead to the fastest modular exponentiations Followed by Montgomery and the other two algorithms 43 44 CHAPTER 6 EXPONENTIATION 6 3 Root Finding int mp_n_root mp_int a mp_digit mp_int This computes c a such that lt a and c 1 gt a The implementation of this function is not ideal for values of b greater than three It will work but become very slow So unless you are working with very small numbers less than 1000 bits avoid b gt 3 situations Will return a positive root only for even roots and return a root with the sign of the input for odd roots For example performing 41 2 will return 2 whereas 8 will return 2 This algorithm use
39. ring from str into a It will stop reading when it reads a character it does not recognize which happens to include th NUL char imagine that A single leading sign can be used to denote a negative number 49 50 CHAPTER 8 INPUT AND OUTPUT 8 2 Binary Conversions Converting an int to and from binary is another keen idea int mp_unsigned_bin_size mp_int This will return the number of bytes octets required to store the unsigned copy of the integer a int mp_to_unsigned_bin mp_int a unsigned char b This will store a into the buffer b in big endian format Fortunately this is exactly what DER or is it ASN requires It does not store the sign of the integer int mp_read_unsigned_bin mp_int a unsigned char b int This will read an unsigned big endian array of bytes octets from b of length c into The resulting integer a will always be positive For those who acknowledge the existence of negative numbers heretic there are signed versions of the previous functions int mp_signed_bin_size mp_int a int mp_read_signed_bin mp_int a unsigned char b int int mp_to_signed_bin mp_int unsigned char b They operate essentially the same as the unsigned copies except they prefix the data with zero or non zero byte depending on the sign If the sign is zpos e g not negative the prefix is zero otherwise the prefix is non zero Chapter 9 Algebraic Functions
40. s 2 6 1 Reducing Memory Usage When an mp int is in a state where it won t be changed again excess digits can be removed to return memory to the heap with the mp shrink function int mp shrink mp int a This will remove excess digits of the mp int a If the operation fails the mp int should be intact without the excess digits being removed Note that you can use a shrunk mp int in further computations however such operations will require heap operations which can be slow It is not ideal to shrink mp int variables that you will further modify in the system unless you are seriously low on memory int main void mp int number int result if result mp_init amp number MP OKAY printf Error initializing the number js mp error to string result return EXIT FAILURE use the number e g pre computation We re done with it for now if result mp shrink amp number MP OKAY printf Error shrinking the number 78 mp error to string result return EXIT FAILURE use it we re done with it 2A Diffie Hellman modulus for instance 16 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH mp_clear number return EXIT_SUCCESS 2 6 2 Adding additional digits Within the mp int structure are two parameters which control the limitations of the array of digits that represent the integer the mp int is meant to equal The used parameter dictates how many d
41. s the Newton Approximation method and will converge on the correct root fairly quickly Since the algorithm requires raising a to the power of b it is not ideal to attempt to find roots for large values of b If particularly large roots are required then a factor method could be used instead 1 16 For example 1 16 is equivalent to 1 4 ue Chapter 7 Prime Numbers 7 1 Trial Division int mp prime is divisible mp int a int result This will attempt to evenly divide a by a list of primed and store the outcome in result That is if result 0 then a is not divisible by the primes otherwise it is Note that if the function does not return MP OKAY the value in result should be considered undefined 7 2 Fermat Test int mp prime fermat mp int a mp int b int result Performs a Fermat primality test to the base b That is it computes b mod a and tests whether the value is equal to b or not If the values are equal then a is probably prime and result is set to one Otherwise result is set to zero 71 3 Miller Rabin Test int mp prime miller rabin mp int a mp int b int result l Default is the first 256 primes 2Currently the default is to set it to zero first 45 46 CHAPTER 7 PRIME NUMBERS Performs a Miller Rabin test to the base b of a This test is much stronger than the Fermat test and is very hard to fool besides with Carmichael numbers If a passes the test therefore is pro
42. tializing the number 4s mp error to string result return EXIT FAILURE use the number done with it mp_clear number return EXIT_SUCCESS 2 5 3 Multiple Initializations Certain algorithms require more than one large integer In these instances it is ideal to initialize all of the int variables in an all or nothing fashion That is they are either all initialized successfully or they are all not initialized The mp init_multi function provides this functionality int mp_init_multi mp_int mp It accepts a NULL terminated list of pointers to mp_int structures It will attempt to initialize them all at once If the function returns MP_OKAY then all of the variables are ready to use otherwise none of them are available for use A complementary mp clear multi function allows multiple mp int variables to be free d from the heap at the same time l The dp member 2 5 INITIALIZATION 13 int main void mp int numi num2 num3 int result if result mp init multi amp numi amp num2 amp num3 NULL MP OKAY printf Error initializing the numbers js mp_error_to_string result return EXIT_FAILURE use the numbers We re done with them mp clear multi numi amp num2 amp num3 NULL return EXIT SUCCESS 2 5 4 Other Initializers To initialized and make a copy of an mp int the mp init copy fu
43. uce 2k mp reduce 2k setup mp reduce setup mp rshd mp set 19 _ int mp shrink mp sqr 33 mp sub mp sub d mp to unsigned bin mp toradix mp unsigned bin size VAL 9 mp xor 29 YES 9 INDEX

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