System.BigInt.debug.js

//=============================================================================
// Jocys.com JavaScript.NET Classes               (In C# Object Oriented Style)
// Created by Evaldas Jocys <evaldas@jocys.com>
//=============================================================================
/// <reference path="System.debug.js" />
//=============================================================================
// Namespaces
//-----------------------------------------------------------------------------
// <PropertyGroup>
//		<RootNamespace>System</RootNamespace>
// <PropertyGroup>
//-----------------------------------------------------------------------------
System.Type.RegisterNamespace("System");
//=============================================================================


//=============================================================================
// Extensions
//-----------------------------------------------------------------------------
System.BigInt = function () {
	/// <summary>
	/// </summary>
	/// <remarks>
	///	var big = new System.Numerics.BigInteger();
	/// Code refactored from MS.NET System.Security.Cryptography.BigInt class
	/// </remarks>
	//---------------------------------------------------------
	// Store numbers
	var u = System.BigInt.Utils;
	this.digits = [];

	this.Clear = function () {
		this.digits = [];
	};

	this.CopyFrom = function (a) {
		this.digits = [a.digits.length];
		System.Array.Copy(a.digits, 0, this.digits, 0, a.digits.length);
	};

	this.Clone = function () {
		var bi = new System.BigInt();
		bi.CopyFrom(this);
		return bi;
	};

	this.Divide = function (b) {
	};

	this.Multiply = function (b) {
		System.BigInt.Multiply(this, b, this);
	};

	this.Equals = function (obj) {
		return System.BigInt.Equals(this, obj);
	};

	this.GetHashCode = function () {
	};

	this.IsNegative = function () {
		return u.IsNegative(this.digits);
	};

	this.IsZero = function () {
		return true;
	};

	//#region Convert
	// Decimal: (mbs) "..." (lbs) - Big Endian
	// Hexadecimal (mbs) 0x... (lbs) - Big Endian
	// HexString (lbs) xx-xx-xx-xx... (mbs) - Little Endian, xx - Big Endian.

	this.FromHex = function (s) { this.FromString(s, 16); };
	this.ToHex = function () { return this.ToString(16); };
	this.FromDecimal = function (s) { this.FromString(a, 10); };
	this.ToDecimal = function () { return this.ToString(10); };

	this.FromString = function (s, base) {
		// if number is negative;
		var isNegative = false;
		if (s.indexOf("-") === 0) {
			isNegative = true;
			s = s.substring(1, s.length);
		}
		if (s.indexOf("x") > -1) {
			s = s.substring(s.indexOf("x") + 1, s.length);
			this.digits = u.FromString(s, 16, 0);
		} else if (typeof base === "undefined") {
			this.digits = u.FromString(s, 10, 0);
		} else {
			this.digits = u.FromString(s, base, 0);
		}
		if (isNegative) {
			u.Negate_(this.digits);
		}
	};

	this.ToString = function (base) {
		var s;
		var d = this.digits;
		var isNegative = this.IsNegative();
		if (isNegative) {
			d = u.Negate(d);
		}
		if (typeof base === "undefined") s = u.ToString(d, base);
		else s = u.ToString(d, base);
		if (isNegative) s = "-" + s;
		return s;
	};

	function GetByteArraySize(array, byteValue) {
		var length = array.length;
		while (length-- > 0) {
			if (array[length] !== byteValue) break;
		}
		return length + 1;
	}

	this.ToByteArray = function () {
		// return 
		var d = u.Clone(this.digits);
		var b = u.ToArray(d, 256);
		// If array is negative.
		var isNegative = this.IsNegative();
		if (isNegative) b[b.length - 1] = 0xFF;
		var size = GetByteArraySize(b, isNegative ? 0xFF : 0x00);
		// If last bit of array is negative ( = 1).
		var bNeg = (b[size - 1] & 0x80) !== 0;
		// If BigInt is negative but byte array is positive then...
		if (isNegative && !bNeg) {
			b.push(0xFF);
			size++;
			// Here you will have byte array where highest bit = 1.
			// You can extend array by adding 0xFF bytes.
		}
		// If BigInt is positive but byte array is negative then...
		if (!isNegative && bNeg) {
			// add positive byte.
			b.push(0x00);
			size++;
			// Here you will have byte array where highest bit = 0.
			// You can extend array by adding 0x00 bytes.
		}
		return b.slice(0, size);
	};

	this.FromByteArray = function (bytes) {
		// If last bit of array is negative (= 1).
		var bNeg = ((bytes[bytes.length - 1]) & 0x80) !== 0;
		//if (bNeg){
		// If last byte is all ones.
		// if (bytes[bytes.length-1] == 0xFF);
		//}
		this.digits = u.FromArray(bytes, 256);
	};

	//---------------------------------------------------------
	function initialize0() {
		m_maxbytes = System.BigInt.MaxBytes;
		this.digits = new System.Byte(1);
	}

	function initialize2(b) {
		m_maxbytes = System.BigInt.MaxBytes;
		this.digits = new System.Byte(1);
		this.SetDigit(0, b);
	}

	function initialize() {
		var a = arguments[0];
		switch (typeof (a)) {
			case "string":
				this.FromString.apply(this, arguments);
				break;
			default:
				this.FromString.apply(this, ["0"]);
		}

		//		if("number" == typeof a) this.fromNumber.apply(this, arguments);
		//		else if(b == null && "string" != typeof a) this.FromString.apply(this, arguments);

		// bigInt  FromString(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

		//		else this.fromString(a,b);
		//		if (arguments.length == 0){
		//			initialize0.apply(this, arguments);
		//		}else if (typeof(arguments[0]) == "string") {
		//			initialize1.apply(this, arguments);
		//		}else if ((typeof(arguments[0]) == "number")){
		//			initialize2.apply(this, arguments);
		//		}else{
		//			initialize0.apply(this, arguments);
		//		}
	}
	initialize.apply(this, arguments);
};

//#region Operators

/// <summary>
/// Compares two numbers and returns an integer that indicates their relationship to one another.
/// </summary>
/// <param name="a">BigInt</param>
/// <param name="b">BigInt</param>
/// <returns>
/// -1 (a) is less than (b).
///  0 (a) is equals (b).
///  1 (a) is greater than (b). 
/// </returns>
System.BigInt.Compare = function (a, b) {
	if (a === null && b === null) return 0;
	if (a === null) return -1;
	if (b === null) return 1;
	var size = a.Size();
	var num2 = b.Size();
	if (size === num2) {
		while (size-- > 0) {
			if (a.digits[size] !== b.digits[size]) {
				return (a.digits[size] < b.digits[size]) ? -1 : 1;
			}
		}
		return 0;
	}
	else {
		return (size < num2) ? -1 : 1;
	}
};

System.BigInt.Equals = function (a, b) {
	return System.BigInt.Compare(a, b) === 0;
};

System.BigInt.MoreThan = function (a, b) {
	return System.BigInt.Compare(a, b) === 1;
};

System.BigInt.LessThan = function (a, b) {
	return System.BigInt.Compare(a, b) === -1;
};

System.BigInt._Utils = function () {

	////////////////////////////////////////////////////////////////////////////////////////
	// Big Integer Library v. 5.4
	// Created 2000, last modified 2009
	// Leemon Baird
	// www.leemon.com
	//
	// Version history:
	// v 5.4  3 Oct 2009
	//   - added "var i" to greaterShift() so i is not global. (Thanks to PŽter Szab— for finding that bug)
	//
	// v 5.3  21 Sep 2009
	//   - added randProbPrime(k) for probable primes
	//   - unrolled loop in mont_ (slightly faster)
	//   - millerRabin now takes a bigInt parameter rather than an int
	//
	// v 5.2  15 Sep 2009
	//   - fixed capitalization in call to int2bigInt in randBigInt
	//     (thanks to Emili Evripidou, Reinhold Behringer, and Samuel Macaleese for finding that bug)
	//
	// v 5.1  8 Oct 2007 
	//   - renamed inverseModInt_ to inverseModInt since it doesn't change its parameters
	//   - added functions GCD and randBigInt, which call GCD_ and randBigInt_
	//   - fixed a bug found by Rob Visser (see comment with his name below)
	//   - improved comments
	//
	// This file is public domain.   You can use it for any purpose without restriction.
	// I do not guarantee that it is correct, so use it at your own risk.  If you use 
	// it for something interesting, I'd appreciate hearing about it.  If you find 
	// any bugs or make any improvements, I'd appreciate hearing about those too.
	// It would also be nice if my name and URL were left in the comments.  But none 
	// of that is required.
	//
	// This code defines a bigInt library for arbitrary-precision integers.
	// A bigInt is an array of integers storing the value in chunks of bpe bits, 
	// little endian (buff[0] is the least significant word).
	// Negative bigInts are stored two's complement.  Almost all the functions treat
	// bigInts as nonnegative.  The few that view them as two's complement say so
	// in their comments.  Some functions assume their parameters have at least one 
	// leading zero element. Functions with an underscore at the end of the name put
	// their answer into one of the arrays passed in, and have unpredictable behavior 
	// in case of overflow, so the caller must make sure the arrays are big enough to 
	// hold the answer.  But the average user should never have to call any of the 
	// underscored functions.  Each important underscored function has a wrapper function 
	// of the same name without the underscore that takes care of the details for you.  
	// For each underscored function where a parameter is modified, that same variable 
	// must not be used as another argument too.  So, you cannot square x by doing 
	// multMod_(x,x,n).  You must use squareMod_(x,n) instead, or do y=dup(x); multMod_(x,y,n).
	// Or simply use the multMod(x,x,n) function without the underscore, where
	// such issues never arise, because non-underscored functions never change
	// their parameters; they always allocate new memory for the answer that is returned.
	//
	// These functions are designed to avoid frequent dynamic memory allocation in the inner loop.
	// For most functions, if it needs a BigInt as a local variable it will actually use
	// a global, and will only allocate to it only when it's not the right size.  This ensures
	// that when a function is called repeatedly with same-sized parameters, it only allocates
	// memory on the first call.
	//
	// Note that for cryptographic purposes, the calls to Math.random() must 
	// be replaced with calls to a better pseudorandom number generator.
	//
	// In the following, "bigInt" means a bigInt with at least one leading zero element,
	// and "integer" means a nonnegative integer less than radix.  In some cases, integer 
	// can be negative.  Negative bigInts are 2s complement.
	// 
	// The following functions do not modify their inputs.
	// Those returning a bigInt, string, or Array will dynamically allocate memory for that value.
	// Those returning a boolean will return the integer 0 (false) or 1 (true).
	// Those returning boolean or int will not allocate memory except possibly on the first 
	// time they're called with a given parameter size.
	// 
	// bigInt  add(x,y)               //return (x+y) for bigInts x and y.  
	// bigInt  addInt(x,n)            //return (x+n) where x is a bigInt and n is an integer.
	// string  bigInt2str(x,base)     //return a string form of bigInt x in a given base, with 2 <= base <= 95
	// int     bitSize(x)             //return how many bits long the bigInt x is, not counting leading zeros
	// bigInt  dup(x)                 //return a copy of bigInt x
	// boolean equals(x,y)            //is the bigInt x equal to the bigint y?
	// boolean equalsInt(x,y)         //is bigint x equal to integer y?
	// bigInt  expand(x,n)            //return a copy of x with at least n elements, adding leading zeros if needed
	// Array   findPrimes(n)          //return array of all primes less than integer n
	// bigInt  GCD(x,y)               //return greatest common divisor of bigInts x and y (each with same number of elements).
	// boolean greater(x,y)           //is x>y?  (x and y are nonnegative bigInts)
	// boolean greaterShift(x,y,shift)//is (x <<(shift*bpe)) > y?
	// bigInt  int2bigInt(t,n,m)      //return a bigInt equal to integer t, with at least n bits and m array elements
	// bigInt  inverseMod(x,n)        //return (x**(-1) mod n) for bigInts x and n.  If no inverse exists, it returns null
	// int     inverseModInt(x,n)     //return x**(-1) mod n, for integers x and n.  Return 0 if there is no inverse
	// boolean isZero(x)              //is the bigInt x equal to zero?
	// boolean millerRabin(x,b)       //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is bigInt, 1<b<x)
	// boolean millerRabinInt(x,b)    //does one round of Miller-Rabin base integer b say that bigInt x is possibly prime? (b is int,    1<b<x)
	// bigInt  mod(x,n)               //return a new bigInt equal to (x mod n) for bigInts x and n.
	// int     modInt(x,n)            //return x mod n for bigInt x and integer n.
	// bigInt  mult(x,y)              //return x*y for bigInts x and y. This is faster when y<x.
	// bigInt  multMod(x,y,n)         //return (x*y mod n) for bigInts x,y,n.  For greater speed, let y<x.
	// boolean negative(x)            //is bigInt x negative?
	// bigInt  powMod(x,y,n)          //return (x**y mod n) where x,y,n are bigInts and ** is exponentiation.  0**0=1. Faster for odd n.
	// bigInt  randBigInt(n,s)        //return an n-bit random BigInt (n>=1).  If s=1, then the most significant of those n bits is set to 1.
	// bigInt  randTruePrime(k)       //return a new, random, k-bit, true prime bigInt using Maurer's algorithm.
	// bigInt  randProbPrime(k)       //return a new, random, k-bit, probable prime bigInt (probability it's composite less than 2^-80).
	// bigInt  str2bigInt(s,b,n,m)    //return a bigInt for number represented in string s in base b with at least n bits and m array elements
	// bigInt  sub(x,y)               //return (x-y) for bigInts x and y.  Negative answers will be 2s complement
	// bigInt  trim(x,k)              //return a copy of x with exactly k leading zero elements
	//
	//
	// The following functions each have a non-underscored version, which most users should call instead.
	// These functions each write to a single parameter, and the caller is responsible for ensuring the array 
	// passed in is large enough to hold the result. 
	//
	// void    addInt_(x,n)          //do x=x+n where x is a bigInt and n is an integer
	// void    add_(x,y)             //do x=x+y for bigInts x and y
	// void    copy_(x,y)            //do x=y on bigInts x and y
	// void    copyInt_(x,n)         //do x=n on bigInt x and integer n
	// void    GCD_(x,y)             //set x to the greatest common divisor of bigInts x and y, (y is destroyed).  (This never overflows its array).
	// boolean inverseMod_(x,n)      //do x=x**(-1) mod n, for bigInts x and n. Returns 1 (0) if inverse does (doesn't) exist
	// void    mod_(x,n)             //do x=x mod n for bigInts x and n. (This never overflows its array).
	// void    mult_(x,y)            //do x=x*y for bigInts x and y.
	// void    multMod_(x,y,n)       //do x=x*y  mod n for bigInts x,y,n.
	// void    powMod_(x,y,n)        //do x=x**y mod n, where x,y,n are bigInts (n is odd) and ** is exponentiation.  0**0=1.
	// void    randBigInt_(b,n,s)    //do b = an n-bit random BigInt. if s=1, then nth bit (most significant bit) is set to 1. n>=1.
	// void    randTruePrime_(ans,k) //do ans = a random k-bit true random prime (not just probable prime) with 1 in the msb.
	// void    sub_(x,y)             //do x=x-y for bigInts x and y. Negative answers will be 2s complement.
	//
	// The following functions do NOT have a non-underscored version. 
	// They each write a bigInt result to one or more parameters.  The caller is responsible for
	// ensuring the arrays passed in are large enough to hold the results. 
	//
	// void addShift_(x,y,ys)       //do x=x+(y<<(ys*bpe))
	// void carry_(x)               //do carries and borrows so each element of the bigInt x fits in bpe bits.
	// void divide_(x,y,q,r)        //divide x by y giving quotient q and remainder r
	// int  divInt_(x,n)            //do x=floor(x/n) for bigInt x and integer n, and return the remainder. (This never overflows its array).
	// int  eGCD_(x,y,d,a,b)        //sets a,b,d to positive bigInts such that d = GCD_(x,y) = a*x-b*y
	// void halve_(x)               //do x=floor(|x|/2)*sgn(x) for bigInt x in 2's complement.  (This never overflows its array).
	// void leftShift_(x,n)         //left shift bigInt x by n bits.  n<bpe.
	// void linComb_(x,y,a,b)       //do x=a*x+b*y for bigInts x and y and integers a and b
	// void linCombShift_(x,y,b,ys) //do x=x+b*(y<<(ys*bpe)) for bigInts x and y, and integers b and ys
	// void mont_(x,y,n,np)         //Montgomery multiplication (see comments where the function is defined)
	// void multInt_(x,n)           //do x=x*n where x is a bigInt and n is an integer.
	// void rightShift_(x,n)        //right shift bigInt x by n bits.  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
	var bpe = 0;         //bits stored per array element
	var mask = 0;        //AND this with an array element to chop it down to bpe bits
	var radix = 0;
	var digitsStr = "";
	var one = [];

	//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_()

	rpprb = t; //used in randProbPrimeRounds() (which also uses "primes")

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


	//return array of all primes less than integer n
	function findPrimes(n) {
		var i, s, p, ans;
		s = new Array(n);
		for (i = 0; i < n; i++)
			s[i] = 0;
		s[0] = 2;
		p = 0;    //first p elements of s are primes, the rest are a sieve
		for (; s[p] < n;) {                  //s[p] is the pth prime
			for (i = s[p] * s[p]; i < n; i += s[p]) //mark multiples of s[p]
				s[i] = 1;
			p++;
			s[p] = s[p - 1] + 1;
			for (; s[p] < n && s[s[p]]; s[p]++); //find next prime (where s[p]==0)
		}
		ans = new Array(p);
		for (i = 0; i < p; i++)
			ans[i] = s[i];
		return ans;
	}


	//does a single round of Miller-Rabin base b consider x to be a possible prime?
	//x is a bigInt, and b is an integer, with b<x
	function millerRabinInt(x, b) {
		if (mr_x1.length !== x.length) {
			mr_x1 = dup(x);
			mr_r = dup(x);
			mr_a = dup(x);
		}

		copyInt_(mr_a, b);
		return millerRabin(x, mr_a);
	}

	//does a single round of Miller-Rabin base b consider x to be a possible prime?
	//x and b are bigInts with b<x
	function millerRabin(x, b) {
		var i, j, k, s;

		if (mr_x1.length !== x.length) {
			mr_x1 = dup(x);
			mr_r = dup(x);
			mr_a = dup(x);
		}

		copy_(mr_a, b);
		copy_(mr_r, x);
		copy_(mr_x1, x);

		addInt_(mr_r, -1);
		addInt_(mr_x1, -1);

		//s=the highest power of two that divides mr_r
		k = 0;
		for (i = 0; i < mr_r.length; i++)
			for (j = 1; j < mask; j <<= 1)
				if (x[i] & j) {
					s = (k < mr_r.length + bpe ? k : 0);
					i = mr_r.length;
					j = mask;
				} else
					k++;

		if (s)
			rightShift_(mr_r, s);

		powMod_(mr_a, mr_r, x);

		if (!equalsInt(mr_a, 1) && !equals(mr_a, mr_x1)) {
			j = 1;
			while (j <= s - 1 && !equals(mr_a, mr_x1)) {
				squareMod_(mr_a, x);
				if (equalsInt(mr_a, 1)) {
					return 0;
				}
				j++;
			}
			if (!equals(mr_a, mr_x1)) {
				return 0;
			}
		}
		return 1;
	}

	//returns how many bits long the bigInt is, not counting leading zeros.
	function bitSize(x) {
		var j, z, w;
		for (j = x.length - 1; (x[j] === 0) && (j > 0); j--);
		for (z = 0, w = x[j]; w; (w >>= 1), z++);
		z += bpe * j;
		return z;
	}

	//return a copy of x with at least n elements, adding leading zeros if needed
	function expand(x, n) {
		var ans = int2bigInt(0, (x.length > n ? x.length : n) * bpe, 0);
		copy_(ans, x);
		return ans;
	}

	//return a k-bit true random prime using Maurer's algorithm.
	function randTruePrime(k) {
		var ans = int2bigInt(0, k, 0);
		randTruePrime_(ans, k);
		return trim(ans, 1);
	}

	//return a k-bit random probable prime with probability of error < 2^-80
	function randProbPrime(k) {
		if (k >= 600) return randProbPrimeRounds(k, 2); //numbers from HAC table 4.3
		if (k >= 550) return randProbPrimeRounds(k, 4);
		if (k >= 500) return randProbPrimeRounds(k, 5);
		if (k >= 400) return randProbPrimeRounds(k, 6);
		if (k >= 350) return randProbPrimeRounds(k, 7);
		if (k >= 300) return randProbPrimeRounds(k, 9);
		if (k >= 250) return randProbPrimeRounds(k, 12); //numbers from HAC table 4.4
		if (k >= 200) return randProbPrimeRounds(k, 15);
		if (k >= 150) return randProbPrimeRounds(k, 18);
		if (k >= 100) return randProbPrimeRounds(k, 27);
		return randProbPrimeRounds(k, 40); //number from HAC remark 4.26 (only an estimate)
	}

	//return a k-bit probable random prime using n rounds of Miller Rabin (after trial division with small primes)	
	function randProbPrimeRounds(k, n) {
		var ans, i, divisible, B;
		B = 30000;  //B is largest prime to use in trial division
		ans = int2bigInt(0, k, 0);

		//optimization: try larger and smaller B to find the best limit.

		if (primes.length === 0)
			primes = findPrimes(30000);  //check for divisibility by primes <=30000

		if (rpprb.length !== ans.length)
			rpprb = dup(ans);

		for (; ;) { //keep trying random values for ans until one appears to be prime
			//optimization: pick a random number times L=2*3*5*...*p, plus a 
			//   random element of the list of all numbers in [0,L) not divisible by any prime up to p.
			//   This can reduce the amount of random number generation.

			randBigInt_(ans, k, 0); //ans = a random odd number to check
			ans[0] |= 1;
			divisible = 0;

			//check ans for divisibility by small primes up to B
			for (i = 0; (i < primes.length) && (primes[i] <= B); i++)
				if (modInt(ans, primes[i]) === 0 && !equalsInt(ans, primes[i])) {
					divisible = 1;
					break;
				}

			//optimization: change millerRabin so the base can be bigger than the number being checked, then eliminate the while here.

			//do n rounds of Miller Rabin, with random bases less than ans
			for (i = 0; i < n && !divisible; i++) {
				randBigInt_(rpprb, k, 0);
				while (!greater(ans, rpprb)) //pick a random rpprb that's < ans
					randBigInt_(rpprb, k, 0);
				if (!millerRabin(ans, rpprb))
					divisible = 1;
			}

			if (!divisible)
				return ans;
		}
	}

	//return a new bigInt equal to (x mod n) for bigInts x and n.
	function mod(x, n) {
		var ans = dup(x);
		mod_(ans, n);
		return trim(ans, 1);
	}

	//return (x+n) where x is a bigInt and n is an integer.
	function addInt(x, n) {
		var ans = expand(x, x.length + 1);
		addInt_(ans, n);
		return trim(ans, 1);
	}

	//return x*y for bigInts x and y. This is faster when y<x.
	function mult(x, y) {
		var ans = expand(x, x.length + y.length);
		mult_(ans, y);
		return trim(ans, 1);
	}

	//return (x**y mod n) where x,y,n are bigInts and ** is exponentiation.  0**0=1. Faster for odd n.
	function powMod(x, y, n) {
		var ans = expand(x, n.length);
		powMod_(ans, trim(y, 2), trim(n, 2), 0);  //this should work without the trim, but doesn't
		return trim(ans, 1);
	}

	//return (x-y) for bigInts x and y.  Negative answers will be 2s complement
	function sub(x, y) {


		var xN = negative(x);
		var yN = negative(y);
		var x1 = x;
		var y1 = y;
		var z;
		// Make positive.
		if (xN) x1 = negate(x);
		if (yN) y1 = negate(y);
		if (xN) {
			if (yN) {
				if (greater(x1, y1)) {
					z = sub(x1, y1);
					negate_(z);
					return z;
				} else {
					return sub(y1, x1);
				}
			} else {
				z = add(x1, y);
				negate_(z);
				return z;
			}
		} else {
			if (yN) {
				return add(x, y1);
			} else {
				if (!greater(x1, y1)) {
					z = sub(y1, x);
					negate_(z);
					return z;
				}
			}
		}

		var ans = expand(x, x.length > y.length ? x.length + 1 : y.length + 1);
		sub_(ans, y);
		return trim(ans, 1);
	}

	//return (x+y) for bigInts x and y.  
	function add(x, y) {
		var xN = negative(x);
		var yN = negative(y);
		var x1 = x;
		var y1 = y;
		var z;
		// Make positive.
		if (xN) x1 = negate(x);
		if (yN) y1 = negate(y);
		if (xN) {
			if (yN) {
				z = add(x1, y1);
				negate_(z);
				return z;
			} else {
				if (greater(y1, x1)) {
					return sub(y1, x1);
				} else {
					z = sub(x1, y1);
					negate_(z);
					return z;
				}
			}
		} else {
			if (yN) {
				if (greater(x1, y1)) {
					return sub(x1, y1);
				} else {
					z = sub(y1, x1);
					negate_(z);
					return z;
				}
			}
		}



		var ans = expand(x, (x.length > y.length ? x.length + 1 : y.length + 1));
		add_(ans, y);
		return trim(ans, 1);
	}

	//return (x**(-1) mod n) for bigInts x and n.  If no inverse exists, it returns null
	function inverseMod(x, n) {
		var ans = expand(x, n.length);
		var s;
		s = inverseMod_(ans, n);
		return s ? trim(ans, 1) : null;
	}

	//return (x*y mod n) for bigInts x,y,n.  For greater speed, let y<x.
	function multMod(x, y, n) {
		var ans = expand(x, n.length);
		multMod_(ans, y, n);
		return trim(ans, 1);
	}

	//generate a k-bit true random prime using Maurer's algorithm,
	//and put it into ans.  The bigInt ans must be large enough to hold it.
	function randTruePrime_(ans, k) {
		var c, 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 && !equalsInt(s_n, primes[j])) {
					divisible = 1;
					break;
				}

			if (!divisible)    //if it passes small primes check, then try a single Miller-Rabin base 2
				if (!millerRabinInt(s_n, 2)) //this line represents 75% of the total runtime for randTruePrime_ 
					divisible = 1;

			if (!divisible) {  //if it passes that test, continue checking s_n
				addInt_(s_n, -3);
				for (j = s_n.length - 1; (s_n[j] === 0) && (j > 0); j--);  //strip leading zeros
				for (zz = 0, w = s_n[j]; w; (w >>= 1), zz++);
				zz += bpe * j;                             //zz=number of bits in s_n, ignoring leading zeros
				for (; ;) {  //generate z-bit numbers until one falls in the range [0,s_n-1]
					randBigInt_(s_a, zz, 0);
					if (greater(s_n, s_a))
						break;
				}                //now s_a is in the range [0,s_n-1]
				addInt_(s_n, 3);  //now s_a is in the range [0,s_n-4]
				addInt_(s_a, 2);  //now s_a is in the range [2,s_n-2]
				copy_(s_b, s_a);
				copy_(s_n1, s_n);
				addInt_(s_n1, -1);
				powMod_(s_b, s_n1, s_n);   //s_b=s_a^(s_n-1) modulo s_n
				addInt_(s_b, -1);
				if (isZero(s_b)) {
					copy_(s_b, s_a);
					powMod_(s_b, s_r2, s_n);
					addInt_(s_b, -1);
					copy_(s_aa, s_n);
					copy_(s_d, s_b);
					GCD_(s_d, s_n);  //if s_b and s_n are relatively prime, then s_n is a prime
					if (equalsInt(s_d, 1)) {
						copy_(ans, s_aa);
						return;     //if we've made it this far, then s_n is absolutely guaranteed to be prime
					}
				}
			}
		}
	}

	//Return an n-bit random BigInt (n>=1).  If s=1, then the most significant of those n bits is set to 1.
	function randBigInt(n, s) {
		var a, b;
		a = Math.floor((n - 1) / bpe) + 2; //# array elements to hold the BigInt with a leading 0 element
		b = int2bigInt(0, 0, a);
		randBigInt_(b, n, s);
		return b;
	}

	//Set b to an n-bit random BigInt.  If s=1, then the most significant of those n bits is set to 1.
	//Array b must be big enough to hold the result. Must have n>=1
	function randBigInt_(b, n, s) {
		var i, a;
		for (i = 0; i < b.length; i++)
			b[i] = 0;
		a = Math.floor((n - 1) / bpe) + 1; //# array elements to hold the BigInt
		for (i = 0; i < a; i++) {
			b[i] = 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);
	}

	function signum(x) {
		return negative(x) ? -1 : 0;
	}


	//is (x << (shift*bpe)) > y?
	//x and y are nonnegative bigInts
	//shift is a nonnegative integer
	function greaterShift(x, y, shift) {
		var i, kx = x.length, ky = y.length;
		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)
	var greater = function (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) {   //stop at first illegal character
				break;
			}
			multInt_(x, base);
			addInt_(x, d);
		}

		for (k = x.length; k > 0 && !x[k - 1]; k--); //strip off leading zeros
		k = minSize > k + 1 ? minSize : k + 1;
		y = new Array(k);
		kk = k < x.length ? k : x.length;
		for (i = 0; i < kk; i++)
			y[i] = x[i];
		for (; i < k; i++)
			y[i] = 0;
		return y;
	}

	//is bigint x equal to integer y?
	//y must have less than bpe bits
	function equalsInt(x, y) {
		var i;
		if (x[0] !== y)
			return 0;
		for (i = 1; i < x.length; i++)
			if (x[i])
				return 0;
		return 1;
	}

	//are bigints x and y equal?
	//this works even if x and y are different lengths and have arbitrarily many leading zeros
	function equals(x, y) {
		var i;
		var k = x.length < y.length ? x.length : y.length;
		for (i = 0; i < k; i++)
			if (x[i] !== y[i])
				return 0;
		if (x.length > y.length) {
			for (; i < x.length; i++)
				if (x[i])
					return 0;
		} else {
			for (; i < y.length; i++)
				if (y[i])
					return 0;
		}
		return 1;
	}

	//is the bigInt x equal to zero?
	function isZero(x) {
		var i;
		for (i = 0; i < x.length; i++)
			if (x[i])
				return 0;
		return 1;
	}

	//convert a bigInt into a string in a given base, from base 2 up to base 95.
	//Base -1 prints the contents of the array representing the number.
	function bigInt2str(x, base) {
		var i, t, s = "";

		if (s6.length !== x.length)
			s6 = dup(x);
		else
			copy_(s6, x);

		if (base === -1) { //return the list of array contents
			for (i = x.length - 1; i > 0; i--)
				s += x[i] + ',';
			s += x[0];
		}
		else { //return it in the given base
			while (!isZero(s6)) {
				t = divInt_(s6, base);  //t=s6 % base; s6=floor(s6/base);
				s = digitsStr.substring(t, t + 1) + s;
			}
		}
		if (s.length === 0)
			s = "0";
		return s;
	}

	//returns a duplicate of bigInt x
	function dup(x) {
		var i;
		buff = 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 xN = negative(x);
		//		var yN = negative(y);
		//		var z, y1;
		//		if (xN) negate_(x);
		//		if (yN) y1 = negate(y);
		//		if (xN){
		//			if (yN){
		//				if (greater(x, y1)){
		//					sub_(x, y1);
		//					negate_(x);
		//					return;
		//				}else{
		//					z = sub(y1, x);
		//					copy_(x, z);
		//					return;
		//				}
		//			}else{
		//				add_(x, y);
		//				negate_(x);
		//				return;
		//			}
		//		}else{
		//			if (yN){
		//				add_(x, y);
		//				return;
		//			}else{
		//				if (!greater(x, y)){
		//					z = sub(y, x);
		//					copy_(x, z);
		//					negate_(x);
		//					return; 
		//				}
		//			}
		//		}


		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 xN = negative(x);
		var yN = negative(y);
		var z, y1;
		//		if (xN) negate_(x);
		if (yN) y1 = negate(y);
		if (xN) {
			//			if (yN){
			//				add_(x, y1);
			//				negate_(x);
			//				return;
			//			}else{
			//				if (greater(y1, x)){
			//					z = sub(y1, x);
			//					copy_(x, z);
			//					return;
			//				}else{
			//					sub_(x, y1);
			//					negate_(x);
			//					return;
			//				}
			//			}	
		} else {
			if (yN) {
				if (greater(x, y1)) {
					sub_(x, y1);
					return;
				} else {
					z = sub(y1, x);
					copy_(x, z);
					negate_(x);
					return;
				}
			}
		}

		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 trim(x, k) {
		var i, y;
		for (i = x.length; i > 0 && !x[i - 1]; i--);
		y = new Array(i + k);
		copy_(y, x);
		return y;
	}

	//do x=x**y mod n, where x,y,n are bigInts and ** is exponentiation.  0**0=1.
	//this is faster when n is odd.  x usually needs to have as many elements as n.
	function powMod_(x, y, n) {
		var k1, k2, kn, np;
		if (s7.length !== n.length)
			s7 = dup(n);

		//for even modulus, use a simple square-and-multiply algorithm,
		//rather than using the more complex Montgomery algorithm.
		if ((n[0] & 1) === 0) {
			copy_(s7, x);
			copyInt_(x, 1);
			while (!equalsInt(y, 0)) {
				if (y[0] & 1)
					multMod_(x, s7, n);
				divInt_(y, 2);
				squareMod_(s7, n);
			}
			return;
		}

		//calculate np from n for the Montgomery multiplications
		copyInt_(s7, 0);
		for (kn = n.length; kn > 0 && !n[kn - 1]; kn--);
		np = radix - inverseModInt(modInt(n, radix), radix);
		s7[kn] = 1;
		multMod_(x, s7, n);   // x = x * 2**(kn*bp) mod n

		if (s3.length !== x.length)
			s3 = dup(x);
		else
			copy_(s3, x);

		for (k1 = y.length - 1; k1 > 0 & !y[k1]; k1--);  //k1=first nonzero element of y
		if (y[k1] === 0) {  //anything to the 0th power is 1
			copyInt_(x, 1);
			return;
		}
		for (k2 = 1 << (bpe - 1); k2 && !(y[k1] & k2); k2 >>= 1);  //k2=position of first 1 bit in y[k1]
		for (; ;) {
			if (!(k2 >>= 1)) {  //look at next bit of y
				k1--;
				if (k1 < 0) {
					mont_(x, one, n, np);
					return;
				}
				k2 = 1 << (bpe - 1);
			}
			mont_(x, x, n, np);

			if (k2 & y[k1]) //if next bit is a 1
				mont_(x, s3, n, np);
		}
	}


	//do x=x*y*Ri mod n for bigInts x,y,n, 
	//  where Ri = 2**(-kn*bpe) mod n, and kn is the 
	//  number of elements in the n array, not 
	//  counting leading zeros.  
	//x array must have at least as many elemnts as the n array
	//It's OK if x and y are the same variable.
	//must have:
	//  x,y < n
	//  n is odd
	//  np = -(n^(-1)) mod radix
	function mont_(x, y, n, np) {
		var i, j, c, ui, t, ks;
		var kn = n.length;
		var ky = y.length;

		if (sa.length !== kn)
			sa = new Array(kn);

		copyInt_(sa, 0);

		for (; kn > 0 && n[kn - 1] === 0; kn--); //ignore leading zeros of n
		for (; ky > 0 && y[ky - 1] === 0; ky--); //ignore leading zeros of y
		ks = sa.length - 1; //sa will never have more than this many nonzero elements.  

		//the following loop consumes 95% of the runtime for randTruePrime_() and powMod_() for large numbers
		for (i = 0; i < kn; i++) {
			t = sa[0] + x[i] * y[0];
			ui = ((t & mask) * np) & mask;  //the inner "& mask" was needed on Safari (but not MSIE) at one time
			c = (t + ui * n[0]) >> bpe;
			t = x[i];

			//do sa=(sa+x[i]*y+ui*n)/b   where b=2**bpe.  Loop is unrolled 5-fold for speed
			j = 1;
			for (; j < ky - 4;) {
				c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
			}
			for (; j < ky;) { c += sa[j] + ui * n[j] + t * y[j]; sa[j - 1] = c & mask; c >>= bpe; j++; }
			for (; j < kn - 4;) {
				c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
				c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++;
			}
			for (; j < kn;) { c += sa[j] + ui * n[j]; sa[j - 1] = c & mask; c >>= bpe; j++; }
			for (; j < ks;) { c += sa[j]; sa[j - 1] = c & mask; c >>= bpe; j++; }
			sa[j - 1] = c & mask;
		}

		if (!greater(n, sa))
			sub_(sa, n);
		copy_(x, sa);
	}

	//------------------------------------------------------------
	// add, add_, sub, sub_ methods were modified to support negative big ints.
	//------------------------------------------------------------

	function negate(x) {
		var y = dup(x);
		multInt_(y, -1);
		return y;
	}

	function negate_(x) {
		multInt_(x, -1);
	}

	this.ToArray = function (x, base) {
		var i, t;
		var s = [];
		if (s6.length !== x.length)
			s6 = dup(x);
		else
			copy_(s6, x);

		if (base === -1) { //return the list of array contents
			for (i = 0; i < x.length; i++) s.push(x[i]);
		}
		else { //return it in the given base
			while (!isZero(s6)) {
				t = divInt_(s6, base);  //t=s6 % base; s6=floor(s6/base);
				s.push(t);
			}
		}
		if (s.length === 0) s.push(0);
		return s;
	};

	this.FromArray = function (s, base, minSize) {
		var d, i, j, x, y, kk;

		var k = s.length;
		x = int2bigInt(0, base * k, 0);
		for (i = 0; i < k; i++) {
			d = s[i];
			if (d >= base || d < 0) {   //stop at first illegal character
				break;
			}
			multInt_(x, base);
			addInt_(x, d);
		}

		for (k = x.length; k > 0 && !x[k - 1]; k--); //strip off leading zeros
		k = minSize > k + 1 ? minSize : k + 1;
		y = new Array(k);
		kk = k < x.length ? k : x.length;
		for (i = 0; i < kk; i++)
			y[i] = x[i];
		for (; i < k; i++)
			y[i] = 0;
		return y;
	};

	var greater2 = greater;

	greater = function (x, y) {

		return greater2(x, y) === 1;

	};

	this.ToBytes = function (x) { return this.ToArray(x, 256); };
	this.FromBytes = function (bytes) { return this.FromArray(bytes, 256, 0); };

	this._initialize = function () {

		this.ElementSize = bpe;
		this.ElementMask = mask;
		this.ElementRadix = radix;

		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_()

		this.Add = add;
		this.AddInt = addInt;
		this.ToString = bigInt2str;
		this.BitCount = bitSize;
		this.Clone = dup;
		this.Equals = equals;
		this.EqualsInt = equalsInt;
		this.Expand = expand;
		this.GetPrimes = findPrimes;
		this.GCD = GCD;
		this.MoreThan = greater;
		this.MoreThanShitf = greaterShift;
		this.FromInt = int2bigInt;
		this.InverseMod = inverseMod;
		this.InverseModInt = inverseModInt;
		this.IsZero = isZero;
		this.IsProbPrime = millerRabin;
		this.IsPronPrimeInt = millerRabinInt;
		this.Mod = mod;
		this.ModInt = modInt;
		this.Multiply = mult;
		this.MultiplyMod = multMod;
		this.IsNegative = negative;
		this.PowMod = powMod;
		this.NewBigInt = randBigInt;
		this.NewPrime = randTruePrime;
		this.NewProbPrime = randProbPrime;
		this.FromString = str2bigInt;
		this.Subtract = sub;
		this.Trim = trim;

		this.Negate = negate;
		this.Negate_ = negate_;

		this.Add_ = add_;
		this.AddInt_ = addInt_;
		this.Clone_ = copy_;
		this.CloneInt_ = copyInt_;
		this.GCD_ = GCD_;
		this.InverseMod_ = inverseMod_;
		this.Mod_ = mod_;
		this.Multiply_ = mult_;
		this.MultiplyMod_ = multMod_;
		this.PowMod_ = powMod_;
		this.NewBigInt_ = randBigInt_;
		this.NewPrime_ = randTruePrime_;
		this.Subtract_ = sub_;

		this.AddShift_ = addShift_;
		this.Carry_ = carry_;
		this.Divide_ = divide_;
		this.DivideInt_ = divInt_;
		this.eGCD_ = eGCD_;
		this.Halve_ = halve_;
		this.LeftShift_ = leftShift_;
		this.LinComb_ = linComb_;
		this.LinCombShift_ = linCombShift_;
		this.MontMultiply_ = mont_;
		this.MultiplyInt_ = multInt_;
		this.RightShift_ = rightShift_;
		this.SquareMod_ = squareMod_;
		this.SubtractShift_ = subShift_;

	};
	this._initialize.apply(this, arguments);

};

System.BigInt.Utils = new System.BigInt._Utils();

System.BigInt.Add = function (a, b) {
	var bi = new System.BigInt();
	bi.digits = System.BigInt.Utils.Add(a.digits, b.digits);
	return bi;
};
System.BigInt.Divide = function (a, b, qBi, rBi) {
	qBi.digits = new Array(a.digits.length);
	rBi.digits = new Array(a.digits.length);
	System.BigInt.Utils.Divide_(a.digits, b.digits, qBi.digits, rBi.digits);
};
System.BigInt.Negate = function (a) {
	System.BigInt.Utils.Negate_(a.digits);
};
System.BigInt.Multiply = function (a, b) {
	var bi = new System.BigInt();
	bi.digits = System.BigInt.Utils.Multiply(a.digits, b.digits);
	return bi;
};
System.BigInt.Subtract = function (a, b) {
	var bi = new System.BigInt();
	bi.digits = System.BigInt.Utils.Subtract(a.digits, b.digits);
	return bi;
};

//==============================================================================
// END
//------------------------------------------------------------------------------