////////////////////////////////////////////////////////////////////////////////////////
// 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);
}