////////////////////////////////////////////////////////////////////////////////////////

// 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);