Telegram Web, preconfigured for usage in I2P.
http://web.telegram.i2p/
You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
1507 lines
48 KiB
1507 lines
48 KiB
//////////////////////////////////////////////////////////////////////////////////////// |
|
// 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. |
|
//////////////////////////////////////////////////////////////////////////////////////// |
|
|
|
//globals |
|
bpe=0; //bits stored per array element |
|
mask=0; //AND this with an array element to chop it down to bpe bits |
|
radix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask. |
|
|
|
//the digits for converting to different bases |
|
digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-'; |
|
|
|
//initialize the global variables |
|
for (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platform |
|
bpe>>=1; //bpe=number of bits in one element of the array representing the bigInt |
|
mask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bits |
|
radix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of mask |
|
one=int2bigInt(1,1,1); //constant used in powMod_() |
|
|
|
//the following global variables are scratchpad memory to |
|
//reduce dynamic memory allocation in the inner loop |
|
t=new Array(0); |
|
ss=t; //used in mult_() |
|
s0=t; //used in multMod_(), squareMod_() |
|
s1=t; //used in powMod_(), multMod_(), squareMod_() |
|
s2=t; //used in powMod_(), multMod_() |
|
s3=t; //used in powMod_() |
|
s4=t; s5=t; //used in mod_() |
|
s6=t; //used in bigInt2str() |
|
s7=t; //used in powMod_() |
|
T=t; //used in GCD_() |
|
sa=t; //used in mont_() |
|
mr_x1=t; mr_r=t; mr_a=t; //used in millerRabin() |
|
eg_v=t; eg_u=t; eg_A=t; eg_B=t; eg_C=t; eg_D=t; //used in eGCD_(), inverseMod_() |
|
md_q1=t; md_q2=t; md_q3=t; md_r=t; md_r1=t; md_r2=t; md_tt=t; //used in mod_() |
|
|
|
primes=t; pows=t; s_i=t; s_i2=t; s_R=t; s_rm=t; s_q=t; s_n1=t; |
|
s_a=t; s_r2=t; s_n=t; s_b=t; s_d=t; s_x1=t; s_x2=t, s_aa=t; //used in randTruePrime_() |
|
|
|
rpprb=t; //used in randProbPrimeRounds() (which also uses "primes") |
|
|
|
//////////////////////////////////////////////////////////////////////////////////////// |
|
|
|
|
|
//return array of all primes less than integer n |
|
function findPrimes(n) { |
|
var i,s,p,ans; |
|
s=new Array(n); |
|
for (i=0;i<n;i++) |
|
s[i]=0; |
|
s[0]=2; |
|
p=0; //first p elements of s are primes, the rest are a sieve |
|
for(;s[p]<n;) { //s[p] is the pth prime |
|
for(i=s[p]*s[p]; i<n; i+=s[p]) //mark multiples of s[p] |
|
s[i]=1; |
|
p++; |
|
s[p]=s[p-1]+1; |
|
for(; s[p]<n && s[s[p]]; s[p]++); //find next prime (where s[p]==0) |
|
} |
|
ans=new Array(p); |
|
for(i=0;i<p;i++) |
|
ans[i]=s[i]; |
|
return ans; |
|
} |
|
|
|
|
|
//does a single round of Miller-Rabin base b consider x to be a possible prime? |
|
//x is a bigInt, and b is an integer, with b<x |
|
function millerRabinInt(x,b) { |
|
if (mr_x1.length!=x.length) { |
|
mr_x1=dup(x); |
|
mr_r=dup(x); |
|
mr_a=dup(x); |
|
} |
|
|
|
copyInt_(mr_a,b); |
|
return millerRabin(x,mr_a); |
|
} |
|
|
|
//does a single round of Miller-Rabin base b consider x to be a possible prime? |
|
//x and b are bigInts with b<x |
|
function millerRabin(x,b) { |
|
var i,j,k,s; |
|
|
|
if (mr_x1.length!=x.length) { |
|
mr_x1=dup(x); |
|
mr_r=dup(x); |
|
mr_a=dup(x); |
|
} |
|
|
|
copy_(mr_a,b); |
|
copy_(mr_r,x); |
|
copy_(mr_x1,x); |
|
|
|
addInt_(mr_r,-1); |
|
addInt_(mr_x1,-1); |
|
|
|
//s=the highest power of two that divides mr_r |
|
k=0; |
|
for (i=0;i<mr_r.length;i++) |
|
for (j=1;j<mask;j<<=1) |
|
if (x[i] & j) { |
|
s=(k<mr_r.length+bpe ? k : 0); |
|
i=mr_r.length; |
|
j=mask; |
|
} else |
|
k++; |
|
|
|
if (s) |
|
rightShift_(mr_r,s); |
|
|
|
powMod_(mr_a,mr_r,x); |
|
|
|
if (!equalsInt(mr_a,1) && !equals(mr_a,mr_x1)) { |
|
j=1; |
|
while (j<=s-1 && !equals(mr_a,mr_x1)) { |
|
squareMod_(mr_a,x); |
|
if (equalsInt(mr_a,1)) { |
|
return 0; |
|
} |
|
j++; |
|
} |
|
if (!equals(mr_a,mr_x1)) { |
|
return 0; |
|
} |
|
} |
|
return 1; |
|
} |
|
|
|
//returns how many bits long the bigInt is, not counting leading zeros. |
|
function bitSize(x) { |
|
var j,z,w; |
|
for (j=x.length-1; (x[j]==0) && (j>0); j--); |
|
for (z=0,w=x[j]; w; (w>>=1),z++); |
|
z+=bpe*j; |
|
return z; |
|
} |
|
|
|
//return a copy of x with at least n elements, adding leading zeros if needed |
|
function expand(x,n) { |
|
var ans=int2bigInt(0,(x.length>n ? x.length : n)*bpe,0); |
|
copy_(ans,x); |
|
return ans; |
|
} |
|
|
|
//return a k-bit true random prime using Maurer's algorithm. |
|
function randTruePrime(k) { |
|
var ans=int2bigInt(0,k,0); |
|
randTruePrime_(ans,k); |
|
return trim(ans,1); |
|
} |
|
|
|
//return a k-bit random probable prime with probability of error < 2^-80 |
|
function randProbPrime(k) { |
|
if (k>=600) return randProbPrimeRounds(k,2); //numbers from HAC table 4.3 |
|
if (k>=550) return randProbPrimeRounds(k,4); |
|
if (k>=500) return randProbPrimeRounds(k,5); |
|
if (k>=400) return randProbPrimeRounds(k,6); |
|
if (k>=350) return randProbPrimeRounds(k,7); |
|
if (k>=300) return randProbPrimeRounds(k,9); |
|
if (k>=250) return randProbPrimeRounds(k,12); //numbers from HAC table 4.4 |
|
if (k>=200) return randProbPrimeRounds(k,15); |
|
if (k>=150) return randProbPrimeRounds(k,18); |
|
if (k>=100) return randProbPrimeRounds(k,27); |
|
return randProbPrimeRounds(k,40); //number from HAC remark 4.26 (only an estimate) |
|
} |
|
|
|
//return a k-bit probable random prime using n rounds of Miller Rabin (after trial division with small primes) |
|
function randProbPrimeRounds(k,n) { |
|
var ans, i, divisible, B; |
|
B=30000; //B is largest prime to use in trial division |
|
ans=int2bigInt(0,k,0); |
|
|
|
//optimization: try larger and smaller B to find the best limit. |
|
|
|
if (primes.length==0) |
|
primes=findPrimes(30000); //check for divisibility by primes <=30000 |
|
|
|
if (rpprb.length!=ans.length) |
|
rpprb=dup(ans); |
|
|
|
for (;;) { //keep trying random values for ans until one appears to be prime |
|
//optimization: pick a random number times L=2*3*5*...*p, plus a |
|
// 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. |
|
function mod(x,n) { |
|
var ans=dup(x); |
|
mod_(ans,n); |
|
return trim(ans,1); |
|
} |
|
|
|
//return (x+n) where x is a bigInt and n is an integer. |
|
function addInt(x,n) { |
|
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. |
|
function mult(x,y) { |
|
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. |
|
function powMod(x,y,n) { |
|
var ans=expand(x,n.length); |
|
powMod_(ans,trim(y,2),trim(n,2),0); //this should work without the trim, but doesn't |
|
return trim(ans,1); |
|
} |
|
|
|
//return (x-y) for bigInts x and y. Negative answers will be 2s complement |
|
function sub(x,y) { |
|
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. |
|
function add(x,y) { |
|
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 |
|
function inverseMod(x,n) { |
|
var ans=expand(x,n.length); |
|
var s; |
|
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. |
|
function multMod(x,y,n) { |
|
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. |
|
function randTruePrime_(ans,k) { |
|
var c,m,pm,dd,j,r,B,divisible,z,zz,recSize; |
|
|
|
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 |
|
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); |
|
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=.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 it passes small primes check, then try a single Miller-Rabin base 2 |
|
if (!millerRabinInt(s_n,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. |
|
function randBigInt(n,s) { |
|
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 |
|
function randBigInt_(b,n,s) { |
|
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). |
|
function GCD(x,y) { |
|
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. |
|
function GCD_(x,y) { |
|
var i,xp,yp,A,B,C,D,q,sing; |
|
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. |
|
function inverseMod_(x,n) { |
|
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 |
|
function inverseModInt(x,n) { |
|
var a=1,b=0,t; |
|
for (;;) { |
|
if (x==1) return a; |
|
if (x==0) return 0; |
|
b-=a*Math.floor(n/x); |
|
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); |
|
x%=n; |
|
} |
|
} |
|
|
|
//this deprecated function is for backward compatibility only. |
|
function inverseModInt_(x,n) { |
|
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. |
|
function eGCD_(x,y,v,a,b) { |
|
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? |
|
function negative(x) { |
|
return ((x[x.length-1]>>(bpe-1))&1); |
|
} |
|
|
|
|
|
//is (x << (shift*bpe)) > y? |
|
//x and y are nonnegative bigInts |
|
//shift is a nonnegative integer |
|
function greaterShift(x,y,shift) { |
|
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) |
|
function greater(x,y) { |
|
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. |
|
function divide_(x,y,q,r) { |
|
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. |
|
function carry_(x) { |
|
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. |
|
function modInt(x,n) { |
|
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. |
|
function int2bigInt(t,bits,minSize) { |
|
var i,k; |
|
k=Math.ceil(bits/bpe)+1; |
|
k=minSize>k ? minSize : k; |
|
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. |
|
function str2bigInt(s,base,minSize) { |
|
var d, i, j, 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); |
|
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=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 |
|
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 |
|
function equalsInt(x,y) { |
|
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 |
|
function equals(x,y) { |
|
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? |
|
function isZero(x) { |
|
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. |
|
function bigInt2str(x,base) { |
|
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; |
|
} |
|
|
|
//returns a duplicate of bigInt x |
|
function dup(x) { |
|
var i; |
|
buff=new 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). |
|
function copy_(x,y) { |
|
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. |
|
function copyInt_(x,n) { |
|
var i,c; |
|
for (c=n,i=0;i<x.length;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. |
|
function addInt_(x,n) { |
|
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. |
|
function rightShift_(x,n) { |
|
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; |
|
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 |
|
function halve_(x) { |
|
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. |
|
function leftShift_(x,n) { |
|
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; |
|
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. |
|
function multInt_(x,n) { |
|
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 |
|
function divInt_(x,n) { |
|
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. |
|
function linComb_(x,y,a,b) { |
|
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. |
|
function linCombShift_(x,y,b,ys) { |
|
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 integers a,b and ys. |
|
//x must be large enough to hold the answer. |
|
function addShift_(x,y,ys) { |
|
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 integers a,b and ys. |
|
//x must be large enough to hold the answer. |
|
function subShift_(x,y,ys) { |
|
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 |
|
function sub_(x,y) { |
|
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. |
|
function add_(x,y) { |
|
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. |
|
function mult_(x,y) { |
|
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. |
|
function mod_(x,n) { |
|
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. |
|
function multMod_(x,y,n) { |
|
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. |
|
function squareMod_(x,n) { |
|
var i,j,d,c,kx,kn,k; |
|
for (kx=x.length; kx>0 && !x[kx-1]; kx--); //ignore leading zeros in x |
|
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 |
|
if (s0.length!=k) |
|
s0=new Array(k); |
|
copyInt_(s0,0); |
|
for (i=0;i<kx;i++) { |
|
c=s0[2*i]+x[i]*x[i]; |
|
s0[2*i]=c & mask; |
|
c>>=bpe; |
|
for (j=i+1;j<kx;j++) { |
|
c=s0[i+j]+2*x[i]*x[j]+c; |
|
s0[i+j]=(c & mask); |
|
c>>=bpe; |
|
} |
|
s0[i+kx]=c; |
|
} |
|
mod_(s0,n); |
|
copy_(x,s0); |
|
} |
|
|
|
//return x with exactly k leading zero elements |
|
function trim(x,k) { |
|
var i,y; |
|
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. |
|
function powMod_(x,y,n) { |
|
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); |
|
|
|
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 |
|
function mont_(x,y,n,np) { |
|
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); |
|
} |
|
|
|
|