LibTomMath User Manual v0.41
Contents
1. 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 1 4 1 Build Depends In the file tommath class h you will see a large list of C defines followed by a series of 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 1 4 BUILD CONFIGURATION 5 the respective file will be compiled and linked into the library Ac cordingly 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 c2. 8 will return 2 This algorithm uses the Newton Approximation method and will converge on the correct root fairly guickly Since the algorithm reguires 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 For example a is 1 4 1 21 1 2 me equivalent to a 4 or simply a Chapter 7 Prime Numbers 7 1 Trial Division int mp prime is divisible mp int x a int result This will attempt to evenly divide a by a list of primeg and store the outcome in result That is if result 0 then a is not divis ible 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 Default is the first 256 primes 2Currently the default is to set it to zero first 49 50 CHAPTER 7 PRIME NUMBERS values are equal then a is probably prime and result is set to one Otherwise result is set to zero 7 3 Miller Rabin Test int mp_prime_miller_rabin mp_int a mp_int b int result 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 f
3. if result mp_mul_2 amp number amp number MP_OKAY 4 printf Error multiplying the number s mp error to stringl result return EXIT FAILURE switch mp_cmp_d amp number 7 case MP_GT printf 2 number gt 7 case MP_EQ printf 2 number 7 break case MP_LT printf 2 number 7 A now divide by two if result mp div 2 knumber amp number MP_OKAY 4 printf Error dividing the number js mp error to stringl 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 lt 7 we re done with it mp clear amp number return EXIT_SUCCESS If this program is successful it will print out the following text 2 number gt 7 2 number 2 lt 7 Since 10 gt 7 and 5 lt 7 To multiply by a power of two the following function can be used int mp_mul_2d mp_int x a int b mp int x c 30 CHAPTER 3 BASIC OPERATIONS 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 x a int b mp int c mp int x d Which will divide a by 2 store the quotient in c and the re mainder in d If b lt 0 then the function simply copie
4. a mp int x 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 numl 2 5 INITIALIZATION 15 We want a copy of numl in num2 now if result mp_init_copy amp num2 amp num1 MP_OKAY printf Error initializing the copy s mp error to stringl result return EXIT FAILURE now num2 is ready and contains a copy of numl We re done with them mp_clear_multi amp num1 amp num2 NULL return EXIT_SUCCESS Another less common initializer is mp init size which allows the user to initialize 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 s mp error to stringl result return EXIT FAILURE 16 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH use the number return EXIT_SUCCESS 2 6 Maintenance Functions 2 6 1 Reducing Memory Usage When an mp int is in a state where i
5. int x c int mp xor mp int a mp int b mp int x c Which compute c 40b where is one of OR AND or XOR 3 4 Addition and Subtraction To compute an addition or subtraction the following two functions can be used int mp add mp int a mp int b mp int x c int mp sub mp int a mp int b mp int x 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 x a mp_int b Which assigns a to b 32 CHAPTER 3 BASIC OPERATIONS 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 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 x 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 reguired 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 fol lowing int mp mul mp int a mp int b mp int x c Which assigns the full signed product ab
6. result mp_neg amp number2 amp number2 MP_OKAY printf Error negating number2 s mp error to stringl result return EXIT FAILURE switch mp_cmp_mag amp number1 amp number2 case MP_GT printf number1 gt number2 break case MP_EQ printf number1 number2 break case MP LT printf number1 lt number2 break we re done with it mp_clear_multi amp number1 amp number2 NULL return EXIT_SUCCESS 3 2 COMPARISONS 25 If this progrand completes successfully it should print the fol lowing number1 lt number2 This is because 6 6 and obviously 5 lt 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 im mediately 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 number1 amp number2 NULL MP_OKAY 4 printf Error initializing the numbers js mp error to stringl result return EXIT FAILURE set the number1 to 5 mp_set amp number1 5 lThis function uses the mp neg function
7. tity divided by another Expressed as 5 1 the modular reduction is equivalent to the remainder of b divided by c a b mod c 5 1 Of particular interest to cryptography are reductions where b is limited to the range 0 lt b lt c since particularly fast reduction algorithms can be written for the limited range Note that one of the four optimized reduction algorithms are au tomatically chosen in the modular exponentiation algorithm mp_exptmod when an appropriate 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 a mp_int b mp_int c 39 40 CHAPTER 5 MODULAR REDUCTION 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 5 2 Barrett Reduction Barrett reduction is a generic optimized reduction algorithm that requires pre computation to achieve a decent speedup over straight division First a y 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 y 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 a mp_int b mp_int c This will reduce a in place modulo b with the precomputed y value in
8. E SIZE is the number of primes in the prime number table by default this is 256 7 5 Next Prime int mp_prime_next_prime mp_int a int t int bbs_style 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 256 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 52 CHAPTER 7 PRIME NUMBERS 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 deprecated 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 prim
9. Signed comparisonhj 3 2 3 Single Digit 3 3 Logical Operations 3 3 1 Multiplication by two 2 3 3 2 Polynomial Basis Operations 3 3 3 AND OR and XOR Operations 3 4 Addition and Subtraction 3 5 Sign Manipulation 3 0 1 Negation 3 5 2 Absolute 3 6 Integer Division and Remainderh 4 Multiplication and Squaring 4 1 Multiplication 4 2 Squaring 4 3 Tuning Polynomial Basis Routines 5 Modular Reduction 5 1 Straight Division 5 2 Barrett Reduction 5 3 Montgomery Reduction 5 4 Restricted Dimminished Radix 5 0 Unrestricted Dimminshed Radix 6 Exponentiation 6 1 Single Digit Exponentiationhj 6 2 Modular Exponentiation 6 3 Root Finding 7 Prime Numbers A A AA ete ea 1 2 Fermat lestt 7 3 Miller Rabin Testt 7 3 1 Required Number of lestsi 7 4 Primality Tlestingl a Gia erase ra MU 7 6 Random Primes 7 6 1 Extended Generation 8 Input and Output 8 1 ASCILIConversionss 811 To ASC so ba be bbb dd ee oo oS 8 1 2 FromASClIH 8 2 Binary Conversions o ooo 9 Algebraic Functions 9 1 Extended Euclidean Algorithm 9 2 Greatest Common Divisorh 9 4
10. This allows operands to be re used which can make program ming 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 x a 12 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH 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 stringl 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 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 pointef 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 1The dp member 2 5 INITIALIZATION 13 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 str
11. amp number MP_OKAY printf Error initializing the number fs mp error to stringl result return EXIT FAILURE set the number to 654321 note this is bigger than 127 if result mp_set_int amp number 654321 MP OKAY 4 printf Error setting the value of the number 4s mp error to stringl result return EXIT FAILURE printf number lu mp get int knumber 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 func tions are available 22 CHAPTER 3 BASIC OPERATIONS 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 a before setting the values int main void mp int numberi number2 int result initialize and set a single digit if result mp_init_set amp number1 100 MP_OKAY printf Error setting numberl s mp_error_to_string result return EXIT_FAILURE initialize and set a long if result mp_init_set_int amp number2 1023 MP_OKAY 4 printf Error setting number2 s mp_error_to_string result return EXIT_FAILURE display printf Number1 Number2 lu lu mp get int gnumberl mp_get_int am
12. single digit can be accomplished with the following func tion 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 19 20 CHAPTER 3 BASIC OPERATIONS 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 printf Error initializing the number fs mp error to stringl result return EXIT FAILURE set the number to 5 mp_set amp 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 3 1 SMALL CONSTANTS 21 unsigned long mp_get_int mp_int a This will return the 32 least significant bits of the mp int a int main void mp_int number int result if result mp_init
13. to c This function actu ally 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 33 34 CHAPTER 4 MULTIPLICATION AND SQUARING determine on its owx 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 amp number2 NULL MP_OKAY printf Error initializing the numbers js mp_error_to_string result return EXIT_FAILURE set the terms if result mp set int knumber 257 MP_OKAY 4 printf Error setting numberl 4s mp error to stringl result return EXIT FAILURE if result mp set int knumber2 1023 MP OKAY printf Error setting number2 fs mp error to stringl result return EXIT FAILURE multiply them if result mp_mul amp number1 amp number2 amp number1 MP_OKAY printf Error multiplying terms s 1Some tweaking may be required 4 2 SQUARING 35 mp_error_to_string result return EXIT_FAILURE d
14. which is discussed in section 3 9 1 26 CHAPTER 3 BASIC OPERATIONS 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 mp error to stringl result return EXIT FAILURE switch mp_cmp amp number1 amp number2 case MP GT printf numberl gt number2 break case MP EO printf numberl number2 break case MP LT printf numberl lt number2 break we re done with it mp_clear_multi amp number1 amp number2 NULL return EXIT_SUCCESS If this progrant completes successfully it should print the fol lowing number1 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 2This function uses the mp neg function which is discussed in section 3 5 1 3 2 COMPARISONS 27 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 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 stringl result
15. wo w w w w N FAST MP INVMOD C Operand Size Related Restriction Undefine Moduli lt 2560 bits MP _MONTGOMERY_REDUCE C JS MP MUL DIGS C S MP MUL HIGH DIGS C S M P SOR C Polynomial Schmolynomial W w y ww w w w z z Z z z z Z Z MP_KARATSUBA_MUL_C MP_KARATSUBA_SQR_C MP_TOOM_MUL_C MP TOOM SOR C 1 5 Purpose of LibTomMath Unlike GNU MP GMP Library LIP OpenSSL or various other commercial kits Miracl LibTomMath was not written with bleed 1 5 PURPOSE OF LIBTOMMATH 7 ing 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 accessible for computer science students attempting to learn BigNum or multiple precision arithmetic techniques LibTomMath was written to be an instructive collection of source code This is why there are many comments only one func tion per source file and often I use a middle road approach where I 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 LibTomMath by compari
16. 53 54 CHAPTER 8 INPUT AND OUTPUT This will read the base radix NUL terminated string from str into a It will stop reading when it reads a character it does not rec ognize which happens to include th NUL char imagine that A single leading sign can be used to denote a negative number 8 2 Binary Conversions Converting an mp int to and from binary is another keen idea int mp_unsigned_bin_size mp_int a 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 c This will read in an unsigned big endian array of bytes octets from b of length c into a 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 ka unsigned char xb int c int mp to signed bin mp int xa unsigned char sb 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 pref
17. H are in the public domain everyone is entitled to do with them 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 C 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 To build as a static library for GCC issue the following make command This will build the library and archive the object files in libtommath 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 tom math lib This has been tested with MSVC version 6 00 with service pack 5 1Note that the MPI files under mtest are copyrighted by Michael Fromberger They are not required to use LibTomMath 1 3 BUILDING LIBTOMMATH 3 1 3 2 Shared Libraries To build as a shared library for GCC 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 libtom math la while the static library called libtommath a Generally you use libtool
18. Jacobi Symbol 9 5 Modular Inverse 9 6 Single Digit Functionsg List of Figures NODOS a e A IE ee 8 2 1 Return Codesk 10 3 1 Comparison Codesfora h 23 el 36 NN 52 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 C compiler In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how to im plement bignum math However the resulting code has proven to be very useful It has been used by numerous universities com mercial 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 based 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 1 2 CHAPTER 1 INTRODUCTION 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 bot
19. LibTomMath User Manual v0 41 Tom St Denis tomstdenisQgmail com March 10 2007 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 ITpX book macro package Open Source Open Academia Open Minds Tom St Denis Ontario Canada Contents 1 Introduction 1 11 What is LiblomMathj 1 1 2 License o s aa a a a 1 1 3 Building LibTomMath 2 1 3 1 Static Librariesk 2 1 3 2 Shared Librariesk 3 1 3 9 Testing abe see ee Aa ee pI ie be ee 3 VE ENE NENA es GANE 4 14 1 Build Dependsg 4 142 Build Uweaksl 22 aa s 6 a a s k ee ea 5 14 3 Build Irims 5 1 5 Purpose of LiblomMathj 6 2 Getting Started with LibTomMath 9 bad dt ch be ee a da ee sina 9 2 2 Return Codes 9 pe A AA an eee 10 Chadha a oad 11 Saf on Aa ta OE eee 11 sagu aa da NOE 11 gen cad tee ap Gana Se eevee as Bae 12 Sig Meas dl RS 13 ill 2 5 4 Other Initializers 2 6 Maintenance Functions 2 6 1 Reducing Memory Usage 2 6 2 Adding additional digits 3 Basic Operations 3 1 Small Constants 3 1 1 Single Digit 3 1 2 Long Constants 3 1 3 Initialize and Setting Constants 3 2 Comparisons 3 2 1 Unsigned comparisonh 3 2 2
20. ad unsigned bin mp reduce mp reduce 2k mp reduce 2k setup mp reduce setup mp rshd mp set 19 mp set int mp shrink mp sqr mp sub mp sub_d mp to unsigned bin mp toradix mp unsigned bin size MP_VAL p mp_xor 81 MP_YES 9
21. ation GCC 3 3 1 and an Athlon XP processor the cutoff 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 110 digits Karat suba 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 XXX is one of the following entries from the table 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 4 3 TUNING POLYNOMIAL BASIS ROUTINES 37 When the program is running it will output a series of measure ments for different cutoff points It will first find good Karatsuba squaring and multiplication 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 38 CHAPTER 4 MULTIPLICATION AND SQUARING Chapter 5 Modular Reduction Modular reduction is process of taking the remainder of one quan
22. bTomMath offers you a totally free public domain well structured math library that is very flexible complete and performs well in resource contrained environments Fast RSA for example can be performed with as little as 8KB of ram for data again depending on build options 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 initialization required and the entire library is thread safe 2 2 Return Codes There are three possible return codes a function may return The last two codes listed are not actually return ed by a func tion They 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 10 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH 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 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 Data Types The basic multiple precision integer type is known as
23. c a must be in the range 0 lt a lt b int main void mp_int a b Cc mu int result initialize a b to desired values mp_init mu x c and set c to 1 we want to compute a 3 mod b get mu value if result mp_reduce_setup amp mu b MP_OKAY printf Error getting mu s 5 2 BARRETT REDUCTION 41 if if if if mp_error_to_string result return EXIT_FAILURE square a to get c a 2 x result mp_sqr amp a amp c MP_OKAY printf Error squaring s mp_error_to_string result return EXIT_FAILURE now reduce c modulo b result mp_reduce amp c amp b amp mu MP_OKAY printf Error reducing s mp_error_to_string result return EXIT_FAILURE multiply a to get c a 3 result mp mul ka amp c amp c MP_OKAY 4 printf Error reducing s mp_error_to_string result return EXIT_FAILURE now reduce c modulo b result mp_reduce amp c amp b amp mu MP_OKAY printf Error reducing s mp_error_to_string result return EXIT_FAILURE c now equals a 3 mod b 42 CHAPTER 5 MODULAR REDUCTION return EXIT_SUCCESS This program will calculate a mod b if all the functions suc ceed 5 3 Montgomery Reduction Montgomery is a specialized reduction algorithm for any odd mod uli Like Barrett reduction a pre computation step is required This is accomplished with
24. down to c a 3R x R 1 a 3 x 5 4 RESTRICTED DIMMINISHED RADIX 45 if result mp_montgomery_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 demon strates 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 mul tiplications In this case the restricted variant refers to moduli of the form 8 p for some k gt 0 and 0 lt p lt B where is the radix default to 278 As in the case of Montgomery reduction there is a pre computation phase required for a given modulus void mp_dr_setup mp_int a mp_digit d 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 pre computation a reduction can be performed with the following 46 CHAPTER 5 MODULAR REDUCTION int mp_dr_reduce mp_int a mp_int b mp_digit mp This reduce
25. e in a using t tests of the primality test ing algorithms The variable size specifies the bit length of the prime desired 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 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 LEM PRIME BBS as well LTM_PRIME_2MSB_OFF Makes sure that the bit adjacent to the most significant Is forced to zero LTM_PRIME_2MSB_ON Makes sure that the bit adjacent to the most significant Is forced to one Figure 7 1 Primality Generation Options Chapter 8 Input and Output 8 1 ASCII Conversions 8 1 1 To ASCII int mp toradix mp int x 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 reguired 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 reguired 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
26. ingl result return EXIT FAILURE use the number We re done with it mp clear amp number return EXIT SUCCESS 2 5 3 Multiple Initializations Certain algorithms reguire more than one large integer In these instances it is ideal to initialize all of the mp 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 struc tures It will attempt to initialize them all at once If the function returns MP OKAY then all of the mp int variables are ready to use otherwise none of them are available for use A complemen tary mp clear multi function allows multiple mp int variables to be free d from the heap at the same time 14 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH int main void mp int numi num2 num3 int result if result mp init multi Knuml amp num2 amp num3 NULL MP_OKAY printf Error initializing the numbers fs mp error to stringl result return EXIT FAILURE use the numbers We re done with them mp clear multi knuml 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 function has been provided int mp_init_copy mp int
27. isplay printf number1 x number2 lu mp_get_int amp number1 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 x number2 262911 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 sgr mp int a mp int b Will sguare a and store it in b Like the case of multiplication there are four different sguaring algorithms all which can be called from mp sgr It is ideal to use mp sgr over mp mul when squaring terms because of the speed difference 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 36 CHAPTER 4 MULTIPLICATION AND SQUARING and baseline algorithms use At O n 464973 and O n 584962 run ning times respectively 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 sin gle precision multiplications with the standard Comba a factor of 138 So why not always use Karatsuba or Toom Cook The sim ple 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 configur
28. ix is non zero Chapter 9 Algebraic Functions 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 eguation holds a U14b U2 lt U3 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 geod mp int a mp int b mp int x c This will compute the greatest common divisor of a and b and store it in c 55 56 CHAPTER 9 ALGEBRAIC FUNCTIONS 9 3 Least Common Multiple int mp lcm mp int a mp int b mp int x c This will compute the least common multiple of a and b and store it in c 9 4 Jacobi Symbol int mp_jacobi mp_int a mp_int p int c 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 c Computes the multiplicative inverse of a modulo b and stores the result in c such that ac 1 mod b 9 6 Single Digit Functions For those using small numbers snicker snicker there are several helpe
29. lass 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 usually between size and space They can be enabled at any pass of the configuration phase Define Purpose BN_MP_DIV_SMALL Enables a slower smaller and equally functional mp div function 1 4 3 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 6 CHAPTER 1 INTRODUCTION Moduli Related Restriction Undefine Exponentiation with odd moduli only N_S_MP_EXPTMOD_C N_MP_REDUCE_C N_MP_REDUCE_SETUP_C N_S_MP_MUL_HIGH_DIGS_C N FAST S MP MUL HIGH DIGS C Exponentiation with random odd moduli The above plus the following N MP REDUCE 2K C N MP REDUCE 2K SETUP C N MP REDUCE IS 2K C N MP DR IS MODULUS C N MP DR REDUCE C N MP DR SETUP C Modular inverse odd moduli only N MP INVMOD SLOW C Modular inverse both smaller slower tv tuj v u tu w u
30. loc parameter dictates how many digits are currently avail able 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 any thing 18 CHAPTER 2 GETTING STARTED WITH LIBTOMMATH 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 stringl result return EXIT FAILURE use the number We need to add 20 digits to the number if result mp grow knumber number alloc 20 lt MP OKAY 4 printf Error growing the number s mp error to stringl result return EXIT FAILURE use the number we re done with it mp_clear amp number return EXIT_SUCCESS Chapter 3 Basic Operations 3 1 Small Constants Setting mp_ints to small constants is a relatively common opera tion To accomodate 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 con stant The reason for both functions is efficiency Setting a single digit is quick but the domain of a digit can change it s always at least 0 127 3 1 1 Single Digit Setting a
31. ng it to the math routines from GnuPG 3GnuPG v1 2 3 versus LibTomMath v0 28 8 CHAPTER 1 INTRODUCTION Criteria Pro Con Notes Few lines of code per file GnuPG 300 9 LibTomMath Commented function prototypes GnuPG function names are cry Speed X LibTomMath is slower Totally free GPL has unfavourable restrictic Large function base GnuPG is barebones Five modular reduction algorithms Faster modular exponentiation Portable GnuPG requires configuration t P gt lt P P lt P lt P lt Figure 1 1 LibTomMath Valuation It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application However LibTomMath 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 appli cation but I think there 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 It depends largely on the processor compiler and the moduli being used Essentially the only time you wouldn t use LibTomMath is when blazing speed is the primary concern However on the other side of the coin Li
32. o 1 get normalization if result mp_montgomery_calc_normalization amp R b MP_OKAY printf Error getting norm s mp_error_to_string result return EXIT_FAILURE get mp value if result mp_montgomery_setup amp c amp mp MP_OKAY printf Error setting up montgomery js mp_error_to_string result return EXIT_FAILURE normalize a so now a is equal to aR if result mp_mulmod amp a amp R amp b amp a MP OKAY 4 44 if if if if CHAPTER 5 MODULAR REDUCTION printf Error computing aR s 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 s mp error to stringl result return EXIT FAILURE now reduce c back down to c a 2R72 x R 1 a 2R result mp montgomery reduce kc Kb mp MP OKAY 4 printf Error reducing s mp error to stringl result return EXIT FAILURE multiply a to get c a 3R 2 result mp mul ka amp c amp c MP OKAY 4 printf Error reducing s mp_error_to_string result return EXIT_FAILURE now reduce c back down to c a 3R72 x R 1 a 3R result mp_montgomery_reduce amp c amp b mp MP_OKAY printf Error reducing s mp_error_to_string result return EXIT_FAILURE now reduce again c back
33. ool besides with Carmichael numbers If a passes the test therefore is probably prime result is set to one Otherwise result is set to zero Note that is suggested that you use the Miller Rabin test in stead of the Fermat test since all of the failures of Miller Rabin are a subset of the failures of the Fermat test 7 3 1 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 a 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 27 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 sl 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 PRIM
34. p number2 clear mp_clear_multi amp number1 amp number2 NULL return EXIT_SUCCESS 3 2 COMPARISONS 23 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 com parison Result Code Meaning MP_GT a gt b MP EQ a lt b MP LT a lt b Figure 3 1 Comparison Codes for a b In figure 3 1 two integers a and b are being compared In this case a is said to be to the left of b 3 2 1 Unsigned comparison An unsigned comparison considers only the digits themselves and not the associated sign flag of the mp_int structures This is analo gous to an absolute comparison The function mp cmp_mag will compare two mp int variables based on their digits only int mp_cmp_mag 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 24 CHAPTER 3 BASIC OPERATIONS int main void mp int numberi number2 int result if result mp_init_multi amp number1 amp number2 NULL lt MP_OKAY 4 printf Error initializing the numbers js mp error to stringl result return EXIT FAILURE set the number1 to 5 mp_set amp number1 5 set the number2 to 6 mp_set amp number2 6 if
35. r functions int mp_add_d mp_int xa mp digit b mp_int c int mp_sub_d mp_int a mp_digit b mp_int c int mp mul d mp int xa mp digit b mp_int c 9 6 SINGLE DIGIT FUNCTIONS 57 int mp_div_d mp_int a mp_digit b mp_int c mp_digit int mp mod d mp int a mp digit b mp digit c These work like the full mp int capable variants except the sec ond parameter b is 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 kd Index mp add mp add d mp and mp clear mp clear_multi mp_cmp 25 mp cmp d mp cmp mag 23 mp div mp dr reduce mp dr setup MP_EQ mp error_to_string 9 mp expt d mp exptmod mp exteuclid mp gcd mp get_int mp grow 17 MP GT mp init mp init copy mp init multi mp init set mp init_set_int mp init size mp int mp invmod mp _jacobi mp lem mp lshd MP LT MP_MEM 9 mp mod mp mod d mp montgomery calc normalization mp montgomery reduce mp montgomery setup 42 mp mul mp mul 2 mp mul 2d mp mul d mp n root 48 mp neg BI B2 MPNO p MP OKAY P INDEX 59 mp or mp prime 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 mp read radix mp re
36. return EXIT FAILURE set the number to 5 mp_set amp number 5 switch mp_cmp_d amp number 7 case MP GT printf number gt case MP_EQ printf number case MP_LT printf number lt 1 NNN UU loz H o w p we re done with it mp clear number return EXIT_SUCCESS If this program functions properly it will print out the following number lt 7 28 CHAPTER 3 BASIC OPERATIONS 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 oper ation 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 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 stringl result return EXIT FAILURE set the number to 5 mp set knumber 5 multiply by two 3 3 LOGICAL OPERATIONS 29
37. s 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 range 0 lt a lt b Dimminished radix reductions are much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time Since the moduli are restricted this algorithm is not particu larly useful for something like Rabin RSA or BBS cryptographic purposes This reduction algorithm is useful for Diffie Hellman and ECC where fixed primes are acceptable Note that unlike Montgomery reduction there is no normaliza tion 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 lt B 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 a mp_digit d 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 Chapter 6 Exponentiation 6 1 Single Digit Exponentiation int mp_expt_d mp int x a mp digit b mp int c This computes c a using a simple binar
38. s a over to c 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 mp 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 Noe 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 0 To multiply by the polynomial g a 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 x 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 4 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 31 3 3 3 AND OR and XOR Operations While AND OR and XOR operations are not typical bignum functions they can be useful in several instances The three func tions are prototyped as follows int mp_or mp int a mp int b mp int x c int mp and mp int a mp int b mp
39. t won t be changed agair excess digits can be removed to return memory to the heap with the mp shrink function int mp shrink mp int x 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 com putations however such operations will reguire 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 4 printf Error initializing the number fs mp error to stringl result return EXIT FAILURE 2A Diffie Hellman modulus for instance 2 6 MAINTENANCE FUNCTIONS 17 use the number e g pre computation We re done with it for now if result mp_shrink amp number MP_OKAY 4 printf Error shrinking the number jis mp error to stringl result return EXIT FAILURE use it we re done with it mp clear amp 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 digits are significant that is contribute to the value of the mp int The al
40. the mp_in within LibTomMath This data type is used to organize all of the data required to manipulate 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 dig its 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 appropri ate macros 2 4 FUNCTION ORGANIZATION 11 All LTM functions that use the mp_int type will expect a pointer to mp int structure You must allocate memory to hold the struc ture itself by yourself whether off stack or heap it doesn t matter The very first thing that must be done to use an mp 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 a t b a k a a b d a mod b mp add ka Kb amp c c mp_mul a Ka dc c mp div ga amp b amp c ka c Another feature of the way the functions have been imple mented is that source operands can be destination operands as well For instance a t b a b c a mod b mp add ka kb amp b b mp div gka kb ka amp c x a
41. the following int mp_montgomery_setup mp_int a mp_digit mp For the given odd moduli a the precomputation value is placed in mp The reduction is computed with the following int mp_montgomery_reduce mp_int a mp_int m mp_digit This reduces a in place modulo m with the pre computed value mp a must be in the range 0 lt a lt b Montgomery reduction is faster than Barrett reduction for mod uli smaller than the comba limit With the default setup for instance the limit is 127 digits 3556 bits Note that this func tion 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 the n number of digits in m and is radix used default is 228 To quickly calculate R the following function was provided mp int mp montgomery calc normalization mp int a mp int b 5 3 MONTGOMERY REDUCTION 43 Which calculates a R for the odd moduli b without using multi plication 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 R c and set c t
42. to link your application against the shared object There is limited support for making a DLL in windows via the makefile 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 Windows program using Cygwin 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 2A copy of MPI is included in the package 4 CHAPTER 1 INTRODUCTION myprng mtest mtest test This will 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 depends
43. y left to right algorithm It is faster than repeated multiplications by a for all values of b greater than three 6 2 Modular Exponentiation int mp_exptmod mp_int G mp_int X mp_int P mp_int Y This computes Y G mod P using a variable width slid ing window algorithm This function will automatically detect the fastest modular reduction technique to use during the oper ation For negative values of X the operation is performed as Y G mod P mod P provided that gcd G P 1 This function is actually a shell around the two internal expo nentiation functions This routine will automatically detect when 47 48 CHAPTER 6 EXPONENTIATION 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 modu lar exponentiations Followed by Montgomery and the other two algorithms 6 3 Root Finding int mp_n_root mp_int x a mp digit b mp_int x c This computes c a such that c 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 I d 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 4 will return 2 whereas
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