2168 lines
58 KiB
TypeScript
2168 lines
58 KiB
TypeScript
'use strict'
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//@flow
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/** * * * * * * * * * *
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* Big Integer Library *
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* Created 2000 *
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* Leemon Baird *
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* www.leemon.com *
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* * * * * * * * * * * */
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////////////////////////////////////////////////////////////////////////////////////////
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// These functions are designed to avoid frequent dynamic memory allocation in the inner loop.
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// For most functions, if it needs a BigInt as a local variable it will actually use
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// a global, and will only allocate to it only when it's not the right size. This ensures
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// that when a function is called repeatedly with same-sized parameters, it only allocates
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// memory on the first call.
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//
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// Note that for cryptographic purposes, the calls to Math.random() must
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// be replaced with calls to a better pseudorandom number generator.
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//
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// In the following, "bigInt" means a bigInt with at least one leading zero element,
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// and "integer" means a nonnegative integer less than radix. In some cases, integer
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// can be negative. Negative bigInts are 2s complement.
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//
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// The following functions do not modify their inputs.
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// Those returning a bigInt, string, or Array will dynamically allocate memory for that value.
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// Those returning a boolean will return the integer 0 (false) or 1 (true).
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// Those returning boolean or int will not allocate memory except possibly on the first
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// time they're called with a given parameter size.
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//
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// bigInt add(x,y) //return (x+y) for bigInts x and y.
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// bigInt addInt(x,n) //return (x+n) where x is a bigInt and n is an integer.
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// string bigInt2str(x,base) //return a string form of bigInt x in a given base, with 2 <= base <= 95
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// int bitSize(x) //return how many bits long the bigInt x is, not counting leading zeros
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// bigInt dup(x) //return a copy of bigInt x
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// boolean equals(x,y) //is the bigInt x equal to the bigint y?
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// boolean equalsInt(x,y) //is bigint x equal to integer y?
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// bigInt expand(x,n) //return a copy of x with at least n elements, adding leading zeros if needed
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// Array findPrimes(n) //return array of all primes less than integer n
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// bigInt GCD(x,y) //return greatest common divisor of bigInts x and y (each with same number of elements).
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// boolean greater(x,y) //is x>y? (x and y are nonnegative bigInts)
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// boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?
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// bigInt int2bigInt(t,n,m) //return a bigInt equal to integer t, with at least n bits and m array elements
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// bigInt inverseMod(x,n) //return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns null
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// int inverseModInt(x,n) //return x**(-1) mod n, for integers x and n. Return 0 if there is no inverse
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// boolean isZero(x) //is the bigInt x equal to zero?
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// 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)
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// 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)
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// bigInt mod(x,n) //return a new bigInt equal to (x mod n) for bigInts x and n.
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// int modInt(x,n) //return x mod n for bigInt x and integer n.
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// bigInt mult(x,y) //return x*y for bigInts x and y. This is faster when y<x.
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// bigInt multMod(x,y,n) //return (x*y mod n) for bigInts x,y,n. For greater speed, let y<x.
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// boolean negative(x) //is bigInt x negative?
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// 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.
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// 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.
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// bigInt randTruePrime(k) //return a new, random, k-bit, true prime bigInt using Maurer's algorithm.
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// bigInt randProbPrime(k) //return a new, random, k-bit, probable prime bigInt (probability it's composite less than 2^-80).
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// 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
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// bigInt sub(x,y) //return (x-y) for bigInts x and y. Negative answers will be 2s complement
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// bigInt trim(x,k) //return a copy of x with exactly k leading zero elements
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//
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//
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// The following functions each have a non-underscored version, which most users should call instead.
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// These functions each write to a single parameter, and the caller is responsible for ensuring the array
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// passed in is large enough to hold the result.
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//
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// void addInt_(x,n) //do x=x+n where x is a bigInt and n is an integer
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// void add_(x,y) //do x=x+y for bigInts x and y
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// void copy_(x,y) //do x=y on bigInts x and y
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// void copyInt_(x,n) //do x=n on bigInt x and integer n
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// void GCD_(x,y) //set x to the greatest common divisor of bigInts x and y, (y is destroyed). (This never overflows its array).
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// boolean inverseMod_(x,n) //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist
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// void mod_(x,n) //do x=x mod n for bigInts x and n. (This never overflows its array).
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// void mult_(x,y) //do x=x*y for bigInts x and y.
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// void multMod_(x,y,n) //do x=x*y mod n for bigInts x,y,n.
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// 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.
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// 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.
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// void randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.
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// void sub_(x,y) //do x=x-y for bigInts x and y. Negative answers will be 2s complement.
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//
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// The following functions do NOT have a non-underscored version.
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// They each write a bigInt result to one or more parameters. The caller is responsible for
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// ensuring the arrays passed in are large enough to hold the results.
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//
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// void addShift_(x,y,ys) //do x=x+(y<<(ys*bpe))
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// void carry_(x) //do carries and borrows so each element of the bigInt x fits in bpe bits.
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// void divide_(x,y,q,r) //divide x by y giving quotient q and remainder r
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// int divInt_(x,n) //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array).
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// void eGCD_(x,y,d,a,b) //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y
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// void halve_(x) //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement. (This never overflows its array).
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// void leftShift_(x,n) //left shift bigInt x by n bits. n<bpe.
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// void linComb_(x,y,a,b) //do x=a*x+b*y for bigInts x and y and integers a and b
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// void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys
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// void mont_(x,y,n,np) //Montgomery multiplication (see comments where the function is defined)
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// void multInt_(x,n) //do x=x*n where x is a bigInt and n is an integer.
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// void rightShift_(x,n) //right shift bigInt x by n bits. 0 <= n < bpe. (This never overflows its array).
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// void squareMod_(x,n) //do x=x*x mod n for bigInts x,n
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// void subShift_(x,y,ys) //do x=x-(y<<(ys*bpe)). Negative answers will be 2s complement.
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//
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// The following functions are based on algorithms from the _Handbook of Applied Cryptography_
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// powMod_() = algorithm 14.94, Montgomery exponentiation
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// eGCD_,inverseMod_() = algorithm 14.61, Binary extended GCD_
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// GCD_() = algorothm 14.57, Lehmer's algorithm
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// mont_() = algorithm 14.36, Montgomery multiplication
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// divide_() = algorithm 14.20 Multiple-precision division
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// squareMod_() = algorithm 14.16 Multiple-precision squaring
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// randTruePrime_() = algorithm 4.62, Maurer's algorithm
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// millerRabin() = algorithm 4.24, Miller-Rabin algorithm
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//
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// Profiling shows:
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// randTruePrime_() spends:
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// 10% of its time in calls to powMod_()
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// 85% of its time in calls to millerRabin()
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// millerRabin() spends:
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// 99% of its time in calls to powMod_() (always with a base of 2)
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// powMod_() spends:
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// 94% of its time in calls to mont_() (almost always with x==y)
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//
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// This suggests there are several ways to speed up this library slightly:
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// - convert powMod_ to use a Montgomery form of k-ary window (or maybe a Montgomery form of sliding window)
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// -- this should especially focus on being fast when raising 2 to a power mod n
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// - convert randTruePrime_() to use a minimum r of 1/3 instead of 1/2 with the appropriate change to the test
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// - tune the parameters in randTruePrime_(), including c, m, and recLimit
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// - speed up the single loop in mont_() that takes 95% of the runtime, perhaps by reducing checking
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// within the loop when all the parameters are the same length.
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//
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// There are several ideas that look like they wouldn't help much at all:
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// - replacing trial division in randTruePrime_() with a sieve (that speeds up something taking almost no time anyway)
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// - 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)
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// - speeding up mont_(x,y,n,np) when x==y by doing a non-modular, non-Montgomery square
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// followed by a Montgomery reduction. The intermediate answer will be twice as long as x, so that
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// method would be slower. This is unfortunate because the code currently spends almost all of its time
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// doing mont_(x,x,...), both for randTruePrime_() and powMod_(). A faster method for Montgomery squaring
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// would have a large impact on the speed of randTruePrime_() and powMod_(). HAC has a couple of poorly-worded
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// sentences that seem to imply it's faster to do a non-modular square followed by a single
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// Montgomery reduction, but that's obviously wrong.
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////////////////////////////////////////////////////////////////////////////////////////
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export type Bool = 1 | 0
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//globals
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export var bpe = 0 //bits stored per array element
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var mask = 0 //AND this with an array element to chop it down to bpe bits
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var radix = mask + 1 //equals 2^bpe. A single 1 bit to the left of the last bit of mask.
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//the digits for converting to different bases
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var digitsStr =
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'0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'"+-'
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//initialize the global variables
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//bpe=number of bits in the mantissa on this platform
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for (bpe = 0; 1 << (bpe + 1) > 1 << bpe; bpe++);
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bpe >>= 1 //bpe=number of bits in one element of the array representing the bigInt
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mask = (1 << bpe) - 1 //AND the mask with an integer to get its bpe least significant bits
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radix = mask + 1 //2^bpe. a single 1 bit to the left of the first bit of mask
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export var one = int2bigInt(1, 1, 1) //constant used in powMod_()
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export var zero = int2bigInt(0, 1, 1)
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//the following global variables are scratchpad memory to
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//reduce dynamic memory allocation in the inner loop
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var t: number[] | number = new Array(0)
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var ss = t //used in mult_()
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var s0 = t //used in multMod_(), squareMod_()
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// var s1=t; //used in powMod_(), multMod_(), squareMod_()
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// var s2=t; //used in powMod_(), multMod_()
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var s3 = t //used in powMod_()
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var s4 = t,
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s5 = t //used in mod_()
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var s6 = t //used in bigInt2str()
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var s7 = t //used in powMod_()
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var T = t //used in GCD_()
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var sa = t //used in mont_()
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var mr_x1 = t,
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mr_r = t,
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mr_a = t, //used in millerRabin()
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eg_v = t,
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eg_u = t,
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eg_A = t,
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eg_B = t,
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eg_C = t,
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eg_D = t, //used in eGCD_(), inverseMod_()
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//, md_q1=t, md_q2=t, md_q3=t, md_r=t, md_r1=t, md_r2=t, md_tt=t, //used in mod_()
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primes = t,
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pows = t,
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s_i = t,
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s_i2 = t,
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s_R = t,
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s_rm = t,
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s_q = t,
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s_n1 = t,
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s_a = t,
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s_r2 = t,
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s_n = t,
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s_b = t,
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s_d = t,
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s_x1 = t,
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s_x2 = t,
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s_aa = t, //used in randTruePrime_()
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rpprb = t //used in randProbPrimeRounds() (which also uses "primes")
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////////////////////////////////////////////////////////////////////////////////////////
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var k, buff
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/**
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* return array of all primes less than integer n
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*
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* @param {number} n
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* @returns {number[]}
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*/
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export function findPrimes(n: number): number[] {
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var i, s, p, ans
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s = new Array(n)
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for (i = 0; i < n; i++) s[i] = 0
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s[0] = 2
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p = 0 //first p elements of s are primes, the rest are a sieve
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for (; s[p] < n; ) {
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//s[p] is the pth prime
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for (
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i = s[p] * s[p];
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i < n;
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i += s[p] //mark multiples of s[p]
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)
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s[i] = 1
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p++
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s[p] = s[p - 1] + 1
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for (; s[p] < n && s[s[p]]; s[p]++); //find next prime (where s[p]==0)
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}
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ans = new Array(p)
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for (i = 0; i < p; i++) ans[i] = s[i]
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return ans
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}
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/**
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* does a single round of Miller-Rabin base b consider x to be a possible prime?
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*
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* x is a bigInt, and b is an integer, with b<x
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*
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* @export
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* @param {number[]} x
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* @param {number} b
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* @returns {(0 | 1)}
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*/
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export function millerRabinInt(x: number[], b: number): Bool {
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if (mr_x1.length !== x.length) {
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mr_x1 = dup(x)
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mr_r = dup(x)
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mr_a = dup(x)
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}
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copyInt_(mr_a, b)
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return millerRabin(x, mr_a)
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}
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/**
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* does a single round of Miller-Rabin base b consider x to be a possible prime?
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*
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* x and b are bigInts with b<x
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*
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* @export
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* @param {number[]} x
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* @param {number[]} b
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* @returns {(0 | 1)}
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*/
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export function millerRabin(x: number[], b: number[]): Bool {
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var i, j, k, s
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if (mr_x1.length !== x.length) {
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mr_x1 = dup(x)
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mr_r = dup(x)
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mr_a = dup(x)
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}
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copy_(mr_a, b)
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copy_(mr_r, x)
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copy_(mr_x1, x)
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addInt_(mr_r, -1)
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addInt_(mr_x1, -1)
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//s=the highest power of two that divides mr_r
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k = 0
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for (i = 0; i < mr_r.length; i++)
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for (j = 1; j < mask; j <<= 1)
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if (x[i] & j) {
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s = k < mr_r.length + bpe ? k : 0
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i = mr_r.length
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j = mask
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} else k++
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if (s) rightShift_(mr_r, s)
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powMod_(mr_a, mr_r, x)
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if (!equalsInt(mr_a, 1) && !equals(mr_a, mr_x1)) {
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j = 1
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//$off
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while (j <= s - 1 && !equals(mr_a, mr_x1)) {
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squareMod_(mr_a, x)
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if (equalsInt(mr_a, 1)) {
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return 0
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}
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j++
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}
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if (!equals(mr_a, mr_x1)) {
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return 0
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}
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}
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return 1
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}
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/**
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* returns how many bits long the bigInt is, not counting leading zeros.
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*
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* @param {number[]} x
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* @returns {number}
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*/
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export function bitSize(x: number[]): number {
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var j, z, w
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for (j = x.length - 1; x[j] == 0 && j > 0; j--);
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for (z = 0, w = x[j]; w; w >>= 1, z++);
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z += bpe * j
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return z
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}
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/**
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* return a copy of x with at least n elements, adding leading zeros if needed
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*
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* @param {number[]} x
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* @param {number} n
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* @returns {number[]}
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*/
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export function expand(x: number[], n: number): number[] {
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var ans = int2bigInt(0, (x.length > n ? x.length : n) * bpe, 0)
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copy_(ans, x)
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return ans
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}
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/**
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* return a k-bit true random prime using Maurer's algorithm.
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*
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* @export
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* @param {number} k
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* @returns {number[]}
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*/
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/* export function randTruePrime(k: number): number[] {
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var ans = int2bigInt(0, k, 0)
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randTruePrime_(ans, k)
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return trim(ans, 1)
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} */
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/**
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* return a k-bit random probable prime with probability of error < 2^-80
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*
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* @export
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* @param {number} k
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* @returns {number[]}
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*/
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/* export function randProbPrime(k: number): number[] {
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if (k >= 600) return randProbPrimeRounds(k, 2) //numbers from HAC table 4.3
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if (k >= 550) return randProbPrimeRounds(k, 4)
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if (k >= 500) return randProbPrimeRounds(k, 5)
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if (k >= 400) return randProbPrimeRounds(k, 6)
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if (k >= 350) return randProbPrimeRounds(k, 7)
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if (k >= 300) return randProbPrimeRounds(k, 9)
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if (k >= 250) return randProbPrimeRounds(k, 12) //numbers from HAC table 4.4
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if (k >= 200) return randProbPrimeRounds(k, 15)
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if (k >= 150) return randProbPrimeRounds(k, 18)
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if (k >= 100) return randProbPrimeRounds(k, 27)
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return randProbPrimeRounds(k, 40) //number from HAC remark 4.26 (only an estimate)
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} */
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/**
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* return a k-bit probable random prime using n rounds of Miller Rabin
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* (after trial division with small primes)
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*
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* @export
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* @param {number} k
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* @param {number} n
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* @returns {number[]}
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*/
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/* export function randProbPrimeRounds(k: number, n: number): number[] {
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var ans, i, divisible, B
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B = 30000 //B is largest prime to use in trial division
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ans = int2bigInt(0, k, 0)
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//optimization: try larger and smaller B to find the best limit.
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if (primes.length === 0) primes = findPrimes(30000) //check for divisibility by primes <=30000
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if (rpprb.length !== ans.length) rpprb = dup(ans)
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for (;;) {
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//keep trying random values for ans until one appears to be prime
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//optimization: pick a random number times L=2*3*5*...*p, plus a
|
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// random element of the list of all numbers in [0,L) not divisible by any prime up to p.
|
|
// This can reduce the amount of random number generation.
|
|
|
|
randBigInt_(ans, k, 0) //ans = a random odd number to check
|
|
ans[0] |= 1
|
|
divisible = 0
|
|
|
|
//check ans for divisibility by small primes up to B
|
|
for (i = 0; i < primes.length && primes[i] <= B; i++)
|
|
if (modInt(ans, primes[i]) === 0 && !equalsInt(ans, primes[i])) {
|
|
divisible = 1
|
|
break
|
|
}
|
|
|
|
//optimization: change millerRabin so the base can be bigger than the number being checked, then eliminate the while here.
|
|
|
|
//do n rounds of Miller Rabin, with random bases less than ans
|
|
for (i = 0; i < n && !divisible; i++) {
|
|
randBigInt_(rpprb, k, 0)
|
|
while (
|
|
!greater(ans, rpprb) //pick a random rpprb that's < ans
|
|
)
|
|
randBigInt_(rpprb, k, 0)
|
|
if (!millerRabin(ans, rpprb)) divisible = 1
|
|
}
|
|
|
|
if (!divisible) return ans
|
|
}
|
|
} */
|
|
|
|
/**
|
|
* return a new bigInt equal to (x mod n) for bigInts x and n.
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} n
|
|
* @returns {number[]}
|
|
*/
|
|
export function mod(x: number[], n: number[]): number[] {
|
|
var ans = dup(x)
|
|
mod_(ans, n)
|
|
return trim(ans, 1)
|
|
}
|
|
|
|
/**
|
|
* return (x+n) where x is a bigInt and n is an integer.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
* @returns {number[]}
|
|
*/
|
|
/* export function addInt(x: number[], n: number): number[] {
|
|
var ans = expand(x, x.length + 1)
|
|
addInt_(ans, n)
|
|
return trim(ans, 1)
|
|
} */
|
|
|
|
/**
|
|
* return x*y for bigInts x and y. This is faster when y<x.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @returns {number[]}
|
|
*/
|
|
export function mult(x: number[], y: number[]): number[] {
|
|
var ans = expand(x, x.length + y.length)
|
|
mult_(ans, y)
|
|
return trim(ans, 1)
|
|
}
|
|
|
|
/**
|
|
* return (x**y mod n) where x,y,n are bigInts and ** is exponentiation.
|
|
*
|
|
* 0**0=1.
|
|
*
|
|
* Faster for odd n.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number[]} n
|
|
* @returns {number[]}
|
|
*/
|
|
export function powMod(x: number[], y: number[], n: number[]): number[] {
|
|
var ans = expand(x, n.length)
|
|
powMod_(
|
|
//this should work without the trim, but doesn't
|
|
ans,
|
|
trim(y, 2),
|
|
trim(n, 2),
|
|
)
|
|
return trim(ans, 1)
|
|
}
|
|
|
|
/**
|
|
* Simple pow with no optimizations (in 40x times slower than jsbn's pow)
|
|
* @param x bigInt
|
|
* @param e
|
|
*/
|
|
export function pow(x: number[], e: number) {
|
|
let ans = dup(x);
|
|
e -= 1;
|
|
for(let i = 0; i < e; ++i) {
|
|
ans = mult(ans, x);
|
|
}
|
|
return trim(ans, 1);
|
|
}
|
|
|
|
/**
|
|
* return (x-y) for bigInts x and y
|
|
*
|
|
* Negative answers will be 2s complement
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @returns {number[]}
|
|
*/
|
|
export function sub(x: number[], y: number[]): number[] {
|
|
var ans = expand(x, x.length > y.length ? x.length + 1 : y.length + 1)
|
|
sub_(ans, y)
|
|
return trim(ans, 1)
|
|
}
|
|
|
|
/**
|
|
* return (x+y) for bigInts x and y
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @returns {number[]}
|
|
*/
|
|
export function add(x: number[], y: number[]): number[] {
|
|
var ans = expand(x, x.length > y.length ? x.length + 1 : y.length + 1)
|
|
add_(ans, y)
|
|
return trim(ans, 1)
|
|
}
|
|
|
|
/**
|
|
* return (x**(-1) mod n) for bigInts x and n.
|
|
*
|
|
* If no inverse exists, it returns null
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} n
|
|
* @returns {(number[] | null)}
|
|
*/
|
|
/* export function inverseMod(x: number[], n: number[]): number[] | null {
|
|
var ans = expand(x, n.length)
|
|
var s = inverseMod_(ans, n)
|
|
return s ? trim(ans, 1) : null
|
|
} */
|
|
|
|
/**
|
|
* return (x*y mod n) for bigInts x,y,n.
|
|
*
|
|
* For greater speed, let y<x.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number[]} n
|
|
* @returns {number[]}
|
|
*/
|
|
export function multMod(x: number[], y: number[], n: number[]): number[] {
|
|
var ans = expand(x, n.length)
|
|
multMod_(ans, y, n)
|
|
return trim(ans, 1)
|
|
}
|
|
|
|
/**
|
|
* generate a k-bit true random prime using Maurer's algorithm, and put it into ans.
|
|
*
|
|
* The bigInt ans must be large enough to hold it.
|
|
*
|
|
* @export
|
|
* @param {number[]} ans
|
|
* @param {number} k
|
|
* @return {void}
|
|
*/
|
|
/* export function randTruePrime_(ans: number[], k: number): void {
|
|
var c, m, pm, dd, j, r, B, divisible, z, zz, recSize
|
|
var w
|
|
if (primes.length == 0) primes = findPrimes(30000) //check for divisibility by primes <=30000
|
|
|
|
if (pows.length == 0) {
|
|
pows = new Array(512)
|
|
for (j = 0; j < 512; j++) {
|
|
pows[j] = Math.pow(2, j / 511 - 1)
|
|
}
|
|
}
|
|
|
|
//c and m should be tuned for a particular machine and value of k, to maximize speed
|
|
c = 0.1 //c=0.1 in HAC
|
|
m = 20 //generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits
|
|
var recLimit = 20 //stop recursion when k <=recLimit. Must have recLimit >= 2
|
|
|
|
if (s_i2.length != ans.length) {
|
|
s_i2 = dup(ans)
|
|
s_R = dup(ans)
|
|
s_n1 = dup(ans)
|
|
s_r2 = dup(ans)
|
|
s_d = dup(ans)
|
|
s_x1 = dup(ans) //TODO Seems like a bug in eslint, reports as unused
|
|
s_x2 = dup(ans)
|
|
s_b = dup(ans)
|
|
s_n = dup(ans)
|
|
s_i = dup(ans)
|
|
s_rm = dup(ans)
|
|
s_q = dup(ans)
|
|
s_a = dup(ans)
|
|
s_aa = dup(ans)
|
|
}
|
|
|
|
if (k <= recLimit) {
|
|
//generate small random primes by trial division up to its square root
|
|
pm = (1 << ((k + 2) >> 1)) - 1 //pm is binary number with all ones, just over sqrt(2^k)
|
|
copyInt_(ans, 0)
|
|
for (dd = 1; dd; ) {
|
|
dd = 0
|
|
ans[0] = 1 | (1 << (k - 1)) | Math.floor(Math.random() * (1 << k)) //random, k-bit, odd integer, with msb 1
|
|
for (j = 1; j < primes.length && (primes[j] & pm) == primes[j]; j++) {
|
|
//trial division by all primes 3...sqrt(2^k)
|
|
if (0 == ans[0] % primes[j]) {
|
|
dd = 1
|
|
break
|
|
}
|
|
}
|
|
}
|
|
carry_(ans)
|
|
return
|
|
}
|
|
|
|
B = c * k * k //try small primes up to B (or all the primes[] array if the largest is less than B).
|
|
if (k > 2 * m)
|
|
//generate this k-bit number by first recursively generating a number that has between k/2 and k-m bits
|
|
for (r = 1; k - k * r <= m; ) r = pows[Math.floor(Math.random() * 512)] //r=Math.pow(2,Math.random()-1);
|
|
else r = 0.5
|
|
|
|
//simulation suggests the more complex algorithm using r=.333 is only slightly faster.
|
|
|
|
recSize = Math.floor(r * k) + 1
|
|
|
|
randTruePrime_(s_q, recSize)
|
|
copyInt_(s_i2, 0)
|
|
s_i2[Math.floor((k - 2) / bpe)] |= 1 << ((k - 2) % bpe) //s_i2=2^(k-2)
|
|
divide_(s_i2, s_q, s_i, s_rm) //s_i=floor((2^(k-1))/(2q))
|
|
|
|
z = bitSize(s_i)
|
|
|
|
for (;;) {
|
|
for (;;) {
|
|
//generate z-bit numbers until one falls in the range [0,s_i-1]
|
|
randBigInt_(s_R, z, 0)
|
|
if (greater(s_i, s_R)) break
|
|
} //now s_R is in the range [0,s_i-1]
|
|
addInt_(s_R, 1) //now s_R is in the range [1,s_i]
|
|
add_(s_R, s_i) //now s_R is in the range [s_i+1,2*s_i]
|
|
|
|
copy_(s_n, s_q)
|
|
mult_(s_n, s_R)
|
|
multInt_(s_n, 2)
|
|
addInt_(s_n, 1) //s_n=2*s_R*s_q+1
|
|
|
|
copy_(s_r2, s_R)
|
|
multInt_(s_r2, 2) //s_r2=2*s_R
|
|
|
|
//check s_n for divisibility by small primes up to B
|
|
for (divisible = 0, j = 0; j < primes.length && primes[j] < B; j++)
|
|
if (modInt(s_n, primes[j]) == 0 && !equalsInt(s_n, primes[j])) {
|
|
divisible = 1
|
|
break
|
|
}
|
|
|
|
if (!divisible)
|
|
if (!millerRabinInt(s_n, 2))
|
|
//if it passes small primes check, then try a single Miller-Rabin base 2
|
|
//this line represents 75% of the total runtime for randTruePrime_
|
|
divisible = 1
|
|
|
|
if (!divisible) {
|
|
//if it passes that test, continue checking s_n
|
|
addInt_(s_n, -3)
|
|
for (j = s_n.length - 1; s_n[j] == 0 && j > 0; j--); //strip leading zeros
|
|
for (zz = 0, w = s_n[j]; w; w >>= 1, zz++);
|
|
zz += bpe * j //zz=number of bits in s_n, ignoring leading zeros
|
|
for (;;) {
|
|
//generate z-bit numbers until one falls in the range [0,s_n-1]
|
|
randBigInt_(s_a, zz, 0)
|
|
if (greater(s_n, s_a)) break
|
|
} //now s_a is in the range [0,s_n-1]
|
|
addInt_(s_n, 3) //now s_a is in the range [0,s_n-4]
|
|
addInt_(s_a, 2) //now s_a is in the range [2,s_n-2]
|
|
copy_(s_b, s_a)
|
|
copy_(s_n1, s_n)
|
|
addInt_(s_n1, -1)
|
|
powMod_(s_b, s_n1, s_n) //s_b=s_a^(s_n-1) modulo s_n
|
|
addInt_(s_b, -1)
|
|
if (isZero(s_b)) {
|
|
copy_(s_b, s_a)
|
|
powMod_(s_b, s_r2, s_n)
|
|
addInt_(s_b, -1)
|
|
copy_(s_aa, s_n)
|
|
copy_(s_d, s_b)
|
|
GCD_(s_d, s_n) //if s_b and s_n are relatively prime, then s_n is a prime
|
|
if (equalsInt(s_d, 1)) {
|
|
copy_(ans, s_aa)
|
|
return //if we've made it this far, then s_n is absolutely guaranteed to be prime
|
|
}
|
|
}
|
|
}
|
|
}
|
|
} */
|
|
|
|
/**
|
|
* Return an n-bit random BigInt (n>=1). If s=1, then the most significant of those n bits is set to 1.
|
|
*
|
|
* @export
|
|
* @param {number} n
|
|
* @param {number} s
|
|
* @returns {number[]}
|
|
*/
|
|
/* export function randBigInt(n: number, s: number): number[] {
|
|
var a, b
|
|
a = Math.floor((n - 1) / bpe) + 2 //# array elements to hold the BigInt with a leading 0 element
|
|
b = int2bigInt(0, 0, a)
|
|
randBigInt_(b, n, s)
|
|
return b
|
|
} */
|
|
|
|
/**
|
|
* Set b to an n-bit random BigInt. If s=1, then the most significant of those n bits is set to 1.
|
|
*
|
|
* Array b must be big enough to hold the result. Must have n>=1
|
|
*
|
|
* @export
|
|
* @param {number[]} b
|
|
* @param {number} n
|
|
* @param {number} s
|
|
* @return {void}
|
|
*/
|
|
/* export function randBigInt_(b: number[], n: number, s: number): void {
|
|
var i, a
|
|
for (i = 0; i < b.length; i++) b[i] = 0
|
|
a = Math.floor((n - 1) / bpe) + 1 //# array elements to hold the BigInt
|
|
for (i = 0; i < a; i++) {
|
|
b[i] = Math.floor(Math.random() * (1 << (bpe - 1)))
|
|
}
|
|
b[a - 1] &= (2 << ((n - 1) % bpe)) - 1
|
|
if (s == 1) b[a - 1] |= 1 << ((n - 1) % bpe)
|
|
} */
|
|
|
|
/**
|
|
* Return the greatest common divisor of bigInts x and y (each with same number of elements).
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @returns {number[]}
|
|
*/
|
|
export function GCD(x: number[], y: number[]): number[] {
|
|
var xc, yc
|
|
xc = dup(x)
|
|
yc = dup(y)
|
|
GCD_(xc, yc)
|
|
return xc
|
|
}
|
|
|
|
/**
|
|
* set x to the greatest common divisor of bigInts x and y (each with same number of elements).
|
|
*
|
|
* y is destroyed.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
*/
|
|
export function GCD_(x: number[], y: number[]): void {
|
|
var i: number, xp: number, yp: number, A: number, B, C: number, D: number, q, sing
|
|
var qp
|
|
if (T.length !== x.length) T = dup(x)
|
|
|
|
sing = 1
|
|
while (sing) {
|
|
//while y has nonzero elements other than y[0]
|
|
sing = 0
|
|
for (
|
|
i = 1;
|
|
i < y.length;
|
|
i++ //check if y has nonzero elements other than 0
|
|
)
|
|
if (y[i]) {
|
|
sing = 1
|
|
break
|
|
}
|
|
if (!sing) break //quit when y all zero elements except possibly y[0]
|
|
|
|
for (i = x.length; !x[i] && i >= 0; i--); //find most significant element of x
|
|
xp = x[i]
|
|
yp = y[i]
|
|
A = 1
|
|
B = 0
|
|
C = 0
|
|
D = 1
|
|
while (yp + C && yp + D) {
|
|
q = Math.floor((xp + A) / (yp + C))
|
|
qp = Math.floor((xp + B) / (yp + D))
|
|
if (q != qp) break
|
|
t = A - q * C
|
|
A = C
|
|
C = t // do (A,B,xp, C,D,yp) = (C,D,yp, A,B,xp) - q*(0,0,0, C,D,yp)
|
|
t = B - q * D
|
|
B = D
|
|
D = t
|
|
t = xp - q * yp
|
|
xp = yp
|
|
yp = t
|
|
}
|
|
if (B) {
|
|
copy_(T, x)
|
|
linComb_(x, y, A, B) //x=A*x+B*y
|
|
linComb_(y, T, D, C) //y=D*y+C*T
|
|
} else {
|
|
mod_(x, y)
|
|
copy_(T, x)
|
|
copy_(x, y)
|
|
copy_(y, T)
|
|
}
|
|
}
|
|
if (y[0] === 0) return
|
|
t = modInt(x, y[0])
|
|
copyInt_(x, y[0])
|
|
y[0] = t
|
|
while (y[0]) {
|
|
x[0] %= y[0]
|
|
t = x[0]
|
|
x[0] = y[0]
|
|
y[0] = t
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x**(-1) mod n, for bigInts x and n.
|
|
*
|
|
* If no inverse exists, it sets x to zero and returns 0, else it returns 1.
|
|
* The x array must be at least as large as the n array.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} n
|
|
* @returns {(0 | 1)}
|
|
*/
|
|
/* export function inverseMod_(x: number[], n: number[]): Bool {
|
|
var k = 1 + 2 * Math.max(x.length, n.length)
|
|
|
|
if (!(x[0] & 1) && !(n[0] & 1)) {
|
|
//if both inputs are even, then inverse doesn't exist
|
|
copyInt_(x, 0)
|
|
return 0
|
|
}
|
|
|
|
if (eg_u.length != k) {
|
|
eg_u = new Array(k)
|
|
eg_v = new Array(k)
|
|
eg_A = new Array(k)
|
|
eg_B = new Array(k)
|
|
eg_C = new Array(k)
|
|
eg_D = new Array(k)
|
|
}
|
|
|
|
copy_(eg_u, x)
|
|
copy_(eg_v, n)
|
|
copyInt_(eg_A, 1)
|
|
copyInt_(eg_B, 0)
|
|
copyInt_(eg_C, 0)
|
|
copyInt_(eg_D, 1)
|
|
for (;;) {
|
|
while (!(eg_u[0] & 1)) {
|
|
//while eg_u is even
|
|
halve_(eg_u)
|
|
if (!(eg_A[0] & 1) && !(eg_B[0] & 1)) {
|
|
//if eg_A==eg_B==0 mod 2
|
|
halve_(eg_A)
|
|
halve_(eg_B)
|
|
} else {
|
|
add_(eg_A, n)
|
|
halve_(eg_A)
|
|
sub_(eg_B, x)
|
|
halve_(eg_B)
|
|
}
|
|
}
|
|
|
|
while (!(eg_v[0] & 1)) {
|
|
//while eg_v is even
|
|
halve_(eg_v)
|
|
if (!(eg_C[0] & 1) && !(eg_D[0] & 1)) {
|
|
//if eg_C==eg_D==0 mod 2
|
|
halve_(eg_C)
|
|
halve_(eg_D)
|
|
} else {
|
|
add_(eg_C, n)
|
|
halve_(eg_C)
|
|
sub_(eg_D, x)
|
|
halve_(eg_D)
|
|
}
|
|
}
|
|
|
|
if (!greater(eg_v, eg_u)) {
|
|
//eg_v <= eg_u
|
|
sub_(eg_u, eg_v)
|
|
sub_(eg_A, eg_C)
|
|
sub_(eg_B, eg_D)
|
|
} else {
|
|
//eg_v > eg_u
|
|
sub_(eg_v, eg_u)
|
|
sub_(eg_C, eg_A)
|
|
sub_(eg_D, eg_B)
|
|
}
|
|
|
|
if (equalsInt(eg_u, 0)) {
|
|
while (
|
|
negative(eg_C) //make sure answer is nonnegative
|
|
)
|
|
add_(eg_C, n)
|
|
copy_(x, eg_C)
|
|
|
|
if (!equalsInt(eg_v, 1)) {
|
|
//if GCD_(x,n)!=1, then there is no inverse
|
|
copyInt_(x, 0)
|
|
return 0
|
|
}
|
|
return 1
|
|
}
|
|
}
|
|
} */
|
|
|
|
/**
|
|
* return x**(-1) mod n, for integers x and n.
|
|
*
|
|
* Return 0 if there is no inverse
|
|
*
|
|
* @param {number} x
|
|
* @param {number} n
|
|
* @returns {number}
|
|
*/
|
|
export function inverseModInt(x: number, n: number): number {
|
|
var a = 1,
|
|
b = 0,
|
|
t
|
|
for (;;) {
|
|
if (x === 1) return a
|
|
if (x === 0) return 0
|
|
b -= a * Math.floor(n / x)
|
|
//$off
|
|
n %= x
|
|
|
|
if (n === 1) return b //to avoid negatives, change this b to n-b, and each -= to +=
|
|
if (n === 0) return 0
|
|
a -= b * Math.floor(x / n)
|
|
//$off
|
|
x %= n
|
|
}
|
|
/*::
|
|
declare var never: empty
|
|
return never
|
|
*/
|
|
}
|
|
|
|
//this deprecated function is for backward compatibility only.
|
|
/* function inverseModInt_(x: number, n: number) {
|
|
return inverseModInt(x, n)
|
|
} */
|
|
|
|
/**
|
|
* Given positive bigInts x and y, change the bigints v, a, and b to positive bigInts such that:
|
|
*
|
|
* v = GCD_(x,y) = a*x-b*y
|
|
*
|
|
* The bigInts v, a, b, must have exactly as many elements as the larger of x and y.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number[]} v
|
|
* @param {number[]} a
|
|
* @param {number[]} b
|
|
* @return {void}
|
|
*/
|
|
export function eGCD_(
|
|
x: number[],
|
|
y: number[],
|
|
v: number[],
|
|
a: number[],
|
|
b: number[],
|
|
): void {
|
|
var g = 0
|
|
var k = Math.max(x.length, y.length)
|
|
if (eg_u.length != k) {
|
|
eg_u = new Array(k)
|
|
eg_A = new Array(k)
|
|
eg_B = new Array(k)
|
|
eg_C = new Array(k)
|
|
eg_D = new Array(k)
|
|
}
|
|
while (!(x[0] & 1) && !(y[0] & 1)) {
|
|
//while x and y both even
|
|
halve_(x)
|
|
halve_(y)
|
|
g++
|
|
}
|
|
copy_(eg_u, x)
|
|
copy_(v, y)
|
|
copyInt_(eg_A, 1)
|
|
copyInt_(eg_B, 0)
|
|
copyInt_(eg_C, 0)
|
|
copyInt_(eg_D, 1)
|
|
for (;;) {
|
|
while (!(eg_u[0] & 1)) {
|
|
//while u is even
|
|
halve_(eg_u)
|
|
if (!(eg_A[0] & 1) && !(eg_B[0] & 1)) {
|
|
//if A==B==0 mod 2
|
|
halve_(eg_A)
|
|
halve_(eg_B)
|
|
} else {
|
|
add_(eg_A, y)
|
|
halve_(eg_A)
|
|
sub_(eg_B, x)
|
|
halve_(eg_B)
|
|
}
|
|
}
|
|
|
|
while (!(v[0] & 1)) {
|
|
//while v is even
|
|
halve_(v)
|
|
if (!(eg_C[0] & 1) && !(eg_D[0] & 1)) {
|
|
//if C==D==0 mod 2
|
|
halve_(eg_C)
|
|
halve_(eg_D)
|
|
} else {
|
|
add_(eg_C, y)
|
|
halve_(eg_C)
|
|
sub_(eg_D, x)
|
|
halve_(eg_D)
|
|
}
|
|
}
|
|
|
|
if (!greater(v, eg_u)) {
|
|
//v<=u
|
|
sub_(eg_u, v)
|
|
sub_(eg_A, eg_C)
|
|
sub_(eg_B, eg_D)
|
|
} else {
|
|
//v>u
|
|
sub_(v, eg_u)
|
|
sub_(eg_C, eg_A)
|
|
sub_(eg_D, eg_B)
|
|
}
|
|
if (equalsInt(eg_u, 0)) {
|
|
while (negative(eg_C)) {
|
|
//make sure a (C) is nonnegative
|
|
add_(eg_C, y)
|
|
sub_(eg_D, x)
|
|
}
|
|
multInt_(eg_D, -1) ///make sure b (D) is nonnegative
|
|
copy_(a, eg_C)
|
|
copy_(b, eg_D)
|
|
leftShift_(v, g)
|
|
return
|
|
}
|
|
}
|
|
}
|
|
|
|
/**
|
|
* is bigInt x negative?
|
|
*
|
|
* @param {number[]} x
|
|
* @returns {(1 | 0)}
|
|
*/
|
|
export function negative(x: number[]) {
|
|
//TODO Flow Bool type inference
|
|
return (x[x.length - 1] >> (bpe - 1)) & 1
|
|
}
|
|
|
|
/**
|
|
* is (x << (shift*bpe)) > y?
|
|
*
|
|
* x and y are nonnegative bigInts
|
|
* shift is a nonnegative integer
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number} shift
|
|
* @returns {(1 | 0)}
|
|
*/
|
|
export function greaterShift(x: number[], y: number[], shift: number): Bool {
|
|
var i,
|
|
kx = x.length,
|
|
ky = y.length
|
|
k = kx + shift < ky ? kx + shift : ky
|
|
for (i = ky - 1 - shift; i < kx && i >= 0; i++) if (x[i] > 0) return 1 //if there are nonzeros in x to the left of the first column of y, then x is bigger
|
|
for (i = kx - 1 + shift; i < ky; i++) if (y[i] > 0) return 0 //if there are nonzeros in y to the left of the first column of x, then x is not bigger
|
|
for (i = k - 1; i >= shift; i--)
|
|
if (x[i - shift] > y[i]) return 1
|
|
else if (x[i - shift] < y[i]) return 0
|
|
return 0
|
|
}
|
|
|
|
/**
|
|
* is x > y?
|
|
*
|
|
* x and y both nonnegative
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @returns {(1 | 0)}
|
|
*/
|
|
export function greater(x: number[], y: number[]): Bool {
|
|
var i
|
|
var k = x.length < y.length ? x.length : y.length
|
|
|
|
for (i = x.length; i < y.length; i++) if (y[i]) return 0 //y has more digits
|
|
|
|
for (i = y.length; i < x.length; i++) if (x[i]) return 1 //x has more digits
|
|
|
|
for (i = k - 1; i >= 0; i--)
|
|
if (x[i] > y[i]) return 1
|
|
else if (x[i] < y[i]) return 0
|
|
return 0
|
|
}
|
|
|
|
/**
|
|
* divide x by y giving quotient q and remainder r.
|
|
*
|
|
* q = floor(x/y)
|
|
* r = x mod y
|
|
*
|
|
* All 4 are bigints.
|
|
*
|
|
* * x must have at least one leading zero element.
|
|
* * y must be nonzero.
|
|
* * q and r must be arrays that are exactly the same length as x. (Or q can have more).
|
|
* * Must have x.length >= y.length >= 2.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number[]} q
|
|
* @param {number[]} r
|
|
* @return {void}
|
|
*/
|
|
export function divide_(
|
|
x: number[],
|
|
y: number[],
|
|
q: number[],
|
|
r: number[],
|
|
): void {
|
|
var kx, ky
|
|
var i, j, y1, y2, c, a, b
|
|
copy_(r, x)
|
|
for (ky = y.length; y[ky - 1] === 0; ky--); //ky is number of elements in y, not including leading zeros
|
|
|
|
//normalize: ensure the most significant element of y has its highest bit set
|
|
b = y[ky - 1]
|
|
for (a = 0; b; a++) b >>= 1
|
|
a = bpe - a //a is how many bits to shift so that the high order bit of y is leftmost in its array element
|
|
leftShift_(y, a) //multiply both by 1<<a now, then divide both by that at the end
|
|
leftShift_(r, a)
|
|
|
|
//Rob Visser discovered a bug: the following line was originally just before the normalization.
|
|
for (kx = r.length; r[kx - 1] === 0 && kx > ky; kx--); //kx is number of elements in normalized x, not including leading zeros
|
|
|
|
copyInt_(q, 0) // q=0
|
|
while (!greaterShift(y, r, kx - ky)) {
|
|
// while (leftShift_(y,kx-ky) <= r) {
|
|
subShift_(r, y, kx - ky) // r=r-leftShift_(y,kx-ky)
|
|
q[kx - ky]++ // q[kx-ky]++;
|
|
} // }
|
|
|
|
for (i = kx - 1; i >= ky; i--) {
|
|
if (r[i] == y[ky - 1]) q[i - ky] = mask
|
|
else q[i - ky] = Math.floor((r[i] * radix + r[i - 1]) / y[ky - 1])
|
|
|
|
//The following for(;;) loop is equivalent to the commented while loop,
|
|
//except that the uncommented version avoids overflow.
|
|
//The commented loop comes from HAC, which assumes r[-1]==y[-1]==0
|
|
// while (q[i-ky]*(y[ky-1]*radix+y[ky-2]) > r[i]*radix*radix+r[i-1]*radix+r[i-2])
|
|
// q[i-ky]--;
|
|
for (;;) {
|
|
y2 = (ky > 1 ? y[ky - 2] : 0) * q[i - ky]
|
|
c = y2 >> bpe
|
|
y2 = y2 & mask
|
|
y1 = c + q[i - ky] * y[ky - 1]
|
|
c = y1 >> bpe
|
|
y1 = y1 & mask
|
|
|
|
if (
|
|
c == r[i]
|
|
? y1 == r[i - 1] ? y2 > (i > 1 ? r[i - 2] : 0) : y1 > r[i - 1]
|
|
: c > r[i]
|
|
)
|
|
q[i - ky]--
|
|
else break
|
|
}
|
|
|
|
linCombShift_(r, y, -q[i - ky], i - ky) //r=r-q[i-ky]*leftShift_(y,i-ky)
|
|
if (negative(r)) {
|
|
addShift_(r, y, i - ky) //r=r+leftShift_(y,i-ky)
|
|
q[i - ky]--
|
|
}
|
|
}
|
|
|
|
rightShift_(y, a) //undo the normalization step
|
|
rightShift_(r, a) //undo the normalization step
|
|
}
|
|
|
|
/**
|
|
* do carries and borrows so each element of the bigInt x fits in bpe bits.
|
|
*
|
|
* @param {number[]} x
|
|
*/
|
|
export function carry_(x: number[]): void {
|
|
var i, k, c, b
|
|
k = x.length
|
|
c = 0
|
|
for (i = 0; i < k; i++) {
|
|
c += x[i]
|
|
b = 0
|
|
if (c < 0) {
|
|
b = -(c >> bpe)
|
|
c += b * radix
|
|
}
|
|
x[i] = c & mask
|
|
c = (c >> bpe) - b
|
|
}
|
|
}
|
|
|
|
/**
|
|
* return x mod n for bigInt x and integer n.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
* @returns {number}
|
|
*/
|
|
export function modInt(x: number[], n: number): number {
|
|
var i,
|
|
c = 0
|
|
for (i = x.length - 1; i >= 0; i--) c = (c * radix + x[i]) % n
|
|
return c
|
|
}
|
|
|
|
/**
|
|
* convert the integer t into a bigInt with at least the given number of bits.
|
|
* the returned array stores the bigInt in bpe-bit chunks, little endian (buff[0] is least significant word)
|
|
* Pad the array with leading zeros so that it has at least minSize elements.
|
|
*
|
|
* There will always be at least one leading 0 element.
|
|
*
|
|
* @export
|
|
* @param {number} t
|
|
* @param {number} bits
|
|
* @param {number} minSize
|
|
* @returns {number[]}
|
|
*/
|
|
export function int2bigInt(t: number, bits: number, minSize: number): number[] {
|
|
var i, k
|
|
k = Math.ceil(bits / bpe) + 1
|
|
k = minSize > k ? minSize : k
|
|
var buff = new Array(k)
|
|
copyInt_(buff, t)
|
|
return buff
|
|
}
|
|
|
|
/**
|
|
* return the bigInt given a string representation in a given base.
|
|
* Pad the array with leading zeros so that it has at least minSize elements.
|
|
* If base=-1, then it reads in a space-separated list of array elements in decimal.
|
|
*
|
|
* The array will always have at least one leading zero, unless base=-1.
|
|
*
|
|
* @export
|
|
* @param {string} s
|
|
* @param {number} base
|
|
* @param {number} [minSize]
|
|
* @returns {number[]}
|
|
*/
|
|
export function str2bigInt(
|
|
s: string,
|
|
base: number,
|
|
minSize?: number,
|
|
): number[] {
|
|
var d, i, x, y, kk
|
|
var k = s.length
|
|
if (base === -1) {
|
|
//comma-separated list of array elements in decimal
|
|
x = new Array(0)
|
|
for (;;) {
|
|
y = new Array(x.length + 1)
|
|
for (i = 0; i < x.length; i++) y[i + 1] = x[i]
|
|
y[0] = parseInt(s, 10) //TODO PERF Should we replace that with ~~ (not not)? https://jsperf.com/number-vs-parseint-vs-plus/7
|
|
x = y
|
|
d = s.indexOf(',', 0)
|
|
if (d < 1) break
|
|
//$off
|
|
s = s.substring(d + 1)
|
|
if (s.length == 0) break
|
|
}
|
|
//$off
|
|
if (x.length < minSize) {
|
|
//$off
|
|
y = new Array(minSize)
|
|
copy_(y, x)
|
|
return y
|
|
}
|
|
return x
|
|
}
|
|
|
|
x = int2bigInt(0, base * k, 0)
|
|
for (i = 0; i < k; i++) {
|
|
d = digitsStr.indexOf(s.substring(i, i + 1), 0)
|
|
if (base <= 36 && d >= 36)
|
|
//convert lowercase to uppercase if base<=36
|
|
d -= 26
|
|
if (d >= base || d < 0) {
|
|
//stop at first illegal character
|
|
break
|
|
}
|
|
multInt_(x, base)
|
|
addInt_(x, d)
|
|
}
|
|
|
|
for (k = x.length; k > 0 && !x[k - 1]; k--); //strip off leading zeros
|
|
//$off
|
|
k = minSize > k + 1 ? minSize : k + 1
|
|
//$off
|
|
y = new Array(k)
|
|
//$off
|
|
kk = k < x.length ? k : x.length
|
|
//$off
|
|
for (i = 0; i < kk; i++) y[i] = x[i]
|
|
//$off
|
|
for (; i < k; i++) y[i] = 0
|
|
return y
|
|
}
|
|
|
|
//return the bigInt given a string representation in a given base.
|
|
//Pad the array with leading zeros so that it has at least minSize elements.
|
|
//If base=-1, then it reads in a space-separated list of array elements in decimal.
|
|
//The array will always have at least one leading zero, unless base=-1.
|
|
// function str2bigInt(s,b,minSize) {
|
|
// var d, i, j, base, str, x, y, kk;
|
|
// if (typeof b === 'string') {
|
|
// base = b.length;
|
|
// str = b;
|
|
// } else {
|
|
// base = b;
|
|
// str = digitsStr;
|
|
// }
|
|
// var k=s.length;
|
|
// if (base==-1) { //comma-separated list of array elements in decimal
|
|
// x=new Array(0);
|
|
// for (;;) {
|
|
// y=new Array(x.length+1);
|
|
// for (i=0;i<x.length;i++)
|
|
// y[i+1]=x[i];
|
|
// y[0]=parseInt(s,10);
|
|
// x=y;
|
|
// d=s.indexOf(',',0);
|
|
// if (d<1)
|
|
// break;
|
|
// s=s.substring(d+1);
|
|
// if (s.length==0)
|
|
// break;
|
|
// }
|
|
// if (x.length<minSize) {
|
|
// y=new Array(minSize);
|
|
// copy_(y,x);
|
|
// return y;
|
|
// }
|
|
// return x;
|
|
// }
|
|
|
|
// x=int2bigInt(0,base*k,0);
|
|
// for (i=0;i<k;i++) {
|
|
// d=str.indexOf(s.substring(i,i+1),0);
|
|
// if (base<=36 && d>=36) { //convert lowercase to uppercase if base<=36
|
|
// d-=26;
|
|
// }
|
|
// if (d>=base || d<0) { //ignore illegal characters
|
|
// continue;
|
|
// }
|
|
// multInt_(x,base);
|
|
// addInt_(x,d);
|
|
// }
|
|
|
|
// for (k=x.length;k>0 && !x[k-1];k--); //strip off leading zeros
|
|
// k=minSize>k+1 ? minSize : k+1;
|
|
// y=new Array(k);
|
|
// kk=k<x.length ? k : x.length;
|
|
// for (i=0;i<kk;i++)
|
|
// y[i]=x[i];
|
|
// for (;i<k;i++)
|
|
// y[i]=0;
|
|
// return y;
|
|
// }
|
|
|
|
/**
|
|
* is bigint x equal to integer y?
|
|
*
|
|
* y must have less than bpe bits
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} y
|
|
* @returns {(1 | 0)}
|
|
*/
|
|
export function equalsInt(x: number[], y: number): Bool {
|
|
var i
|
|
if (x[0] != y) return 0
|
|
for (i = 1; i < x.length; i++) if (x[i]) return 0
|
|
return 1
|
|
}
|
|
|
|
/**
|
|
* are bigints x and y equal?
|
|
*
|
|
* this works even if x and y are different lengths and have arbitrarily many leading zeros
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @returns {(1 | 0)}
|
|
*/
|
|
export function equals(x: number[], y: number[]): Bool {
|
|
var i
|
|
var k = x.length < y.length ? x.length : y.length
|
|
for (i = 0; i < k; i++) if (x[i] !== y[i]) return 0
|
|
if (x.length > y.length) {
|
|
for (; i < x.length; i++) if (x[i]) return 0
|
|
} else {
|
|
for (; i < y.length; i++) if (y[i]) return 0
|
|
}
|
|
return 1
|
|
}
|
|
|
|
/**
|
|
* is the bigInt x equal to zero?
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @returns {(1 | 0)}
|
|
*/
|
|
export function isZero(x: number[]): Bool {
|
|
var i
|
|
for (i = 0; i < x.length; i++) if (x[i]) return 0
|
|
return 1
|
|
}
|
|
|
|
/**
|
|
* Convert a bigInt into a string in a given base, from base 2 up to base 95.
|
|
*
|
|
* Base -1 prints the contents of the array representing the number.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} base
|
|
* @returns {string}
|
|
*/
|
|
export function bigInt2str(x: number[], base: number): string {
|
|
var i,
|
|
t,
|
|
s = ''
|
|
|
|
if (s6.length !== x.length) s6 = dup(x)
|
|
else copy_(s6, x)
|
|
|
|
if (base === -1) {
|
|
//return the list of array contents
|
|
for (i = x.length - 1; i > 0; i--) s += x[i] + ','
|
|
s += x[0]
|
|
} else {
|
|
//return it in the given base
|
|
while (!isZero(s6)) {
|
|
t = divInt_(s6, base) //t=s6 % base; s6=floor(s6/base);
|
|
s = digitsStr.substring(t, t + 1) + s
|
|
}
|
|
}
|
|
if (s.length === 0) s = '0'
|
|
return s
|
|
}
|
|
|
|
/**
|
|
* Convert a bigInt into bytes
|
|
* @param x bigInt
|
|
* @param littleEndian byte order by default
|
|
*/
|
|
export function bigInt2bytes(x: number[], littleEndian = true) {
|
|
if(s6.length !== x.length) s6 = dup(x);
|
|
else copy_(s6, x);
|
|
|
|
const out: number[] = [];
|
|
|
|
//console.log('bigInt2bytes');
|
|
while(!isZero(s6)) {
|
|
t = divInt_(s6, 256); //t=s6 % base; s6=floor(s6/base);
|
|
out.push(t);
|
|
//console.log('bigInt2bytes', t);
|
|
}
|
|
|
|
if(littleEndian) {
|
|
out.reverse();
|
|
}
|
|
|
|
//console.log('bigInt2bytes', out);
|
|
|
|
return out;
|
|
}
|
|
|
|
/**
|
|
* Compare two bigInts and return -1 if x is less, 0 if equals, 1 if greater
|
|
* @param x bigInt
|
|
* @param y bigInt
|
|
*/
|
|
export function cmp(x: number[], y: number[]) {
|
|
return greater(x, y) ? 1 : (equals(x, y) ? 0 : -1);
|
|
}
|
|
|
|
/* Object.assign(self, {
|
|
cmp,
|
|
str2bigInt,
|
|
int2bigInt,
|
|
bigInt2str,
|
|
one,
|
|
divide_,
|
|
divInt_,
|
|
dup,
|
|
negative
|
|
}); */
|
|
|
|
/**
|
|
* Returns a duplicate of bigInt x
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @returns {number[]}
|
|
*/
|
|
export function dup(x: number[]): number[] {
|
|
var i
|
|
buff = Array(x.length)
|
|
copy_(buff, x)
|
|
return buff
|
|
}
|
|
|
|
/**
|
|
* do x=y on bigInts x and y.
|
|
*
|
|
* x must be an array at least as big as y (not counting the leading zeros in y).
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @returns {void}
|
|
*/
|
|
export function copy_(x: number[], y: number[]): void {
|
|
var i
|
|
var k = x.length < y.length ? x.length : y.length
|
|
for (i = 0; i < k; i++) x[i] = y[i]
|
|
for (i = k; i < x.length; i++) x[i] = 0
|
|
}
|
|
|
|
/**
|
|
* do x=y on bigInt x and integer y.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
* @returns {void}
|
|
*/
|
|
export function copyInt_(x: number[], n: number): void {
|
|
var i, c
|
|
var len = x.length //TODO .length in for loop have perfomance costs. Bench this
|
|
for (c = n, i = 0; i < len; i++) {
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x+n where x is a bigInt and n is an integer.
|
|
*
|
|
* x must be large enough to hold the result.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
* @returns {void}
|
|
*/
|
|
export function addInt_(x: number[], n: number): void {
|
|
var i, k, c, b
|
|
x[0] += n
|
|
k = x.length
|
|
c = 0
|
|
for (i = 0; i < k; i++) {
|
|
c += x[i]
|
|
b = 0
|
|
if (c < 0) {
|
|
b = -(c >> bpe)
|
|
c += b * radix
|
|
}
|
|
x[i] = c & mask
|
|
c = (c >> bpe) - b
|
|
if (!c) return //stop carrying as soon as the carry is zero
|
|
}
|
|
}
|
|
|
|
/**
|
|
* right shift bigInt x by n bits.
|
|
*
|
|
* 0 <= n < bpe.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
*/
|
|
export function rightShift_(x: number[], n: number): void {
|
|
var i
|
|
var k = Math.floor(n / bpe)
|
|
if (k) {
|
|
for (
|
|
i = 0;
|
|
i < x.length - k;
|
|
i++ //right shift x by k elements
|
|
)
|
|
x[i] = x[i + k]
|
|
for (; i < x.length; i++) x[i] = 0
|
|
//$off
|
|
n %= bpe
|
|
}
|
|
for (i = 0; i < x.length - 1; i++) {
|
|
x[i] = mask & ((x[i + 1] << (bpe - n)) | (x[i] >> n))
|
|
}
|
|
x[i] >>= n
|
|
}
|
|
|
|
/**
|
|
* do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement
|
|
*
|
|
* @param {number[]} x
|
|
* @returns {void}
|
|
*/
|
|
export function halve_(x: number[]): void {
|
|
var i
|
|
for (i = 0; i < x.length - 1; i++) {
|
|
x[i] = mask & ((x[i + 1] << (bpe - 1)) | (x[i] >> 1))
|
|
}
|
|
x[i] = (x[i] >> 1) | (x[i] & (radix >> 1)) //most significant bit stays the same
|
|
}
|
|
|
|
/**
|
|
* left shift bigInt x by n bits
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
* @returns {void}
|
|
*/
|
|
export function leftShift_(x: number[], n: number): void {
|
|
var i
|
|
var k = Math.floor(n / bpe)
|
|
if (k) {
|
|
for (
|
|
i = x.length;
|
|
i >= k;
|
|
i-- //left shift x by k elements
|
|
)
|
|
x[i] = x[i - k]
|
|
for (; i >= 0; i--) x[i] = 0
|
|
//$off
|
|
n %= bpe
|
|
}
|
|
if (!n) return
|
|
for (i = x.length - 1; i > 0; i--) {
|
|
x[i] = mask & ((x[i] << n) | (x[i - 1] >> (bpe - n)))
|
|
}
|
|
x[i] = mask & (x[i] << n)
|
|
}
|
|
|
|
/**
|
|
* do x=x*n where x is a bigInt and n is an integer.
|
|
*
|
|
* x must be large enough to hold the result.
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
* @returns {void}
|
|
*/
|
|
export function multInt_(x: number[], n: number): void {
|
|
var i, k, c, b
|
|
if (!n) return
|
|
k = x.length
|
|
c = 0
|
|
for (i = 0; i < k; i++) {
|
|
c += x[i] * n
|
|
b = 0
|
|
if (c < 0) {
|
|
b = -(c >> bpe)
|
|
c += b * radix
|
|
}
|
|
x[i] = c & mask
|
|
c = (c >> bpe) - b
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=floor(x/n) for bigInt x and integer n, and return the remainder
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number} n
|
|
* @returns {number} remainder
|
|
*/
|
|
export function divInt_(x: number[], n: number): number {
|
|
var i,
|
|
r = 0,
|
|
s
|
|
for (i = x.length - 1; i >= 0; i--) {
|
|
s = r * radix + x[i]
|
|
x[i] = Math.floor(s / n)
|
|
r = s % n
|
|
}
|
|
return r
|
|
}
|
|
|
|
/**
|
|
* do the linear combination x=a*x+b*y for bigInts x and y, and integers a and b.
|
|
*
|
|
* x must be large enough to hold the answer.
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number} a
|
|
* @param {number} b
|
|
* @returns {void}
|
|
*/
|
|
export function linComb_(x: number[], y: number[], a: number, b: number): void {
|
|
var i, c, k, kk
|
|
k = x.length < y.length ? x.length : y.length
|
|
kk = x.length
|
|
for (c = 0, i = 0; i < k; i++) {
|
|
c += a * x[i] + b * y[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
for (i = k; i < kk; i++) {
|
|
c += a * x[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do the linear combination x=a*x+b*(y<<(ys*bpe)) for bigInts x and y, and integers a, b and ys.
|
|
*
|
|
* x must be large enough to hold the answer.
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number} b
|
|
* @param {number} ys
|
|
* @returns {void}
|
|
*/
|
|
export function linCombShift_(
|
|
x: number[],
|
|
y: number[],
|
|
b: number,
|
|
ys: number,
|
|
): void {
|
|
var i, c, k, kk
|
|
k = x.length < ys + y.length ? x.length : ys + y.length
|
|
kk = x.length
|
|
for (c = 0, i = ys; i < k; i++) {
|
|
c += x[i] + b * y[i - ys]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
for (i = k; c && i < kk; i++) {
|
|
c += x[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x+(y<<(ys*bpe)) for bigInts x and y, and integer ys.
|
|
*
|
|
* x must be large enough to hold the answer.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number} ys
|
|
* @return {void}
|
|
*/
|
|
export function addShift_(x: number[], y: number[], ys: number): void {
|
|
var i, c, k, kk
|
|
k = x.length < ys + y.length ? x.length : ys + y.length
|
|
kk = x.length
|
|
for (c = 0, i = ys; i < k; i++) {
|
|
c += x[i] + y[i - ys]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
for (i = k; c && i < kk; i++) {
|
|
c += x[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x-(y<<(ys*bpe)) for bigInts x and y, and integer ys
|
|
*
|
|
* x must be large enough to hold the answer
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number} ys
|
|
* @return {void}
|
|
*/
|
|
export function subShift_(x: number[], y: number[], ys: number): void {
|
|
var i, c, k, kk
|
|
k = x.length < ys + y.length ? x.length : ys + y.length
|
|
kk = x.length
|
|
for (c = 0, i = ys; i < k; i++) {
|
|
c += x[i] - y[i - ys]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
for (i = k; c && i < kk; i++) {
|
|
c += x[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x-y for bigInts x and y
|
|
*
|
|
* x must be large enough to hold the answer
|
|
*
|
|
* negative answers will be 2s complement
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @return {void}
|
|
*/
|
|
export function sub_(x: number[], y: number[]): void {
|
|
var i, c, k, kk
|
|
k = x.length < y.length ? x.length : y.length
|
|
for (c = 0, i = 0; i < k; i++) {
|
|
c += x[i] - y[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
for (i = k; c && i < x.length; i++) {
|
|
c += x[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x+y for bigInts x and y
|
|
*
|
|
* x must be large enough to hold the answer
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @return {void}
|
|
*/
|
|
export function add_(x: number[], y: number[]): void {
|
|
var i, c, k, kk
|
|
k = x.length < y.length ? x.length : y.length
|
|
for (c = 0, i = 0; i < k; i++) {
|
|
c += x[i] + y[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
for (i = k; c && i < x.length; i++) {
|
|
c += x[i]
|
|
x[i] = c & mask
|
|
c >>= bpe
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x*y for bigInts x and y.
|
|
*
|
|
* This is faster when y<x.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @return {void}
|
|
*/
|
|
export function mult_(x: number[], y: number[]): void {
|
|
var i
|
|
if (ss.length != 2 * x.length) ss = new Array(2 * x.length)
|
|
copyInt_(ss, 0)
|
|
for (i = 0; i < y.length; i++) if (y[i]) linCombShift_(ss, x, y[i], i) //ss=1*ss+y[i]*(x<<(i*bpe))
|
|
copy_(x, ss)
|
|
}
|
|
|
|
/**
|
|
* do x=x mod n for bigInts x and n
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} n
|
|
* @return {void}
|
|
*/
|
|
export function mod_(x: number[], n: number[]): void {
|
|
if (s4.length !== x.length) s4 = dup(x)
|
|
else copy_(s4, x)
|
|
if (s5.length !== x.length) s5 = dup(x)
|
|
divide_(s4, n, s5, x) //x = remainder of s4 / n
|
|
}
|
|
|
|
/**
|
|
* do x=x*y mod n for bigInts x,y,n.
|
|
*
|
|
* for greater speed, let y<x.
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number[]} n
|
|
* @return {void}
|
|
*/
|
|
export function multMod_(x: number[], y: number[], n: number[]): void {
|
|
var i
|
|
if (s0.length != 2 * x.length) s0 = new Array(2 * x.length)
|
|
copyInt_(s0, 0)
|
|
for (i = 0; i < y.length; i++) if (y[i]) linCombShift_(s0, x, y[i], i) //s0=1*s0+y[i]*(x<<(i*bpe))
|
|
mod_(s0, n)
|
|
copy_(x, s0)
|
|
}
|
|
|
|
/**
|
|
* do x=x*x mod n for bigInts x,n.
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} n
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* @return {void}
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|
*/
|
|
export function squareMod_(x: number[], n: number[]): void {
|
|
var i, j, d, c, kx, kn, k
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for (kx = x.length; kx > 0 && !x[kx - 1]; kx--); //ignore leading zeros in x
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k = kx > n.length ? 2 * kx : 2 * n.length //k=# elements in the product, which is twice the elements in the larger of x and n
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if (s0.length != k) s0 = new Array(k)
|
|
copyInt_(s0, 0)
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|
for (i = 0; i < kx; i++) {
|
|
c = s0[2 * i] + x[i] * x[i]
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s0[2 * i] = c & mask
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|
c >>= bpe
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|
for (j = i + 1; j < kx; j++) {
|
|
c = s0[i + j] + 2 * x[i] * x[j] + c
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|
s0[i + j] = c & mask
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|
c >>= bpe
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|
}
|
|
s0[i + kx] = c
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|
}
|
|
mod_(s0, n)
|
|
copy_(x, s0)
|
|
}
|
|
|
|
/**
|
|
* return x with exactly k leading zero elements
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number} k
|
|
* @returns {number[]}
|
|
*/
|
|
export function trim(x: number[], k: number): number[] {
|
|
var i, y
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|
for (i = x.length; i > 0 && !x[i - 1]; i--);
|
|
y = new Array(i + k)
|
|
copy_(y, x)
|
|
return y
|
|
}
|
|
|
|
/**
|
|
* do `x=x**y mod n`, where x,y,n are bigInts and `**` is exponentiation. `0**0=1`.
|
|
*
|
|
* this is faster when n is odd.
|
|
*
|
|
* x usually needs to have as many elements as n.
|
|
*
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number[]} n
|
|
* @return {void}
|
|
*/
|
|
export function powMod_(x: number[], y: number[], n: number[]): void {
|
|
var k1, k2, kn, np
|
|
if (s7.length != n.length) s7 = dup(n)
|
|
|
|
//for even modulus, use a simple square-and-multiply algorithm,
|
|
//rather than using the more complex Montgomery algorithm.
|
|
if ((n[0] & 1) == 0) {
|
|
copy_(s7, x)
|
|
copyInt_(x, 1)
|
|
while (!equalsInt(y, 0)) {
|
|
if (y[0] & 1) multMod_(x, s7, n)
|
|
divInt_(y, 2)
|
|
squareMod_(s7, n)
|
|
}
|
|
return
|
|
}
|
|
|
|
//calculate np from n for the Montgomery multiplications
|
|
copyInt_(s7, 0)
|
|
for (kn = n.length; kn > 0 && !n[kn - 1]; kn--);
|
|
np = radix - inverseModInt(modInt(n, radix), radix)
|
|
s7[kn] = 1
|
|
multMod_(x, s7, n) // x = x * 2**(kn*bp) mod n
|
|
|
|
if (s3.length != x.length) s3 = dup(x)
|
|
else copy_(s3, x)
|
|
//$off
|
|
// @ts-ignore
|
|
for (k1 = y.length - 1; (k1 > 0) & !y[k1]; k1--); //k1=first nonzero element of y
|
|
if (y[k1] == 0) {
|
|
//anything to the 0th power is 1
|
|
copyInt_(x, 1)
|
|
return
|
|
}
|
|
for (k2 = 1 << (bpe - 1); k2 && !(y[k1] & k2); k2 >>= 1); //k2=position of first 1 bit in y[k1]
|
|
for (;;) {
|
|
if (!(k2 >>= 1)) {
|
|
//look at next bit of y
|
|
k1--
|
|
if (k1 < 0) {
|
|
mont_(x, one, n, np)
|
|
return
|
|
}
|
|
k2 = 1 << (bpe - 1)
|
|
}
|
|
mont_(x, x, n, np)
|
|
|
|
if (k2 & y[k1])
|
|
//if next bit is a 1
|
|
mont_(x, s3, n, np)
|
|
}
|
|
}
|
|
|
|
/**
|
|
* do x=x*y*Ri mod n for bigInts x,y,n,
|
|
* where Ri = 2**(-kn*bpe) mod n, and kn is the
|
|
* number of elements in the n array, not
|
|
* counting leading zeros.
|
|
*
|
|
* x array must have at least as many elemnts as the n array
|
|
* It's OK if x and y are the same variable.
|
|
*
|
|
* must have:
|
|
* * x,y < n
|
|
* * n is odd
|
|
* * np = -(n^(-1)) mod radix
|
|
*
|
|
* @export
|
|
* @param {number[]} x
|
|
* @param {number[]} y
|
|
* @param {number[]} n
|
|
* @param {number} np
|
|
* @return {void}
|
|
*/
|
|
export function mont_(x: number[], y: number[], n: number[], np: number): void {
|
|
var i, j, c, ui, t, ks
|
|
var kn = n.length
|
|
var ky = y.length
|
|
|
|
if (sa.length != kn) sa = new Array(kn)
|
|
|
|
copyInt_(sa, 0)
|
|
|
|
for (; kn > 0 && n[kn - 1] == 0; kn--); //ignore leading zeros of n
|
|
for (; ky > 0 && y[ky - 1] == 0; ky--); //ignore leading zeros of y
|
|
ks = sa.length - 1 //sa will never have more than this many nonzero elements.
|
|
|
|
//the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large numbers
|
|
for (i = 0; i < kn; i++) {
|
|
t = sa[0] + x[i] * y[0]
|
|
ui = ((t & mask) * np) & mask //the inner "& mask" was needed on Safari (but not MSIE) at one time
|
|
c = (t + ui * n[0]) >> bpe
|
|
t = x[i]
|
|
|
|
//do sa=(sa+x[i]*y+ui*n)/b where b=2**bpe. Loop is unrolled 5-fold for speed
|
|
j = 1
|
|
for (; j < ky - 4; ) {
|
|
c += sa[j] + ui * n[j] + t * y[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j] + t * y[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j] + t * y[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j] + t * y[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j] + t * y[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
}
|
|
for (; j < ky; ) {
|
|
c += sa[j] + ui * n[j] + t * y[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
}
|
|
for (; j < kn - 4; ) {
|
|
c += sa[j] + ui * n[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
c += sa[j] + ui * n[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
}
|
|
for (; j < kn; ) {
|
|
c += sa[j] + ui * n[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
}
|
|
for (; j < ks; ) {
|
|
c += sa[j]
|
|
sa[j - 1] = c & mask
|
|
c >>= bpe
|
|
j++
|
|
}
|
|
sa[j - 1] = c & mask
|
|
}
|
|
|
|
if (!greater(n, sa)) sub_(sa, n)
|
|
copy_(x, sa)
|
|
} |