biginteger.js

// Upstream 'BigInteger' here:
// Original Author: http://www-cs-students.stanford.edu/~tjw/jsbn/
// Follows 'jsbn' on Github: https://github.com/jasondavies/jsbn
// Review and Testing: https://github.com/cryptocoinjs/bigi/
/*!
* Basic JavaScript BN library - subset useful for RSA encryption. v1.4
* 
* Copyright (c) 2005  Tom Wu
* All Rights Reserved.
* BSD License
* http://www-cs-students.stanford.edu/~tjw/jsbn/LICENSE
*
* Copyright Stephan Thomas
* Copyright pointbiz
*/

(function () {

	// (public) Constructor function of Global BigInteger object
	var BigInteger = window.BigInteger = function BigInteger(a, b, c) {
		if (!(this instanceof BigInteger))
			return new BigInteger(a, b, c);

		if (a != null)
			if ("number" == typeof a) this.fromNumber(a, b, c);
			else if (b == null && "string" != typeof a) this.fromString(a, 256);
			else this.fromString(a, b);
	};

	// Bits per digit
	var dbits;

	// JavaScript engine analysis
	var canary = 0xdeadbeefcafe;
	var j_lm = ((canary & 0xffffff) == 0xefcafe);

	// return new, unset BigInteger
	function nbi() { return new BigInteger(null); }

	// am: Compute w_j += (x*this_i), propagate carries,
	// c is initial carry, returns final carry.
	// c < 3*dvalue, x < 2*dvalue, this_i < dvalue
	// We need to select the fastest one that works in this environment.

	// am1: use a single mult and divide to get the high bits,
	// max digit bits should be 26 because
	// max internal value = 2*dvalue^2-2*dvalue (< 2^53)
	function am1(i, x, w, j, c, n) {
		while (--n >= 0) {
			var v = x * this[i++] + w[j] + c;
			c = Math.floor(v / 0x4000000);
			w[j++] = v & 0x3ffffff;
		}
		return c;
	}
	// am2 avoids a big mult-and-extract completely.
	// Max digit bits should be <= 30 because we do bitwise ops
	// on values up to 2*hdvalue^2-hdvalue-1 (< 2^31)
	function am2(i, x, w, j, c, n) {
		var xl = x & 0x7fff, xh = x >> 15;
		while (--n >= 0) {
			var l = this[i] & 0x7fff;
			var h = this[i++] >> 15;
			var m = xh * l + h * xl;
			l = xl * l + ((m & 0x7fff) << 15) + w[j] + (c & 0x3fffffff);
			c = (l >>> 30) + (m >>> 15) + xh * h + (c >>> 30);
			w[j++] = l & 0x3fffffff;
		}
		return c;
	}
	// Alternately, set max digit bits to 28 since some
	// browsers slow down when dealing with 32-bit numbers.
	function am3(i, x, w, j, c, n) {
		var xl = x & 0x3fff, xh = x >> 14;
		while (--n >= 0) {
			var l = this[i] & 0x3fff;
			var h = this[i++] >> 14;
			var m = xh * l + h * xl;
			l = xl * l + ((m & 0x3fff) << 14) + w[j] + c;
			c = (l >> 28) + (m >> 14) + xh * h;
			w[j++] = l & 0xfffffff;
		}
		return c;
	}
	if (j_lm && (navigator.appName == "Microsoft Internet Explorer")) {
		BigInteger.prototype.am = am2;
		dbits = 30;
	}
	else if (j_lm && (navigator.appName != "Netscape")) {
		BigInteger.prototype.am = am1;
		dbits = 26;
	}
	else { // Mozilla/Netscape seems to prefer am3
		BigInteger.prototype.am = am3;
		dbits = 28;
	}

	BigInteger.prototype.DB = dbits;
	BigInteger.prototype.DM = ((1 << dbits) - 1);
	BigInteger.prototype.DV = (1 << dbits);

	var BI_FP = 52;
	BigInteger.prototype.FV = Math.pow(2, BI_FP);
	BigInteger.prototype.F1 = BI_FP - dbits;
	BigInteger.prototype.F2 = 2 * dbits - BI_FP;

	// Digit conversions
	var BI_RM = "0123456789abcdefghijklmnopqrstuvwxyz";
	var BI_RC = new Array();
	var rr, vv;
	rr = "0".charCodeAt(0);
	for (vv = 0; vv <= 9; ++vv) BI_RC[rr++] = vv;
	rr = "a".charCodeAt(0);
	for (vv = 10; vv < 36; ++vv) BI_RC[rr++] = vv;
	rr = "A".charCodeAt(0);
	for (vv = 10; vv < 36; ++vv) BI_RC[rr++] = vv;

	function int2char(n) { return BI_RM.charAt(n); }
	function intAt(s, i) {
		var c = BI_RC[s.charCodeAt(i)];
		return (c == null) ? -1 : c;
	}



	// return bigint initialized to value
	function nbv(i) { var r = nbi(); r.fromInt(i); return r; }


	// returns bit length of the integer x
	function nbits(x) {
		var r = 1, t;
		if ((t = x >>> 16) != 0) { x = t; r += 16; }
		if ((t = x >> 8) != 0) { x = t; r += 8; }
		if ((t = x >> 4) != 0) { x = t; r += 4; }
		if ((t = x >> 2) != 0) { x = t; r += 2; }
		if ((t = x >> 1) != 0) { x = t; r += 1; }
		return r;
	}







	// (protected) copy this to r
	BigInteger.prototype.copyTo = function (r) {
		for (var i = this.t - 1; i >= 0; --i) r[i] = this[i];
		r.t = this.t;
		r.s = this.s;
	};


	// (protected) set from integer value x, -DV <= x < DV
	BigInteger.prototype.fromInt = function (x) {
		this.t = 1;
		this.s = (x < 0) ? -1 : 0;
		if (x > 0) this[0] = x;
		else if (x < -1) this[0] = x + this.DV;
		else this.t = 0;
	};

	// (protected) set from string and radix
	BigInteger.prototype.fromString = function (s, b) {
		var k;
		if (b == 16) k = 4;
		else if (b == 8) k = 3;
		else if (b == 256) k = 8; // byte array
		else if (b == 2) k = 1;
		else if (b == 32) k = 5;
		else if (b == 4) k = 2;
		else { this.fromRadix(s, b); return; }
		this.t = 0;
		this.s = 0;
		var i = s.length, mi = false, sh = 0;
		while (--i >= 0) {
			var x = (k == 8) ? s[i] & 0xff : intAt(s, i);
			if (x < 0) {
				if (s.charAt(i) == "-") mi = true;
				continue;
			}
			mi = false;
			if (sh == 0)
				this[this.t++] = x;
			else if (sh + k > this.DB) {
				this[this.t - 1] |= (x & ((1 << (this.DB - sh)) - 1)) << sh;
				this[this.t++] = (x >> (this.DB - sh));
			}
			else
				this[this.t - 1] |= x << sh;
			sh += k;
			if (sh >= this.DB) sh -= this.DB;
		}
		if (k == 8 && (s[0] & 0x80) != 0) {
			this.s = -1;
			if (sh > 0) this[this.t - 1] |= ((1 << (this.DB - sh)) - 1) << sh;
		}
		this.clamp();
		if (mi) BigInteger.ZERO.subTo(this, this);
	};


	// (protected) clamp off excess high words
	BigInteger.prototype.clamp = function () {
		var c = this.s & this.DM;
		while (this.t > 0 && this[this.t - 1] == c) --this.t;
	};

	// (protected) r = this << n*DB
	BigInteger.prototype.dlShiftTo = function (n, r) {
		var i;
		for (i = this.t - 1; i >= 0; --i) r[i + n] = this[i];
		for (i = n - 1; i >= 0; --i) r[i] = 0;
		r.t = this.t + n;
		r.s = this.s;
	};

	// (protected) r = this >> n*DB
	BigInteger.prototype.drShiftTo = function (n, r) {
		for (var i = n; i < this.t; ++i) r[i - n] = this[i];
		r.t = Math.max(this.t - n, 0);
		r.s = this.s;
	};


	// (protected) r = this << n
	BigInteger.prototype.lShiftTo = function (n, r) {
		var bs = n % this.DB;
		var cbs = this.DB - bs;
		var bm = (1 << cbs) - 1;
		var ds = Math.floor(n / this.DB), c = (this.s << bs) & this.DM, i;
		for (i = this.t - 1; i >= 0; --i) {
			r[i + ds + 1] = (this[i] >> cbs) | c;
			c = (this[i] & bm) << bs;
		}
		for (i = ds - 1; i >= 0; --i) r[i] = 0;
		r[ds] = c;
		r.t = this.t + ds + 1;
		r.s = this.s;
		r.clamp();
	};


	// (protected) r = this >> n
	BigInteger.prototype.rShiftTo = function (n, r) {
		r.s = this.s;
		var ds = Math.floor(n / this.DB);
		if (ds >= this.t) { r.t = 0; return; }
		var bs = n % this.DB;
		var cbs = this.DB - bs;
		var bm = (1 << bs) - 1;
		r[0] = this[ds] >> bs;
		for (var i = ds + 1; i < this.t; ++i) {
			r[i - ds - 1] |= (this[i] & bm) << cbs;
			r[i - ds] = this[i] >> bs;
		}
		if (bs > 0) r[this.t - ds - 1] |= (this.s & bm) << cbs;
		r.t = this.t - ds;
		r.clamp();
	};


	// (protected) r = this - a
	BigInteger.prototype.subTo = function (a, r) {
		var i = 0, c = 0, m = Math.min(a.t, this.t);
		while (i < m) {
			c += this[i] - a[i];
			r[i++] = c & this.DM;
			c >>= this.DB;
		}
		if (a.t < this.t) {
			c -= a.s;
			while (i < this.t) {
				c += this[i];
				r[i++] = c & this.DM;
				c >>= this.DB;
			}
			c += this.s;
		}
		else {
			c += this.s;
			while (i < a.t) {
				c -= a[i];
				r[i++] = c & this.DM;
				c >>= this.DB;
			}
			c -= a.s;
		}
		r.s = (c < 0) ? -1 : 0;
		if (c < -1) r[i++] = this.DV + c;
		else if (c > 0) r[i++] = c;
		r.t = i;
		r.clamp();
	};


	// (protected) r = this * a, r != this,a (HAC 14.12)
	// "this" should be the larger one if appropriate.
	BigInteger.prototype.multiplyTo = function (a, r) {
		var x = this.abs(), y = a.abs();
		var i = x.t;
		r.t = i + y.t;
		while (--i >= 0) r[i] = 0;
		for (i = 0; i < y.t; ++i) r[i + x.t] = x.am(0, y[i], r, i, 0, x.t);
		r.s = 0;
		r.clamp();
		if (this.s != a.s) BigInteger.ZERO.subTo(r, r);
	};


	// (protected) r = this^2, r != this (HAC 14.16)
	BigInteger.prototype.squareTo = function (r) {
		var x = this.abs();
		var i = r.t = 2 * x.t;
		while (--i >= 0) r[i] = 0;
		for (i = 0; i < x.t - 1; ++i) {
			var c = x.am(i, x[i], r, 2 * i, 0, 1);
			if ((r[i + x.t] += x.am(i + 1, 2 * x[i], r, 2 * i + 1, c, x.t - i - 1)) >= x.DV) {
				r[i + x.t] -= x.DV;
				r[i + x.t + 1] = 1;
			}
		}
		if (r.t > 0) r[r.t - 1] += x.am(i, x[i], r, 2 * i, 0, 1);
		r.s = 0;
		r.clamp();
	};



	// (protected) divide this by m, quotient and remainder to q, r (HAC 14.20)
	// r != q, this != m.  q or r may be null.
	BigInteger.prototype.divRemTo = function (m, q, r) {
		var pm = m.abs();
		if (pm.t <= 0) return;
		var pt = this.abs();
		if (pt.t < pm.t) {
			if (q != null) q.fromInt(0);
			if (r != null) this.copyTo(r);
			return;
		}
		if (r == null) r = nbi();
		var y = nbi(), ts = this.s, ms = m.s;
		var nsh = this.DB - nbits(pm[pm.t - 1]); // normalize modulus
		if (nsh > 0) { pm.lShiftTo(nsh, y); pt.lShiftTo(nsh, r); }
		else { pm.copyTo(y); pt.copyTo(r); }
		var ys = y.t;
		var y0 = y[ys - 1];
		if (y0 == 0) return;
		var yt = y0 * (1 << this.F1) + ((ys > 1) ? y[ys - 2] >> this.F2 : 0);
		var d1 = this.FV / yt, d2 = (1 << this.F1) / yt, e = 1 << this.F2;
		var i = r.t, j = i - ys, t = (q == null) ? nbi() : q;
		y.dlShiftTo(j, t);
		if (r.compareTo(t) >= 0) {
			r[r.t++] = 1;
			r.subTo(t, r);
		}
		BigInteger.ONE.dlShiftTo(ys, t);
		t.subTo(y, y); // "negative" y so we can replace sub with am later
		while (y.t < ys) y[y.t++] = 0;
		while (--j >= 0) {
			// Estimate quotient digit
			var qd = (r[--i] == y0) ? this.DM : Math.floor(r[i] * d1 + (r[i - 1] + e) * d2);
			if ((r[i] += y.am(0, qd, r, j, 0, ys)) < qd) {	// Try it out
				y.dlShiftTo(j, t);
				r.subTo(t, r);
				while (r[i] < --qd) r.subTo(t, r);
			}
		}
		if (q != null) {
			r.drShiftTo(ys, q);
			if (ts != ms) BigInteger.ZERO.subTo(q, q);
		}
		r.t = ys;
		r.clamp();
		if (nsh > 0) r.rShiftTo(nsh, r); // Denormalize remainder
		if (ts < 0) BigInteger.ZERO.subTo(r, r);
	};


	// (protected) return "-1/this % 2^DB"; useful for Mont. reduction
	// justification:
	//         xy == 1 (mod m)
	//         xy =  1+km
	//   xy(2-xy) = (1+km)(1-km)
	// x[y(2-xy)] = 1-k^2m^2
	// x[y(2-xy)] == 1 (mod m^2)
	// if y is 1/x mod m, then y(2-xy) is 1/x mod m^2
	// should reduce x and y(2-xy) by m^2 at each step to keep size bounded.
	// JS multiply "overflows" differently from C/C++, so care is needed here.
	BigInteger.prototype.invDigit = function () {
		if (this.t < 1) return 0;
		var x = this[0];
		if ((x & 1) == 0) return 0;
		var y = x & 3; 	// y == 1/x mod 2^2
		y = (y * (2 - (x & 0xf) * y)) & 0xf; // y == 1/x mod 2^4
		y = (y * (2 - (x & 0xff) * y)) & 0xff; // y == 1/x mod 2^8
		y = (y * (2 - (((x & 0xffff) * y) & 0xffff))) & 0xffff; // y == 1/x mod 2^16
		// last step - calculate inverse mod DV directly;
		// assumes 16 < DB <= 32 and assumes ability to handle 48-bit ints
		y = (y * (2 - x * y % this.DV)) % this.DV; 	// y == 1/x mod 2^dbits
		// we really want the negative inverse, and -DV < y < DV
		return (y > 0) ? this.DV - y : -y;
	};


	// (protected) true iff this is even
	BigInteger.prototype.isEven = function () { return ((this.t > 0) ? (this[0] & 1) : this.s) == 0; };


	// (protected) this^e, e < 2^32, doing sqr and mul with "r" (HAC 14.79)
	BigInteger.prototype.exp = function (e, z) {
		if (e > 0xffffffff || e < 1) return BigInteger.ONE;
		var r = nbi(), r2 = nbi(), g = z.convert(this), i = nbits(e) - 1;
		g.copyTo(r);
		while (--i >= 0) {
			z.sqrTo(r, r2);
			if ((e & (1 << i)) > 0) z.mulTo(r2, g, r);
			else { var t = r; r = r2; r2 = t; }
		}
		return z.revert(r);
	};


	// (public) return string representation in given radix
	BigInteger.prototype.toString = function (b) {
		if (this.s < 0) return "-" + this.negate().toString(b);
		var k;
		if (b == 16) k = 4;
		else if (b == 8) k = 3;
		else if (b == 2) k = 1;
		else if (b == 32) k = 5;
		else if (b == 4) k = 2;
		else return this.toRadix(b);
		var km = (1 << k) - 1, d, m = false, r = "", i = this.t;
		var p = this.DB - (i * this.DB) % k;
		if (i-- > 0) {
			if (p < this.DB && (d = this[i] >> p) > 0) { m = true; r = int2char(d); }
			while (i >= 0) {
				if (p < k) {
					d = (this[i] & ((1 << p) - 1)) << (k - p);
					d |= this[--i] >> (p += this.DB - k);
				}
				else {
					d = (this[i] >> (p -= k)) & km;
					if (p <= 0) { p += this.DB; --i; }
				}
				if (d > 0) m = true;
				if (m) r += int2char(d);
			}
		}
		return m ? r : "0";
	};


	// (public) -this
	BigInteger.prototype.negate = function () { var r = nbi(); BigInteger.ZERO.subTo(this, r); return r; };

	// (public) |this|
	BigInteger.prototype.abs = function () { return (this.s < 0) ? this.negate() : this; };

	// (public) return + if this > a, - if this < a, 0 if equal
	BigInteger.prototype.compareTo = function (a) {
		var r = this.s - a.s;
		if (r != 0) return r;
		var i = this.t;
		r = i - a.t;
		if (r != 0) return (this.s < 0) ? -r : r;
		while (--i >= 0) if ((r = this[i] - a[i]) != 0) return r;
		return 0;
	}

	// (public) return the number of bits in "this"
	BigInteger.prototype.bitLength = function () {
		if (this.t <= 0) return 0;
		return this.DB * (this.t - 1) + nbits(this[this.t - 1] ^ (this.s & this.DM));
	};

	// (public) this mod a
	BigInteger.prototype.mod = function (a) {
		var r = nbi();
		this.abs().divRemTo(a, null, r);
		if (this.s < 0 && r.compareTo(BigInteger.ZERO) > 0) a.subTo(r, r);
		return r;
	}

	// (public) this^e % m, 0 <= e < 2^32
	BigInteger.prototype.modPowInt = function (e, m) {
		var z;
		if (e < 256 || m.isEven()) z = new Classic(m); else z = new Montgomery(m);
		return this.exp(e, z);
	};

	// "constants"
	BigInteger.ZERO = nbv(0);
	BigInteger.ONE = nbv(1);







	// Copyright (c) 2005-2009  Tom Wu
	// All Rights Reserved.
	// See "LICENSE" for details.
	// Extended JavaScript BN functions, required for RSA private ops.
	// Version 1.1: new BigInteger("0", 10) returns "proper" zero
	// Version 1.2: square() API, isProbablePrime fix


	// return index of lowest 1-bit in x, x < 2^31
	function lbit(x) {
		if (x == 0) return -1;
		var r = 0;
		if ((x & 0xffff) == 0) { x >>= 16; r += 16; }
		if ((x & 0xff) == 0) { x >>= 8; r += 8; }
		if ((x & 0xf) == 0) { x >>= 4; r += 4; }
		if ((x & 3) == 0) { x >>= 2; r += 2; }
		if ((x & 1) == 0) ++r;
		return r;
	}

	// return number of 1 bits in x
	function cbit(x) {
		var r = 0;
		while (x != 0) { x &= x - 1; ++r; }
		return r;
	}

	var lowprimes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997];
	var lplim = (1 << 26) / lowprimes[lowprimes.length - 1];



	// (protected) return x s.t. r^x < DV
	BigInteger.prototype.chunkSize = function (r) { return Math.floor(Math.LN2 * this.DB / Math.log(r)); };

	// (protected) convert to radix string
	BigInteger.prototype.toRadix = function (b) {
		if (b == null) b = 10;
		if (this.signum() == 0 || b < 2 || b > 36) return "0";
		var cs = this.chunkSize(b);
		var a = Math.pow(b, cs);
		var d = nbv(a), y = nbi(), z = nbi(), r = "";
		this.divRemTo(d, y, z);
		while (y.signum() > 0) {
			r = (a + z.intValue()).toString(b).substr(1) + r;
			y.divRemTo(d, y, z);
		}
		return z.intValue().toString(b) + r;
	};

	// (protected) convert from radix string
	BigInteger.prototype.fromRadix = function (s, b) {
		this.fromInt(0);
		if (b == null) b = 10;
		var cs = this.chunkSize(b);
		var d = Math.pow(b, cs), mi = false, j = 0, w = 0;
		for (var i = 0; i < s.length; ++i) {
			var x = intAt(s, i);
			if (x < 0) {
				if (s.charAt(i) == "-" && this.signum() == 0) mi = true;
				continue;
			}
			w = b * w + x;
			if (++j >= cs) {
				this.dMultiply(d);
				this.dAddOffset(w, 0);
				j = 0;
				w = 0;
			}
		}
		if (j > 0) {
			this.dMultiply(Math.pow(b, j));
			this.dAddOffset(w, 0);
		}
		if (mi) BigInteger.ZERO.subTo(this, this);
	};

	// (protected) alternate constructor
	BigInteger.prototype.fromNumber = function (a, b, c) {
		if ("number" == typeof b) {
			// new BigInteger(int,int,RNG)
			if (a < 2) this.fromInt(1);
			else {
				this.fromNumber(a, c);
				if (!this.testBit(a - 1))	// force MSB set
					this.bitwiseTo(BigInteger.ONE.shiftLeft(a - 1), op_or, this);
				if (this.isEven()) this.dAddOffset(1, 0); // force odd
				while (!this.isProbablePrime(b)) {
					this.dAddOffset(2, 0);
					if (this.bitLength() > a) this.subTo(BigInteger.ONE.shiftLeft(a - 1), this);
				}
			}
		}
		else {
			// new BigInteger(int,RNG)
			var x = new Array(), t = a & 7;
			x.length = (a >> 3) + 1;
			b.nextBytes(x);
			if (t > 0) x[0] &= ((1 << t) - 1); else x[0] = 0;
			this.fromString(x, 256);
		}
	};

	// (protected) r = this op a (bitwise)
	BigInteger.prototype.bitwiseTo = function (a, op, r) {
		var i, f, m = Math.min(a.t, this.t);
		for (i = 0; i < m; ++i) r[i] = op(this[i], a[i]);
		if (a.t < this.t) {
			f = a.s & this.DM;
			for (i = m; i < this.t; ++i) r[i] = op(this[i], f);
			r.t = this.t;
		}
		else {
			f = this.s & this.DM;
			for (i = m; i < a.t; ++i) r[i] = op(f, a[i]);
			r.t = a.t;
		}
		r.s = op(this.s, a.s);
		r.clamp();
	};

	// (protected) this op (1<<n)
	BigInteger.prototype.changeBit = function (n, op) {
		var r = BigInteger.ONE.shiftLeft(n);
		this.bitwiseTo(r, op, r);
		return r;
	};

	// (protected) r = this + a
	BigInteger.prototype.addTo = function (a, r) {
		var i = 0, c = 0, m = Math.min(a.t, this.t);
		while (i < m) {
			c += this[i] + a[i];
			r[i++] = c & this.DM;
			c >>= this.DB;
		}
		if (a.t < this.t) {
			c += a.s;
			while (i < this.t) {
				c += this[i];
				r[i++] = c & this.DM;
				c >>= this.DB;
			}
			c += this.s;
		}
		else {
			c += this.s;
			while (i < a.t) {
				c += a[i];
				r[i++] = c & this.DM;
				c >>= this.DB;
			}
			c += a.s;
		}
		r.s = (c < 0) ? -1 : 0;
		if (c > 0) r[i++] = c;
		else if (c < -1) r[i++] = this.DV + c;
		r.t = i;
		r.clamp();
	};

	// (protected) this *= n, this >= 0, 1 < n < DV
	BigInteger.prototype.dMultiply = function (n) {
		this[this.t] = this.am(0, n - 1, this, 0, 0, this.t);
		++this.t;
		this.clamp();
	};

	// (protected) this += n << w words, this >= 0
	BigInteger.prototype.dAddOffset = function (n, w) {
		if (n == 0) return;
		while (this.t <= w) this[this.t++] = 0;
		this[w] += n;
		while (this[w] >= this.DV) {
			this[w] -= this.DV;
			if (++w >= this.t) this[this.t++] = 0;
			++this[w];
		}
	};

	// (protected) r = lower n words of "this * a", a.t <= n
	// "this" should be the larger one if appropriate.
	BigInteger.prototype.multiplyLowerTo = function (a, n, r) {
		var i = Math.min(this.t + a.t, n);
		r.s = 0; // assumes a,this >= 0
		r.t = i;
		while (i > 0) r[--i] = 0;
		var j;
		for (j = r.t - this.t; i < j; ++i) r[i + this.t] = this.am(0, a[i], r, i, 0, this.t);
		for (j = Math.min(a.t, n); i < j; ++i) this.am(0, a[i], r, i, 0, n - i);
		r.clamp();
	};


	// (protected) r = "this * a" without lower n words, n > 0
	// "this" should be the larger one if appropriate.
	BigInteger.prototype.multiplyUpperTo = function (a, n, r) {
		--n;
		var i = r.t = this.t + a.t - n;
		r.s = 0; // assumes a,this >= 0
		while (--i >= 0) r[i] = 0;
		for (i = Math.max(n - this.t, 0); i < a.t; ++i)
			r[this.t + i - n] = this.am(n - i, a[i], r, 0, 0, this.t + i - n);
		r.clamp();
		r.drShiftTo(1, r);
	};

	// (protected) this % n, n < 2^26
	BigInteger.prototype.modInt = function (n) {
		if (n <= 0) return 0;
		var d = this.DV % n, r = (this.s < 0) ? n - 1 : 0;
		if (this.t > 0)
			if (d == 0) r = this[0] % n;
			else for (var i = this.t - 1; i >= 0; --i) r = (d * r + this[i]) % n;
		return r;
	};


	// (protected) true if probably prime (HAC 4.24, Miller-Rabin)
	BigInteger.prototype.millerRabin = function (t) {
		var n1 = this.subtract(BigInteger.ONE);
		var k = n1.getLowestSetBit();
		if (k <= 0) return false;
		var r = n1.shiftRight(k);
		t = (t + 1) >> 1;
		if (t > lowprimes.length) t = lowprimes.length;
		var a = nbi();
		for (var i = 0; i < t; ++i) {
			//Pick bases at random, instead of starting at 2
			a.fromInt(lowprimes[Math.floor(Math.random() * lowprimes.length)]);
			var y = a.modPow(r, this);
			if (y.compareTo(BigInteger.ONE) != 0 && y.compareTo(n1) != 0) {
				var j = 1;
				while (j++ < k && y.compareTo(n1) != 0) {
					y = y.modPowInt(2, this);
					if (y.compareTo(BigInteger.ONE) == 0) return false;
				}
				if (y.compareTo(n1) != 0) return false;
			}
		}
		return true;
	};



	// (public)
	BigInteger.prototype.clone = function () { var r = nbi(); this.copyTo(r); return r; };

	// (public) return value as integer
	BigInteger.prototype.intValue = function () {
		if (this.s < 0) {
			if (this.t == 1) return this[0] - this.DV;
			else if (this.t == 0) return -1;
		}
		else if (this.t == 1) return this[0];
		else if (this.t == 0) return 0;
		// assumes 16 < DB < 32
		return ((this[1] & ((1 << (32 - this.DB)) - 1)) << this.DB) | this[0];
	};


	// (public) return value as byte
	BigInteger.prototype.byteValue = function () { return (this.t == 0) ? this.s : (this[0] << 24) >> 24; };

	// (public) return value as short (assumes DB>=16)
	BigInteger.prototype.shortValue = function () { return (this.t == 0) ? this.s : (this[0] << 16) >> 16; };

	// (public) 0 if this == 0, 1 if this > 0
	BigInteger.prototype.signum = function () {
		if (this.s < 0) return -1;
		else if (this.t <= 0 || (this.t == 1 && this[0] <= 0)) return 0;
		else return 1;
	};


	// (public) convert to bigendian byte array
	BigInteger.prototype.toByteArray = function () {
		var i = this.t, r = new Array();
		r[0] = this.s;
		var p = this.DB - (i * this.DB) % 8, d, k = 0;
		if (i-- > 0) {
			if (p < this.DB && (d = this[i] >> p) != (this.s & this.DM) >> p)
				r[k++] = d | (this.s << (this.DB - p));
			while (i >= 0) {
				if (p < 8) {
					d = (this[i] & ((1 << p) - 1)) << (8 - p);
					d |= this[--i] >> (p += this.DB - 8);
				}
				else {
					d = (this[i] >> (p -= 8)) & 0xff;
					if (p <= 0) { p += this.DB; --i; }
				}
				if ((d & 0x80) != 0) d |= -256;
				if (k == 0 && (this.s & 0x80) != (d & 0x80)) ++k;
				if (k > 0 || d != this.s) r[k++] = d;
			}
		}
		return r;
	};

	BigInteger.prototype.equals = function (a) { return (this.compareTo(a) == 0); };
	BigInteger.prototype.min = function (a) { return (this.compareTo(a) < 0) ? this : a; };
	BigInteger.prototype.max = function (a) { return (this.compareTo(a) > 0) ? this : a; };

	// (public) this & a
	function op_and(x, y) { return x & y; }
	BigInteger.prototype.and = function (a) { var r = nbi(); this.bitwiseTo(a, op_and, r); return r; };

	// (public) this | a
	function op_or(x, y) { return x | y; }
	BigInteger.prototype.or = function (a) { var r = nbi(); this.bitwiseTo(a, op_or, r); return r; };

	// (public) this ^ a
	function op_xor(x, y) { return x ^ y; }
	BigInteger.prototype.xor = function (a) { var r = nbi(); this.bitwiseTo(a, op_xor, r); return r; };

	// (public) this & ~a
	function op_andnot(x, y) { return x & ~y; }
	BigInteger.prototype.andNot = function (a) { var r = nbi(); this.bitwiseTo(a, op_andnot, r); return r; };

	// (public) ~this
	BigInteger.prototype.not = function () {
		var r = nbi();
		for (var i = 0; i < this.t; ++i) r[i] = this.DM & ~this[i];
		r.t = this.t;
		r.s = ~this.s;
		return r;
	};

	// (public) this << n
	BigInteger.prototype.shiftLeft = function (n) {
		var r = nbi();
		if (n < 0) this.rShiftTo(-n, r); else this.lShiftTo(n, r);
		return r;
	};

	// (public) this >> n
	BigInteger.prototype.shiftRight = function (n) {
		var r = nbi();
		if (n < 0) this.lShiftTo(-n, r); else this.rShiftTo(n, r);
		return r;
	};

	// (public) returns index of lowest 1-bit (or -1 if none)
	BigInteger.prototype.getLowestSetBit = function () {
		for (var i = 0; i < this.t; ++i)
			if (this[i] != 0) return i * this.DB + lbit(this[i]);
		if (this.s < 0) return this.t * this.DB;
		return -1;
	};

	// (public) return number of set bits
	BigInteger.prototype.bitCount = function () {
		var r = 0, x = this.s & this.DM;
		for (var i = 0; i < this.t; ++i) r += cbit(this[i] ^ x);
		return r;
	};

	// (public) true iff nth bit is set
	BigInteger.prototype.testBit = function (n) {
		var j = Math.floor(n / this.DB);
		if (j >= this.t) return (this.s != 0);
		return ((this[j] & (1 << (n % this.DB))) != 0);
	};

	// (public) this | (1<<n)
	BigInteger.prototype.setBit = function (n) { return this.changeBit(n, op_or); };
	// (public) this & ~(1<<n)
	BigInteger.prototype.clearBit = function (n) { return this.changeBit(n, op_andnot); };
	// (public) this ^ (1<<n)
	BigInteger.prototype.flipBit = function (n) { return this.changeBit(n, op_xor); };
	// (public) this + a
	BigInteger.prototype.add = function (a) { var r = nbi(); this.addTo(a, r); return r; };
	// (public) this - a
	BigInteger.prototype.subtract = function (a) { var r = nbi(); this.subTo(a, r); return r; };
	// (public) this * a
	BigInteger.prototype.multiply = function (a) { var r = nbi(); this.multiplyTo(a, r); return r; };
	// (public) this / a
	BigInteger.prototype.divide = function (a) { var r = nbi(); this.divRemTo(a, r, null); return r; };
	// (public) this % a
	BigInteger.prototype.remainder = function (a) { var r = nbi(); this.divRemTo(a, null, r); return r; };
	// (public) [this/a,this%a]
	BigInteger.prototype.divideAndRemainder = function (a) {
		var q = nbi(), r = nbi();
		this.divRemTo(a, q, r);
		return new Array(q, r);
	};

	// (public) this^e % m (HAC 14.85)
	BigInteger.prototype.modPow = function (e, m) {
		var i = e.bitLength(), k, r = nbv(1), z;
		if (i <= 0) return r;
		else if (i < 18) k = 1;
		else if (i < 48) k = 3;
		else if (i < 144) k = 4;
		else if (i < 768) k = 5;
		else k = 6;
		if (i < 8)
			z = new Classic(m);
		else if (m.isEven())
			z = new Barrett(m);
		else
			z = new Montgomery(m);

		// precomputation
		var g = new Array(), n = 3, k1 = k - 1, km = (1 << k) - 1;
		g[1] = z.convert(this);
		if (k > 1) {
			var g2 = nbi();
			z.sqrTo(g[1], g2);
			while (n <= km) {
				g[n] = nbi();
				z.mulTo(g2, g[n - 2], g[n]);
				n += 2;
			}
		}

		var j = e.t - 1, w, is1 = true, r2 = nbi(), t;
		i = nbits(e[j]) - 1;
		while (j >= 0) {
			if (i >= k1) w = (e[j] >> (i - k1)) & km;
			else {
				w = (e[j] & ((1 << (i + 1)) - 1)) << (k1 - i);
				if (j > 0) w |= e[j - 1] >> (this.DB + i - k1);
			}

			n = k;
			while ((w & 1) == 0) { w >>= 1; --n; }
			if ((i -= n) < 0) { i += this.DB; --j; }
			if (is1) {	// ret == 1, don't bother squaring or multiplying it
				g[w].copyTo(r);
				is1 = false;
			}
			else {
				while (n > 1) { z.sqrTo(r, r2); z.sqrTo(r2, r); n -= 2; }
				if (n > 0) z.sqrTo(r, r2); else { t = r; r = r2; r2 = t; }
				z.mulTo(r2, g[w], r);
			}

			while (j >= 0 && (e[j] & (1 << i)) == 0) {
				z.sqrTo(r, r2); t = r; r = r2; r2 = t;
				if (--i < 0) { i = this.DB - 1; --j; }
			}
		}
		return z.revert(r);
	};

	// (public) 1/this % m (HAC 14.61)
	BigInteger.prototype.modInverse = function (m) {
		var ac = m.isEven();
		if (this.signum() === 0) throw new Error('division by zero');
		if ((this.isEven() && ac) || m.signum() == 0) return BigInteger.ZERO;
		var u = m.clone(), v = this.clone();
		var a = nbv(1), b = nbv(0), c = nbv(0), d = nbv(1);
		while (u.signum() != 0) {
			while (u.isEven()) {
				u.rShiftTo(1, u);
				if (ac) {
					if (!a.isEven() || !b.isEven()) { a.addTo(this, a); b.subTo(m, b); }
					a.rShiftTo(1, a);
				}
				else if (!b.isEven()) b.subTo(m, b);
				b.rShiftTo(1, b);
			}
			while (v.isEven()) {
				v.rShiftTo(1, v);
				if (ac) {
					if (!c.isEven() || !d.isEven()) { c.addTo(this, c); d.subTo(m, d); }
					c.rShiftTo(1, c);
				}
				else if (!d.isEven()) d.subTo(m, d);
				d.rShiftTo(1, d);
			}
			if (u.compareTo(v) >= 0) {
				u.subTo(v, u);
				if (ac) a.subTo(c, a);
				b.subTo(d, b);
			}
			else {
				v.subTo(u, v);
				if (ac) c.subTo(a, c);
				d.subTo(b, d);
			}
		}
		if (v.compareTo(BigInteger.ONE) != 0) return BigInteger.ZERO;
		while (d.compareTo(m) >= 0) d.subTo(m, d);
		while (d.signum() < 0) d.addTo(m, d);
		return d;
	};


	// (public) this^e
	BigInteger.prototype.pow = function (e) { return this.exp(e, new NullExp()); };

	// (public) gcd(this,a) (HAC 14.54)
	BigInteger.prototype.gcd = function (a) {
		var x = (this.s < 0) ? this.negate() : this.clone();
		var y = (a.s < 0) ? a.negate() : a.clone();
		if (x.compareTo(y) < 0) { var t = x; x = y; y = t; }
		var i = x.getLowestSetBit(), g = y.getLowestSetBit();
		if (g < 0) return x;
		if (i < g) g = i;
		if (g > 0) {
			x.rShiftTo(g, x);
			y.rShiftTo(g, y);
		}
		while (x.signum() > 0) {
			if ((i = x.getLowestSetBit()) > 0) x.rShiftTo(i, x);
			if ((i = y.getLowestSetBit()) > 0) y.rShiftTo(i, y);
			if (x.compareTo(y) >= 0) {
				x.subTo(y, x);
				x.rShiftTo(1, x);
			}
			else {
				y.subTo(x, y);
				y.rShiftTo(1, y);
			}
		}
		if (g > 0) y.lShiftTo(g, y);
		return y;
	};

	// (public) test primality with certainty >= 1-.5^t
	BigInteger.prototype.isProbablePrime = function (t) {
		var i, x = this.abs();
		if (x.t == 1 && x[0] <= lowprimes[lowprimes.length - 1]) {
			for (i = 0; i < lowprimes.length; ++i)
				if (x[0] == lowprimes[i]) return true;
			return false;
		}
		if (x.isEven()) return false;
		i = 1;
		while (i < lowprimes.length) {
			var m = lowprimes[i], j = i + 1;
			while (j < lowprimes.length && m < lplim) m *= lowprimes[j++];
			m = x.modInt(m);
			while (i < j) if (m % lowprimes[i++] == 0) return false;
		}
		return x.millerRabin(t);
	};


	// JSBN-specific extension

	// (public) this^2
	BigInteger.prototype.square = function () { var r = nbi(); this.squareTo(r); return r; };


	// NOTE: BigInteger interfaces not implemented in jsbn:
	// BigInteger(int signum, byte[] magnitude)
	// double doubleValue()
	// float floatValue()
	// int hashCode()
	// long longValue()
	// static BigInteger valueOf(long val)



	// Copyright Stephan Thomas (start) --- //
	// https://raw.github.com/bitcoinjs/bitcoinjs-lib/07f9d55ccb6abd962efb6befdd37671f85ea4ff9/src/util.js
	// BigInteger monkey patching
	BigInteger.valueOf = nbv;

	/**
	* Returns a byte array representation of the big integer.
	*
	* This returns the absolute of the contained value in big endian
	* form. A value of zero results in an empty array.
	*/
	BigInteger.prototype.toByteArrayUnsigned = function () {
		var ba = this.abs().toByteArray();
		if (ba.length) {
			if (ba[0] == 0) {
				ba = ba.slice(1);
			}
			return ba.map(function (v) {
				return (v < 0) ? v + 256 : v;
			});
		} else {
			// Empty array, nothing to do
			return ba;
		}
	};

	/**
	* Turns a byte array into a big integer.
	*
	* This function will interpret a byte array as a big integer in big
	* endian notation and ignore leading zeros.
	*/
	BigInteger.fromByteArrayUnsigned = function (ba) {
		if (!ba.length) {
			return ba.valueOf(0);
		} else if (ba[0] & 0x80) {
			// Prepend a zero so the BigInteger class doesn't mistake this
			// for a negative integer.
			return new BigInteger([0].concat(ba));
		} else {
			return new BigInteger(ba);
		}
	};

	/**
	* Converts big integer to signed byte representation.
	*
	* The format for this value uses a the most significant bit as a sign
	* bit. If the most significant bit is already occupied by the
	* absolute value, an extra byte is prepended and the sign bit is set
	* there.
	*
	* Examples:
	*
	*      0 =>     0x00
	*      1 =>     0x01
	*     -1 =>     0x81
	*    127 =>     0x7f
	*   -127 =>     0xff
	*    128 =>   0x0080
	*   -128 =>   0x8080
	*    255 =>   0x00ff
	*   -255 =>   0x80ff
	*  16300 =>   0x3fac
	* -16300 =>   0xbfac
	*  62300 => 0x00f35c
	* -62300 => 0x80f35c
	*/
	BigInteger.prototype.toByteArraySigned = function () {
		var val = this.abs().toByteArrayUnsigned();
		var neg = this.compareTo(BigInteger.ZERO) < 0;

		if (neg) {
			if (val[0] & 0x80) {
				val.unshift(0x80);
			} else {
				val[0] |= 0x80;
			}
		} else {
			if (val[0] & 0x80) {
				val.unshift(0x00);
			}
		}

		return val;
	};

	/**
	* Parse a signed big integer byte representation.
	*
	* For details on the format please see BigInteger.toByteArraySigned.
	*/
	BigInteger.fromByteArraySigned = function (ba) {
		// Check for negative value
		if (ba[0] & 0x80) {
			// Remove sign bit
			ba[0] &= 0x7f;

			return BigInteger.fromByteArrayUnsigned(ba).negate();
		} else {
			return BigInteger.fromByteArrayUnsigned(ba);
		}
	};
	// Copyright Stephan Thomas (end) --- //




	// ****** REDUCTION ******* //

	// Modular reduction using "classic" algorithm
	var Classic = window.Classic = function Classic(m) { this.m = m; }
	Classic.prototype.convert = function (x) {
		if (x.s < 0 || x.compareTo(this.m) >= 0) return x.mod(this.m);
		else return x;
	};
	Classic.prototype.revert = function (x) { return x; };
	Classic.prototype.reduce = function (x) { x.divRemTo(this.m, null, x); };
	Classic.prototype.mulTo = function (x, y, r) { x.multiplyTo(y, r); this.reduce(r); };
	Classic.prototype.sqrTo = function (x, r) { x.squareTo(r); this.reduce(r); };





	// Montgomery reduction
	var Montgomery = window.Montgomery = function Montgomery(m) {
		this.m = m;
		this.mp = m.invDigit();
		this.mpl = this.mp & 0x7fff;
		this.mph = this.mp >> 15;
		this.um = (1 << (m.DB - 15)) - 1;
		this.mt2 = 2 * m.t;
	}
	// xR mod m
	Montgomery.prototype.convert = function (x) {
		var r = nbi();
		x.abs().dlShiftTo(this.m.t, r);
		r.divRemTo(this.m, null, r);
		if (x.s < 0 && r.compareTo(BigInteger.ZERO) > 0) this.m.subTo(r, r);
		return r;
	}
	// x/R mod m
	Montgomery.prototype.revert = function (x) {
		var r = nbi();
		x.copyTo(r);
		this.reduce(r);
		return r;
	};
	// x = x/R mod m (HAC 14.32)
	Montgomery.prototype.reduce = function (x) {
		while (x.t <= this.mt2)	// pad x so am has enough room later
			x[x.t++] = 0;
		for (var i = 0; i < this.m.t; ++i) {
			// faster way of calculating u0 = x[i]*mp mod DV
			var j = x[i] & 0x7fff;
			var u0 = (j * this.mpl + (((j * this.mph + (x[i] >> 15) * this.mpl) & this.um) << 15)) & x.DM;
			// use am to combine the multiply-shift-add into one call
			j = i + this.m.t;
			x[j] += this.m.am(0, u0, x, i, 0, this.m.t);
			// propagate carry
			while (x[j] >= x.DV) { x[j] -= x.DV; x[++j]++; }
		}
		x.clamp();
		x.drShiftTo(this.m.t, x);
		if (x.compareTo(this.m) >= 0) x.subTo(this.m, x);
	};
	// r = "xy/R mod m"; x,y != r
	Montgomery.prototype.mulTo = function (x, y, r) { x.multiplyTo(y, r); this.reduce(r); };
	// r = "x^2/R mod m"; x != r
	Montgomery.prototype.sqrTo = function (x, r) { x.squareTo(r); this.reduce(r); };





	// A "null" reducer
	var NullExp = window.NullExp = function NullExp() { }
	NullExp.prototype.convert = function (x) { return x; };
	NullExp.prototype.revert = function (x) { return x; };
	NullExp.prototype.mulTo = function (x, y, r) { x.multiplyTo(y, r); };
	NullExp.prototype.sqrTo = function (x, r) { x.squareTo(r); };





	// Barrett modular reduction
	var Barrett = window.Barrett = function Barrett(m) {
		// setup Barrett
		this.r2 = nbi();
		this.q3 = nbi();
		BigInteger.ONE.dlShiftTo(2 * m.t, this.r2);
		this.mu = this.r2.divide(m);
		this.m = m;
	}
	Barrett.prototype.convert = function (x) {
		if (x.s < 0 || x.t > 2 * this.m.t) return x.mod(this.m);
		else if (x.compareTo(this.m) < 0) return x;
		else { var r = nbi(); x.copyTo(r); this.reduce(r); return r; }
	};
	Barrett.prototype.revert = function (x) { return x; };
	// x = x mod m (HAC 14.42)
	Barrett.prototype.reduce = function (x) {
		x.drShiftTo(this.m.t - 1, this.r2);
		if (x.t > this.m.t + 1) { x.t = this.m.t + 1; x.clamp(); }
		this.mu.multiplyUpperTo(this.r2, this.m.t + 1, this.q3);
		this.m.multiplyLowerTo(this.q3, this.m.t + 1, this.r2);
		while (x.compareTo(this.r2) < 0) x.dAddOffset(1, this.m.t + 1);
		x.subTo(this.r2, x);
		while (x.compareTo(this.m) >= 0) x.subTo(this.m, x);
	};
	// r = x*y mod m; x,y != r
	Barrett.prototype.mulTo = function (x, y, r) { x.multiplyTo(y, r); this.reduce(r); };
	// r = x^2 mod m; x != r
	Barrett.prototype.sqrTo = function (x, r) { x.squareTo(r); this.reduce(r); };

})();