////////////////////////////////////////////////////////////////////////////////////////// Big Integer Library v. 5.1// Created 2000, last modified 2007// Leemon Baird// www.leemon.com//// Version history://// 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 address 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.// Some functions assume their parameters have at least one leading zero element.// Functions with an underscore at the end of the name have unpredictable behavior in case of overflow, // so the caller must make sure the arrays must be big enough to hold the answer.// For each 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).//// 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 (as opposed to definitely composite)?// 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 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 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.//////////////////////////////////////////////////////////////////////////////////////////globalsbpe=0; //bits stored per array elementmask=0; //AND this with an array element to chop it down to bpe bitsradix=mask+1; //equals 2^bpe. A single 1 bit to the left of the last bit of mask.//the digits for converting to different basesdigitsStr='0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz_=!@#$%^&*()[]{}|;:,.<>/?`~ \\\'\"+-';//initialize the global variablesfor (bpe=0; (1<<(bpe+1)) > (1<<bpe); bpe++); //bpe=number of bits in the mantissa on this platformbpe>>=1; //bpe=number of bits in one element of the array representing the bigIntmask=(1<<bpe)-1; //AND the mask with an integer to get its bpe least significant bitsradix=mask+1; //2^bpe. a single 1 bit to the left of the first bit of maskone=int2bigInt(1,1,1); //constant used in powMod_()//the following global variables are scratchpad memory to //reduce dynamic memory allocation in the inner loopt=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_()//////////////////////////////////////////////////////////////////////////////////////////return array of all primes less than integer nfunction 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 integerfunction 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); } copyInt_(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 neededfunction 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 bigint_trim(ans,1);}//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 bigint_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 bigint_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 bigint_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,bigint_trim(y,2),bigint_trim(n,2),0); //this should work without the trim, but doesn't return bigint_trim(ans,1);}//return (x-y) for bigInts x and y. Negative answers will be 2s complementfunction sub(x,y) { var ans=expand(x,(x.length>y.length ? x.length+1 : y.length+1)); sub_(ans,y); return bigint_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 bigint_trim(ans,1);}//return (x**(-1) mod n) for bigInts x and n. If no inverse exists, it returns nullfunction inverseMod(x,n) { var ans=expand(x,n.length); var s; s=inverseMod_(ans,n); return s ? bigint_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 bigint_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) { divisible=1; break; } if (!divisible) //if it passes small primes check, then try a single Miller-Rabin base 2 if (!millerRabin(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>=1function 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)) { if (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 inversefunction 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)) { if (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 integerfunction greaterShift(x,y,shift) { var 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) { //ignore illegal characters 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 bitsfunction 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 zerosfunction 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 xfunction 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 complementfunction 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 remainderfunction 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 complementfunction 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 elementsfunction bigint_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 must be large enough to hold the answer.//It's OK if x and y are the same variable.//must have:// x,y < n// n is odd// np = -(n^(-1)) mod radixfunction mont_(x,y,n,np) { var i,j,c,ui,t; var kn=n.length; var ky=y.length; if (sa.length!=kn) sa=new Array(kn); for (;kn>0 && n[kn-1]==0;kn--); //ignore leading zeros of n //this function sometimes gives wrong answers when the next line is uncommented //for (;ky>0 && y[ky-1]==0;ky--); //ignore leading zeros of y copyInt_(sa,0); //the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large keys for (i=0; i<kn; i++) { t=sa[0]+x[i]*y[0]; ui=((t & mask) * np) & mask; //the inner "& mask" is needed on Macintosh MSIE, but not windows MSIE c=(t+ui*n[0]) >> bpe; t=x[i]; //do sa=(sa+x[i]*y+ui*n)/b where b=2**bpe for (j=1;j<ky;j++) { c+=sa[j]+t*y[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; } for (;j<kn;j++) { c+=sa[j]+ui*n[j]; sa[j-1]=c & mask; c>>=bpe; } sa[j-1]=c & mask; } if (!greater(n,sa)) sub_(sa,n); copy_(x,sa);}