buffa/out_take.js

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// Big Integer Library v. 5.5
// Created 2000, last modified 2013
// Leemon Baird
// www.leemon.com
//
// Version history:
// v 5.5  17 Mar 2013
//   - two lines of a form like "if (x<0) x+=n" had the "if" changed to "while" to 
//     handle the case when x<-n. (Thanks to James Ansell for finding that bug)
// v 5.4  3 Oct 2009
//   - added "var i" to greaterShift() so i is not global. (Thanks to PŽter Szab— for finding that bug)
//
// v 5.3  21 Sep 2009
//   - added randProbPrime(k) for probable primes
//   - unrolled loop in mont_ (slightly faster)
//   - millerRabin now takes a bigInt parameter rather than an int
//
// v 5.2  15 Sep 2009
//   - fixed capitalization in call to int2bigInt in randBigInt
//     (thanks to Emili Evripidou, Reinhold Behringer, and Samuel Macaleese for finding that bug)
//
// v 5.1  8 Oct 2007 
//   - renamed inverseModInt_ to inverseModInt since it doesn't change its parameters
//   - added functions GCD and randBigInt, which call GCD_ and randBigInt_
//   - fixed a bug found by Rob Visser (see comment with his name below)
//   - improved comments
//
// This file is public domain.   You can use it for any purpose without restriction.
// I do not guarantee that it is correct, so use it at your own risk.  If you use 
// it for something interesting, I'd appreciate hearing about it.  If you find 
// any bugs or make any improvements, I'd appreciate hearing about those too.
// It would also be nice if my name and URL were left in the comments.  But none 
// of that is required.
//
// This code defines a bigInt library for arbitrary-precision integers.
// A bigInt is an array of integers storing the value in chunks of bpe bits, 
// little endian (buff[0] is the least significant word).
// Negative bigInts are stored two's complement.  Almost all the functions treat
// bigInts as nonnegative.  The few that view them as two's complement say so
// in their comments.  Some functions assume their parameters have at least one 
// leading zero element. Functions with an underscore at the end of the name put
// their answer into one of the arrays passed in, and have unpredictable behavior 
// in case of overflow, so the caller must make sure the arrays are big enough to 
// hold the answer.  But the average user should never have to call any of the 
// underscored functions.  Each important underscored function has a wrapper function 
// of the same name without the underscore that takes care of the details for you.  
// For each underscored function where a parameter is modified, that same variable 
// must not be used as another argument too.  So, you cannot square x by doing 
// multMod_(x,x,n).  You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n).
// Or simply use the multMod(x,x,n) function without the underscore, where
// such issues never arise, because non-underscored functions never change
// their parameters; they always allocate new memory for the answer that is returned.
//
// These functions are designed to avoid frequent dynamic memory allocation in the inner loop.
// For most functions, if it needs a BigInt as a local variable it will actually use
// a global, and will only allocate to it only when it's not the right size.  This ensures
// that when a function is called repeatedly with same-sized parameters, it only allocates
// memory on the first call.
//
// Note that for cryptographic purposes, the calls to Math.random() must 
// be replaced with calls to a better pseudorandom number generator.
//
// In the following, "bigInt" means a bigInt with at least one leading zero element,
// and "integer" means a nonnegative integer less than radix.  In some cases, integer 
// can be negative.  Negative bigInts are 2s complement.
// 
// The following functions do not modify their inputs.
// Those returning a bigInt, string, or Array will dynamically allocate memory for that value.
// Those returning a boolean will return the integer 0 (false) or 1 (true).
// Those returning boolean or int will not allocate memory except possibly on the first 
// time they're called with a given parameter size.
// 
// bigInt  add(x,y)               //return (x+y) for bigInts x and y.  
// bigInt  addInt(x,n)            //return (x+n) where x is a bigInt and n is an integer.
// string  bigInt2str(x,base)     //return a string form of bigInt x in a given base, with 2 <= base <= 95
// int     bitSize(x)             //return how many bits long the bigInt x is, not counting leading zeros
// bigInt  dup(x)                 //return a copy of bigInt x
// boolean equals(x,y)            //is the bigInt x equal to the bigint y?
// boolean equalsInt(x,y)         //is bigint x equal to integer y?
// bigInt  expand(x,n)            //return a copy of x with at least n elements, adding leading zeros if needed
// Array   findPrimes(n)          //return array of all primes less than integer n
// bigInt  GCD(x,y)               //return greatest common divisor of bigInts x and y (each with same number of elements).
// boolean greater(x,y)           //is x>y?  (x and y are nonnegative bigInts)
// boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?
// bigInt  int2bigInt(t,n,m)      //return a bigInt equal to integer t, with at least n bits and m array elements
// bigInt  inverseMod(x,n)        //return (x**(-1) mod n) for bigInts x and n.  If no inverse exists, it returns null
// int     inverseModInt(x,n)     //return x**(-1) mod n, for integers x and n.  Return 0 if there is no inverse
// boolean isZero(x)              //is the bigInt x equal to zero?
// boolean millerRabin(x,b)       //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is bigInt, 1<b<x)
// boolean millerRabinInt(x,b)    //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is int,    1<b<x)
// bigInt  mod(x,n)               //return a new bigInt equal to (x mod n) for bigInts x and n.
// int     modInt(x,n)            //return x mod n for bigInt x and integer n.
// bigInt  mult(x,y)              //return x*y for bigInts x and y. This is faster when y<x.
// bigInt  multMod(x,y,n)         //return (x*y mod n) for bigInts x,y,n.  For greater speed, let y<x.
// boolean negative(x)            //is bigInt x negative?
// bigInt  powMod(x,y,n)          //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation.  0**0=1. Faster for odd n.
// bigInt  randBigInt(n,s)        //return an n-bit random BigInt (n>=1).  If s=1, then the most significant of those n bits is set to 1.
// bigInt  randTruePrime(k)       //return a new, random, k-bit, true prime bigInt using Maurer's algorithm.
// bigInt  randProbPrime(k)       //return a new, random, k-bit, probable prime bigInt (probability it's composite less than 2^-80).
// bigInt  str2bigInt(s,b,n,m)    //return a bigInt for number represented in string s in base b with at least n bits and m array elements
// bigInt  sub(x,y)               //return (x-y) for bigInts x and y.  Negative answers will be 2s complement
// bigInt  trim(x,k)              //return a copy of x with exactly k leading zero elements
//
//
// The following functions each have a non-underscored version, which most users should call instead.
// These functions each write to a single parameter, and the caller is responsible for ensuring the array 
// passed in is large enough to hold the result. 
//
// void    addInt_(x,n)          //do x=x+n where x is a bigInt and n is an integer
// void    add_(x,y)             //do x=x+y for bigInts x and y
// void    copy_(x,y)            //do x=y on bigInts x and y
// void    copyInt_(x,n)         //do x=n on bigInt x and integer n
// void    GCD_(x,y)             //set x to the greatest common divisor of bigInts x and y, (y is destroyed).  (This never overflows its array).
// boolean inverseMod_(x,n)      //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist
// void    mod_(x,n)             //do x=x mod n for bigInts x and n. (This never overflows its array).
// void    mult_(x,y)            //do x=x*y for bigInts x and y.
// void    multMod_(x,y,n)       //do x=x*y  mod n for bigInts x,y,n.
// void    powMod_(x,y,n)        //do x=x**y mod n, where x,y,n are bigInts (n is odd) and ** is exponentiation.  0**0=1.
// void    randBigInt_(b,n,s)    //do b = an n-bit random BigInt. if s=1, then nth bit (most significant bit) is set to 1. n>=1.
// void    randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.
// void    sub_(x,y)             //do x=x-y for bigInts x and y. Negative answers will be 2s complement.
//
// The following functions do NOT have a non-underscored version. 
// They each write a bigInt result to one or more parameters.  The caller is responsible for
// ensuring the arrays passed in are large enough to hold the results. 
//
// void addShift_(x,y,ys)       //do x=x+(y<<(ys*bpe))
// void carry_(x)               //do carries and borrows so each element of the bigInt x fits in bpe bits.
// void divide_(x,y,q,r)        //divide x by y giving quotient q and remainder r
// int  divInt_(x,n)            //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array).
// int  eGCD_(x,y,d,a,b)        //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y
// void halve_(x)               //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement.  (This never overflows its array).
// void leftShift_(x,n)         //left shift bigInt x by n bits.  n<bpe.
// void linComb_(x,y,a,b)       //do x=a*x+b*y for bigInts x and y and integers a and b
// void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys
// void mont_(x,y,n,np)         //Montgomery multiplication (see comments where the function is defined)
// void multInt_(x,n)           //do x=x*n where x is a bigInt and n is an integer.
// void rightShift_(x,n)        //right shift bigInt x by n bits.  0 <= n < bpe. (This never overflows its array).
// void squareMod_(x,n)         //do x=x*x  mod n for bigInts x,n
// void subShift_(x,y,ys)       //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement.
//
// The following functions are based on algorithms from the _Handbook of Applied Cryptography_
//    powMod_()           = algorithm 14.94, Montgomery exponentiation
//    eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_
//    GCD_()              = algorothm 14.57, Lehmer's algorithm
//    mont_()             = algorithm 14.36, Montgomery multiplication
//    divide_()           = algorithm 14.20  Multiple-precision division
//    squareMod_()        = algorithm 14.16  Multiple-precision squaring
//    randTruePrime_()    = algorithm  4.62, Maurer's algorithm
//    millerRabin()       = algorithm  4.24, Miller-Rabin algorithm
//
// Profiling shows:
//     randTruePrime_() spends:
//         10% of its time in calls to powMod_()
//         85% of its time in calls to millerRabin()
//     millerRabin() spends:
//         99% of its time in calls to powMod_()   (always with a base of 2)
//     powMod_() spends:
//         94% of its time in calls to mont_()  (almost always with x==y)
//
// This suggests there are several ways to speed up this library slightly:
//     - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window)
//         -- this should especially focus on being fast when raising 2 to a power mod n
//     - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test
//     - tune the parameters in randTruePrime_(), including c, m, and recLimit
//     - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking
//       within the loop when all the parameters are the same length.
//
// There are several ideas that look like they wouldn't help much at all:
//     - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway)
//     - increase bpe from 15 to 30 (that would help if we had a 32*32->64 multiplier, but not with JavaScript's 32*32->32)
//     - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square
//       followed by a Montgomery reduction.  The intermediate answer will be twice as long as x, so that
//       method would be slower.  This is unfortunate because the code currently spends almost all of its time
//       doing mont_(x,x,...), both for randTruePrime_() and powMod_().  A faster method for Montgomery squaring
//       would have a large impact on the speed of randTruePrime_() and powMod_().  HAC has a couple of poorly-worded
//       sentences that seem to imply it's faster to do a non-modular square followed by a single
//       Montgomery reduction, but that's obviously wrong.
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