HCG Test Tools with GCD totient Graphics User Manual V1
Contents
1. Usage Press the button Analyze then the result will be expressed on the List within an appropriate time For example like Q 2 kg 1 Q 3 kg 271 Q 6 kg 271 that result will be appeared In this List Q x is a sub group of HCG in which you select the base point P x and kg is a rank of its sub group On these group operations with the adaptation of a possible base point the all result are expressed serially on the List This means the total number of the possible base point P x is equal to its rank of HCG Be attention to remember the element of Q x like mP x may not be a candidate of the base point in place of P x in this sub group Q x This stage is to put the resultant data in order Usage Press the button DataSum so the data in the List are estimated and listed again This function is very convenient and you need not to count it For example the List show you like The numbers of the sub group of a rank kg 271 is 270 The numbers of the sub group of a rank kg 1 is 1 In the world of HCG the rank k does not depend on its curve parameters directly and is constant k ptl 2 And the numbers of the sub group Q x with kg is exactly G kg You can confirm these truths on your calculations Adding this we can also confirm the data of the sub group Q x precisely Usage Firstly select the one line of the sub group Q x on the List then press the button Precis2. Design Tool for HCG can design the curve parameters of HCG with the modulus p You can make the curve parameters a b c on the integer function under this tool Firstly these parameters must satisfy the condition using Jacobi s symbol oe where D is a discriminant of an order 2 and have a relation with the parameters as D c 4a Secondary a group condition must be satisfied as x cx a p 1 for the parameters a andc and also the parameters b d must be determined in terms OF xX The group under these conditions named a quadratic hyperbolic curve group which is consisted of the element P x s on the modulus p and the group operation The whole group definition with modulus n is expressed as 2 P x xX CX a Pi G E for Vi on Z nZ n v i for which has been studied in recent years and be resulted in Above conditions are adopted to limit Gn for making a quadratic hyperbolic curve group to use this software A quadratic hyperbolic curve group is now known as a kind of Logarithmic Group Q Firstly we select the parameter c which has an optional in its selection You can select the parameter c within a range of 0 lt c lt p at leisure Usage Input trial number c then press the button Select c That action will result in like parameter c is adopted Yellow on the List and also display few candidates of a parameter a on the next List under the
3. System Context Menu This tool has another function to use a context menu by clicking right on a mouse Copy context menu Copy context menu copies the selected items to the Clip Board So you can take it when you want to use it externally Paste context menu Paste context menu paste the contained text on the Clip Board to the designated List So you can take it when you want to add something from the external Select All context menu Select All context menu can select all lines on the designated List It is very convenient when the List data is enormous Copy Right remarks This English Manual and Software for HCG Test Tool was originated from Kimito Horie and All things in relation with the Copy Right Law are belong to himself Without his permission or his grant any copies modification deformation and revision etc are prohibited This English Manual was made on 14 Jun 2013 Version V1 02E_ 20130614 and revised later End
4. the contents just expressed on the Lists If you want to change the memory place conserved Please press the button NumUP or NumDown Another way of transportation on the memory is opening the data file like DataHCG_01 hcp which will recover ascend parameters on this software and express its file name on the front panel Determine the set of HCG parameters which is used on the group operations Usage Firstly select the data number and express the set of HCG parameters on the List and Press the button SetData to use this set This set is used on the group operations by the next tool for a HCG analytics This software contain its editor to edit directly the parameters which you want to change its values appropriately according to the rules of HCG Please press the brown button EditParam then the next graphics with an Excel base will appear ParamList hitial Data Set PE Param d 0 2 0 3 o 4 0 5 o 6 0 7 0 8 0 9 0 4 OJ OLO 61 9 5 010 ORS i J o This edit tool has 9 context menu which appears by clicking mouse right button ClearRowData will clear that all row data to be 0 ResoreRowData will recover the row data which were exist before cleaning DeleteSelectedRow will delete the selected row data and the rest of the data later will follow just after that row CopySelectedRow will copy the selected row data to the inner memory PasteSelectedRow will paste the dat
5. the values of kq G kq and k with a modulus X Reset button makes this tool be initialized Close button makes this tool be terminated 5 System Menu This tool can conserve parameters of HCG and etc made by itself as a file File gt Open menu Open menu has a function to open the file and read parameters of HCG and etc Such parameter file has an extension hcp on default This file consists of a text file basically you can make it originally by yourself Try to open data_Hcg_01 hcp file the parameters of HCG and etc can restore again on its data numbers File gt Save menu Save menu has a function to save the file with the parameters of HCG and etc If the same name of the file exists on the System the new data will overwrite its contents after confirmation of paying attentions File gt Save As menu Save As menu has a function to save the file with the parameters of HCG and etc with a new file name If the same name of the file exists on the System the new data will overwrite its contents after confirmation of paying attentions File gt Close menu Close menu makes this file be terminated Help gt How to use menu How to use menu will guide you to operate this tool to read and refer this manual Help gt About menu About menu discloses the version of this manual and will link you to my private WEB site the Room of HCG made by myself 6
6. x with a rank kg which is equal to G kg For example this List is expressed as below 1 Basics EHCG modulus n 34 2 1 17 1 n 2 GruopRank k 9 1 9 Max rank of sub groups kq 9 2 SubGroups The number of Q x with kg is total G kg G X GCD totient o 9 6 G 9 6 Ratio l G 9 9 1 3 1 3 2 G 3 2 Ratio l G 3 3 1 141 1 1 G 1 1 Ratio l G 1 1 1 3 Theories ka d 9 k XG d 9 1 kq 1 kq d G d 6 2 1 9 kq odd 4 Presentation of the GCD Set Diml h 1 1 h 2 9 Dim2 h 12 1 5 GCD totientG x with modulus N 34 16 G 34 mod No modulus_N found G 34 mod No modulus_N found In this List you can find the EHCG function to be expressed as the sum of integer factors like G 9 9 1 3 1 as ordered with the dividend of kg And also find the theory of EHCG 1 kq d G d kq J is calculated and confirmed to be divided by the kq in any case The GCD Set shows the structure of the k as the geometrical figure of the ki s in terms of the prime number product of each ki which is the sub group s total elements number order and generated from the prime pi EHCG function is the multi value integer one and has several different modulus for a one value So using a new original method of finding a modulus n this list express the modulus n s with the same value found by this method only in the case of successfully having fo
7. HCG Test Tools with GCD totient Graphics User Manual V1 O2E_20130711 Copy Right remarks This English Manual and Software for HCG Test Tools with GCD totient Graphics was originated from Kimito Horie and All things in relation with the Copy Right Law are belong to himself Without his permission or his grant any copies modification deformation and revision etc are prohibited 1 To start This User Manual has made for using HCG Test Tools smoothly and has a purpose for generalizing the Hyper Curve Group for Cryptography and Mathematics HCG Test Tools with GCD totient Graphics is a software available for the cryptosystem design the signature design and the primary tests without difficulties of a mathematical complex and can easily make an original design with the determination of curve parameters modulus p and the group characteristics of HCG This software is also convenient for the tool of Mathematics spatially for testing conformities of the theory of HCG Your download of this software is available on the download WEB site in free this means you need not pay any fees to use this software http www din or jp horie Please remind this software is made only for an academic use not for a professional designer use So the bits numbers of a modulus p keys so on can not select bigger as you would expect 2 Summary of HCG Test Tools with GCD totient Graphics This software consists of the few t
8. a to the selected row from the inner memory InsertNewRow will add a new row at the position of the selected row Then all row data will shift one to the bottom from that row and last low will disappear RecreateAllData will create an appropriate set of parameters on which the group condition are satisfied according to its Modulus p on the List So you must select a modulus p at first Other parameters like InitialOpCycles KeyMargin InitialWorkingKey InotialBP and EncryptionMethod are related to the encryption applications with HCG and have no relation to the HCG design itself 2 Analytic Tool for HCG Left Middle Analytic Tool for HCG is supplied for confirming the structure of HCG with the set of parameters just made on the Design Tool for HCG Firstly reconfirm your selection of the modulus p and parameters which will be used on the Analytic Tool for HCG This operation is prepared for your convenience Usage Press the button Confirm then a modulus p and parameters which will be used on the Analytic Tool for HCG are expressed on the List If the modulus p or parameters does not coincide with that you expected please press the button SetData again to prepare another set of parameters after changing the data number Finally we execute the analyzing of that HCG This might have enormous time to complete if you choose a large number as the modulus p Do nt miss it
9. discriminant D Next we select the parameter a from the candidates expressed on the List as a result of the operation Usage Click the one of candidates of a parameter a on the List and press the button Select a Then that List show you a message like parameter a is adopted Yellow If you want to select another candidates then press the button Select a again So the new candidates appear in place of the first candidates on the List Next we select the parameter b and d from the candidates expressed on the List The parameter b and d are the integer point of the HCG and must satisfy the condition x cx a p in terms of x We must select the parameter b and d as a different point on the above condition And also we can select the valuable x from that List to determine the base point P x Usage Click the one of candidates of a parameter b or d as displayed on the List and press the button Select b or Select d The result show you on the List like parameter b is selected yellow or parameter d is selected Yellow Determined Parameters can be conserved ina temporally memory for future operations Max set of that conservation is 20 s Usage Firstly select the data number and press the button Import then the set of the parameters on that data number is conserved on the temporal memory Before that operation be sure to confirm
10. eeding and not to be frozen And if you want to resume the group operations again please push the button Resume Then the button changes its name to Stop and suspended task works again If you wan to quit the operations or to clear the List then press Clear Quit button as you like 3 Calculation Tools of Extended HCG Left Down On an Extended HCG we know that the numbers of the sub group Q x with a rank kg under a modulus X is G kg which is called the extended Euler s function or I named it as GCD totient But the extended Euler s function is a new integer function discovered in 2005 not so clear in precise yet only few theories were proved till now We must establish a calculation method for a future tool of Integer Theory The extended Euler s function G n GCD totient is presented as a x Dla Toco k or simply a x Dua T s dikq i dikq 1 with X p pop and k p Bet And u d is Mebius function This tool gives you many information about the EHCG on checking for example the expression of the EHCG with the form of a sum as the order of the dividend of kg in compared with the Euler s function EHCG theory calculations presentations of the GCD set and the modulus n which give us a same value or its multiples Usage Firstly input a modulus n and press the button CalcEHCG then the List show you the whole calculation results with the numbers of the sub group Q
11. ely This operation results the precise data of the whole sub group Q x with a rank of kg on the List For example the List show you like The numbers of the sub group of a rank kg 271 is 270 Q 3 Q 6 Q 14 r Furthermore we can also confirm a one of the sub group Q x precisely Usage Firstly select the one line of the sub group Q x on the List then press the button Precisely again This operation results the precise data of a one of the sub group Q x with a rank of kg on the List For example the List show you like The sub group Q 6 consists of 1P 6 434 2P 300 127 3P 508 227 AP 68 55 5P 14 411 In this case 1P means the base point P x itself and the rank means itself as kgP 0 This List show you the whole results of kg times group operation and each data is equivalent to the elements of a HCG that is mP x on each and the value of x is a point of the selected quadratic hyperbolic curve During the group operations if you want to confirm its process on time then you make it by Stop Red button which is next to the button Analyze Usage Pushing the button Stop during the group operations then the calculations are stopped and the button name is changed to Resume Furthermore the process s percentage at that moment is expressed on the Status upper part of the List So you can confirm the processes how going on This is also useful for confirming the process is proc
12. ools for the HCG and each tool is very useful in a stand alone And this is a second version manual of the HCG Test Tools having strong graphical tools for the GCD totient and HCG characteristics Calculation Tool of Exten 1 Prime Test Tool Upper Left Prime Test Tool is normally used for determining a modulus p as a prime The modulus X composition is not treated here This tool can test the trial number n whether n is a prime or a composition for using a so called sieve method Usage Input a trial number n and press the button TEST The result will be displayed on the List as a Prime Factor Decomposition of an Integer For example if n 541 then represented as n 541 1 541 is a prime This tool also has a function of determining a selected prime whether it is a Sophie Germann prime or not Usage Click the line Ex 541 is a prime in that List and press the button Judge Such action result the expression in the List like p 541 is Germann prime q 271 is also prime If you want to adopt the selected trial number as a modulus press the radio button Yes Then the adoption is confirmed like as Modulus p 541 is adopted Yellow Attention If you select a large number as a modulus the group operation on HCG will dissipate enormous time to complete it that means you will be at fault Please select a modulus p lt 65536 16bits for your convenience 2 Design Tool for HCG Right
13. q based On this Graphics if you click on any place along with X axis then X value of this place is displayed with EHCG characteristics on the top Those are X value k value kq value kq value G kq value ratio k kq and ratio G kq ka If you select X value input on the text box and push Express button then the same shape markers with different color appears on the graphs simultaneously the top EHCG characteristics change its values Ratio button displays the ratio G kq ka as red tiny circles and k kq as blue tiny circles though theoretically these are equal or bigger than 1 Other buttons are the tools of an assistant Avaragelt button displays the average curve of the data LSMofIt button displays the least square mean curves of the data GetF X button displays the linier which pass though two points that are made from clicking a mouse on the common logarithmic axis Ratio button displays the ratio of the k kq and the ratio G kq kq with the kq value Using rough hypothesis and the geometric mean we know k log n gt k loglog n q oln n gt 00 when n going infinite AsyncF X button is for asymptotic functions of the EHCG characteristics that are calculated counting on GCD theories with some hypothesis This is only available for the Ratio for the time being We also give a convenient function on which a mouse click on the graphics can give
14. und it 4 Graphics Tool for the Extended HCG Left Down This version of HCG Test Tools has a strong graphical tool for the GCD characteristics spatially for GCD totient The brown button Graphics give you the graphical tool for the Extended HCG by pressing it That Graphics is expressed as follows Yellow frame with a red line shows GCD totient definitions in brief And the title of the graph is expressed on upper region Other parameters expressed on the top are EHCG characteristics which has a unique value depends on the X axis He GOD Totrent Graphics ee 8 Ssmeo ag e a F a bombas o 0006 0 o oo gt t co E T TE a ee E e E E E i OOD O00 eD mE OI0OTO o 0 ooo i o moo woso mw eo o o 2 Com o ome COMO lomo aoo 0 o o oo io j CeePo co omoo cooo Cm ooo oolio o i i H Usage If you press the button DispAll then the calculated pictures appeared with the red tiny circles of the extended Euler s function G kq though the blue tiny circles are the Euler s function kq on the common logarithmic axis of X and Y In this case kq LCM k is a max value of the rank of the sub group Q x You can make an axis slide view of the graphics appropriate times by directly changing the values of X Slide or Y Slide And the expressions are also modified by using Cross or Line If you click 0n kq button X coordinate is changed from the n based to the k
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