Mercurial > sat_docs
view scripts/minifier/otr/dep/bigint.js @ 98:349cbfea2596
docker (prosody): restored logs in /var/log + info logs to console
author | Goffi <goffi@goffi.org> |
---|---|
date | Fri, 26 Feb 2016 16:09:01 +0100 |
parents | 1596660ddf72 |
children |
line wrap: on
line source
;(function (root, factory) { if (typeof define === 'function' && define.amd) { define(factory.bind(root, root.crypto || root.msCrypto)) } else if (typeof module !== 'undefined' && module.exports) { module.exports = factory(require('crypto')) } else { root.BigInt = factory(root.crypto || root.msCrypto) } }(this, function (crypto) { //////////////////////////////////////////////////////////////////////////////////////// // 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 Péter Szabó 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. (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 // The number of significant bits in the fraction of a JavaScript // floating-point number is 52, independent of platform. // See: https://github.com/arlolra/otr/issues/41 var bpe = 26; // bits stored per array element var radix = 1 << bpe; // equals 2^bpe var mask = radix - 1; // AND this with an array element to chop it down to bpe bits //the digits for converting to different bases var digitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-'; var 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 var t=new Array(0); var ss=t; //used in mult_() var s0=t; //used in multMod_(), squareMod_() var s1=t; //used in powMod_(), multMod_(), squareMod_() var s2=t; //used in powMod_(), multMod_() var s3=t; //used in powMod_() var s4=t, s5=t; //used in mod_() var s6=t; //used in bigInt2str() var s7=t; //used in powMod_() var T=t; //used in GCD_() var sa=t; //used in mont_() var mr_x1=t, mr_r=t, mr_a=t; //used in millerRabin() var eg_v=t, eg_u=t, eg_A=t, eg_B=t, eg_C=t, eg_D=t; //used in eGCD_(), inverseMod_() var md_q1=t, md_q2=t, md_q3=t, md_r=t, md_r1=t, md_r2=t, md_tt=t; //used in mod_() var primes=t, pows=t, s_i=t, s_i2=t, s_R=t, s_rm=t, s_q=t, s_n1=t; var 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_() var 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++; */ /* http://www.javascripter.net/math/primes/millerrabinbug-bigint54.htm */ if (isZero(mr_r)) return 0; for (k=0; mr_r[k]==0; k++); for (i=1,j=2; mr_r[k]%j==0; j*=2,i++ ); s = k*bpe + i - 1; /* end */ 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,w,m,pm,dd,j,r,B,divisible,z,zz,recSize,recLimit; 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/511.0-1.0); } } //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)) | randomBitInt(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[randomBitInt(9)]; //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 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]=randomBitInt(bpe); } 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,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. 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; var 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; y2=y2 & mask; c = (c - y2) / radix; y1=c+q[i-ky]*y[ky-1]; c=y1; y1=y1 & mask; c = (c - y1) / radix; 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 & mask; b = -((c - b) / radix); c+=b*radix; } x[i]=c & mask; c = ((c - x[i]) / radix) - 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, buff; 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; } // log2(base)*k var bb = base, p = 0; var b = base == 1 ? k : 0; while (bb > 1) { if (bb & 1) p = 1; b += k; bb >>= 1; } b += p*k; x=int2bigInt(0,b,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; 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 & mask; b = -((c - b) / radix); c+=b*radix; } x[i]=c & mask; c = ((c - x[i]) / radix) - b; if (!c) return; //stop carrying as soon as the carry is zero } } //right shift bigInt x by n bits. 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 & mask; b = -((c - b) / radix); c+=b*radix; } x[i]=c & mask; c = ((c - x[i]) / radix) - 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 = (c - x[i]) / radix; } for (i=k;i<kk;i++) { c+=a*x[i]; x[i]=c & mask; c = (c - x[i]) / radix; } } //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 = (c - x[i]) / radix; } for (i=k;c && i<kk;i++) { c+=x[i]; x[i]=c & mask; c = (c - x[i]) / radix; } } //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 = (c - x[i]) / radix; } for (i=k;c && i<kk;i++) { c+=x[i]; x[i]=c & mask; c = (c - x[i]) / radix; } } //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 = (c - x[i]) / radix; } for (i=k;c && i<kk;i++) { c+=x[i]; x[i]=c & mask; c = (c - x[i]) / radix; } } //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 = (c - x[i]) / radix; } for (i=k;c && i<x.length;i++) { c+=x[i]; x[i]=c & mask; c = (c - x[i]) / radix; } } //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 = (c - x[i]) / radix; } for (i=k;c && i<x.length;i++) { c+=x[i]; x[i]=c & mask; c = (c - x[i]) / radix; } } //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 = (c - s0[2*i]) / radix; for (j=i+1;j<kx;j++) { c=s0[i+j]+2*x[i]*x[j]+c; s0[i+j]=(c & mask); c = (c - s0[i+j]) / radix; } 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,t2,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]); c = (c - (c & mask)) / radix; 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]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; } for (;j<ky;) { c+=sa[j]+ui*n[j]+t*y[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; } for (;j<kn-4;) { c+=sa[j]+ui*n[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; c+=sa[j]+ui*n[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; } for (;j<kn;) { c+=sa[j]+ui*n[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; } for (;j<ks;) { c+=sa[j]; t2=sa[j-1]=c & mask; c=(c-t2)/radix; j++; } sa[j-1]=c & mask; } if (!greater(n,sa)) sub_(sa,n); copy_(x,sa); } // otr.js additions // computes num / den mod n function divMod(num, den, n) { return multMod(num, inverseMod(den, n), n) } // computes one - two mod n function subMod(one, two, n) { one = mod(one, n) two = mod(two, n) if (greater(two, one)) one = add(one, n) return sub(one, two) } // computes 2^m as a bigInt function twoToThe(m) { var b = Math.floor(m / bpe) + 2 var t = new Array(b) for (var i = 0; i < b; i++) t[i] = 0 t[b - 2] = 1 << (m % bpe) return t } // cache these results for faster lookup var _num2bin = (function () { var i = 0, _num2bin= {} for (; i < 0x100; ++i) { _num2bin[i] = String.fromCharCode(i) // 0 -> "\00" } return _num2bin }()) // serialize a bigInt to an ascii string // padded up to pad length function bigInt2bits(bi, pad) { pad || (pad = 0) bi = dup(bi) var ba = '' while (!isZero(bi)) { ba = _num2bin[bi[0] & 0xff] + ba rightShift_(bi, 8) } while (ba.length < pad) { ba = '\x00' + ba } return ba } // converts a byte array to a bigInt function ba2bigInt(data) { var mpi = str2bigInt('0', 10, data.length) data.forEach(function (d, i) { if (i) leftShift_(mpi, 8) mpi[0] |= d }) return mpi } // returns a function that returns an array of n bytes var randomBytes = (function () { // in node if ( typeof crypto !== 'undefined' && typeof crypto.randomBytes === 'function' ) { return function (n) { try { var buf = crypto.randomBytes(n) } catch (e) { throw e } return Array.prototype.slice.call(buf, 0) } } // in browser else if ( typeof crypto !== 'undefined' && typeof crypto.getRandomValues === 'function' ) { return function (n) { var buf = new Uint8Array(n) crypto.getRandomValues(buf) return Array.prototype.slice.call(buf, 0) } } // err else { throw new Error('Keys should not be generated without CSPRNG.') } }()) // Salsa 20 in webworker needs a 40 byte seed function getSeed() { return randomBytes(40) } // returns a single random byte function randomByte() { return randomBytes(1)[0] } // returns a k-bit random integer function randomBitInt(k) { if (k > 31) throw new Error("Too many bits.") var i = 0, r = 0 var b = Math.floor(k / 8) var mask = (1 << (k % 8)) - 1 if (mask) r = randomByte() & mask for (; i < b; i++) r = (256 * r) + randomByte() return r } return { str2bigInt : str2bigInt , bigInt2str : bigInt2str , int2bigInt : int2bigInt , multMod : multMod , powMod : powMod , inverseMod : inverseMod , randBigInt : randBigInt , randBigInt_ : randBigInt_ , equals : equals , equalsInt : equalsInt , sub : sub , mod : mod , modInt : modInt , mult : mult , divInt_ : divInt_ , rightShift_ : rightShift_ , dup : dup , greater : greater , add : add , isZero : isZero , bitSize : bitSize , millerRabin : millerRabin , divide_ : divide_ , trim : trim , primes : primes , findPrimes : findPrimes , getSeed : getSeed , divMod : divMod , subMod : subMod , twoToThe : twoToThe , bigInt2bits : bigInt2bits , ba2bigInt : ba2bigInt } }))