+ − ////////////////////////////////////////////////////////////////////////////////////////
+ − // 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.
+ − ////////////////////////////////////////////////////////////////////////////////////////
+ −
+ − //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_()
+ −
+ − ////////////////////////////////////////////////////////////////////////////////////////
+ −
+ − //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
+ − 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);
+ − }
+ −
+ − 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 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 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 complement
+ − function 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 null
+ − function 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>=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)) {
+ − 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 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)) {
+ − 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 integer
+ − function 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 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 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 radix
+ − function 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);
+ − }
+ −
+ −