/**
* Constructs a new bignum from another bignum, a number or a hex string.
* @constructor
*/
sjcl.bn = function(it) {
this.initWith(it);
};
sjcl.bn.prototype = {
radix: 24,
maxMul: 8,
_class: sjcl.bn,
copy: function() {
return new this._class(this);
},
/**
* Initializes this with it, either as a bn, a number, or a hex string.
*/
initWith: function(it) {
var i=0, k;
switch(typeof it) {
case "object":
this.limbs = it.limbs.slice(0);
break;
case "number":
this.limbs = [it];
this.normalize();
break;
case "string":
it = it.replace(/^0x/, '');
this.limbs = [];
// hack
k = this.radix / 4;
for (i=0; i < it.length; i+=k) {
this.limbs.push(parseInt(it.substring(Math.max(it.length - i - k, 0), it.length - i),16));
}
break;
default:
this.limbs = [0];
}
return this;
},
/**
* Returns true if "this" and "that" are equal. Calls fullReduce().
* Equality test is in constant time.
*/
equals: function(that) {
if (typeof that === "number") { that = new this._class(that); }
var difference = 0, i;
this.fullReduce();
that.fullReduce();
for (i = 0; i < this.limbs.length || i < that.limbs.length; i++) {
difference |= this.getLimb(i) ^ that.getLimb(i);
}
return (difference === 0);
},
/**
* Get the i'th limb of this, zero if i is too large.
*/
getLimb: function(i) {
return (i >= this.limbs.length) ? 0 : this.limbs[i];
},
/**
* Constant time comparison function.
* Returns 1 if this >= that, or zero otherwise.
*/
greaterEquals: function(that) {
if (typeof that === "number") { that = new this._class(that); }
var less = 0, greater = 0, i, a, b;
i = Math.max(this.limbs.length, that.limbs.length) - 1;
for (; i>= 0; i--) {
a = this.getLimb(i);
b = that.getLimb(i);
greater |= (b - a) & ~less;
less |= (a - b) & ~greater;
}
return (greater | ~less) >>> 31;
},
/**
* Convert to a hex string.
*/
toString: function() {
this.fullReduce();
var out="", i, s, l = this.limbs;
for (i=0; i < this.limbs.length; i++) {
s = l[i].toString(16);
while (i < this.limbs.length - 1 && s.length < 6) {
s = "0" + s;
}
out = s + out;
}
return "0x"+out;
},
/** this += that. Does not normalize. */
addM: function(that) {
if (typeof(that) !== "object") { that = new this._class(that); }
var i, l=this.limbs, ll=that.limbs;
for (i=l.length; i<ll.length; i++) {
l[i] = 0;
}
for (i=0; i<ll.length; i++) {
l[i] += ll[i];
}
return this;
},
/** this *= 2. Requires normalized; ends up normalized. */
doubleM: function() {
var i, carry=0, tmp, r=this.radix, m=this.radixMask, l=this.limbs;
for (i=0; i<l.length; i++) {
tmp = l[i];
tmp = tmp+tmp+carry;
l[i] = tmp & m;
carry = tmp >> r;
}
if (carry) {
l.push(carry);
}
return this;
},
/** this /= 2, rounded down. Requires normalized; ends up normalized. */
halveM: function() {
var i, carry=0, tmp, r=this.radix, l=this.limbs;
for (i=l.length-1; i>=0; i--) {
tmp = l[i];
l[i] = (tmp+carry)>>1;
carry = (tmp&1) << r;
}
if (!l[l.length-1]) {
l.pop();
}
return this;
},
/** this -= that. Does not normalize. */
subM: function(that) {
if (typeof(that) !== "object") { that = new this._class(that); }
var i, l=this.limbs, ll=that.limbs;
for (i=l.length; i<ll.length; i++) {
l[i] = 0;
}
for (i=0; i<ll.length; i++) {
l[i] -= ll[i];
}
return this;
},
mod: function(that) {
var neg = !this.greaterEquals(new sjcl.bn(0));
that = new sjcl.bn(that).normalize(); // copy before we begin
var out = new sjcl.bn(this).normalize(), ci=0;
if (neg) out = (new sjcl.bn(0)).subM(out).normalize();
for (; out.greaterEquals(that); ci++) {
that.doubleM();
}
if (neg) out = that.sub(out).normalize();
for (; ci > 0; ci--) {
that.halveM();
if (out.greaterEquals(that)) {
out.subM(that).normalize();
}
}
return out.trim();
},
/** return inverse mod prime p. p must be odd. Binary extended Euclidean algorithm mod p. */
inverseMod: function(p) {
var a = new sjcl.bn(1), b = new sjcl.bn(0), x = new sjcl.bn(this), y = new sjcl.bn(p), tmp, i, nz=1;
if (!(p.limbs[0] & 1)) {
throw (new sjcl.exception.invalid("inverseMod: p must be odd"));
}
// invariant: y is odd
do {
if (x.limbs[0] & 1) {
if (!x.greaterEquals(y)) {
// x < y; swap everything
tmp = x; x = y; y = tmp;
tmp = a; a = b; b = tmp;
}
x.subM(y);
x.normalize();
if (!a.greaterEquals(b)) {
a.addM(p);
}
a.subM(b);
}
// cut everything in half
x.halveM();
if (a.limbs[0] & 1) {
a.addM(p);
}
a.normalize();
a.halveM();
// check for termination: x ?= 0
for (i=nz=0; i<x.limbs.length; i++) {
nz |= x.limbs[i];
}
} while(nz);
if (!y.equals(1)) {
throw (new sjcl.exception.invalid("inverseMod: p and x must be relatively prime"));
}
return b;
},
/** this + that. Does not normalize. */
add: function(that) {
return this.copy().addM(that);
},
/** this - that. Does not normalize. */
sub: function(that) {
return this.copy().subM(that);
},
/** this * that. Normalizes and reduces. */
mul: function(that) {
if (typeof(that) === "number") { that = new this._class(that); }
var i, j, a = this.limbs, b = that.limbs, al = a.length, bl = b.length, out = new this._class(), c = out.limbs, ai, ii=this.maxMul;
for (i=0; i < this.limbs.length + that.limbs.length + 1; i++) {
c[i] = 0;
}
for (i=0; i<al; i++) {
ai = a[i];
for (j=0; j<bl; j++) {
c[i+j] += ai * b[j];
}
if (!--ii) {
ii = this.maxMul;
out.cnormalize();
}
}
return out.cnormalize().reduce();
},
/** this ^ 2. Normalizes and reduces. */
square: function() {
return this.mul(this);
},
/** this ^ n. Uses square-and-multiply. Normalizes and reduces. */
power: function(l) {
l = new sjcl.bn(l).normalize().trim().limbs;
var i, j, out = new this._class(1), pow = this;
for (i=0; i<l.length; i++) {
for (j=0; j<this.radix; j++) {
if (l[i] & (1<<j)) { out = out.mul(pow); }
if (i == (l.length - 1) && l[i]>>(j + 1) == 0) { break; }
pow = pow.square();
}
}
return out;
},
/** this * that mod N */
mulmod: function(that, N) {
return this.mod(N).mul(that.mod(N)).mod(N);
},
/** this ^ x mod N */
powermod: function(x, N) {
x = new sjcl.bn(x);
N = new sjcl.bn(N);
// Jump to montpowermod if possible.
if ((N.limbs[0] & 1) == 1) {
var montOut = this.montpowermod(x, N);
if (montOut != false) { return montOut; } // else go to slow powermod
}
var i, j, l = x.normalize().trim().limbs, out = new this._class(1), pow = this;
for (i=0; i<l.length; i++) {
for (j=0; j<this.radix; j++) {
if (l[i] & (1<<j)) { out = out.mulmod(pow, N); }
if (i == (l.length - 1) && l[i]>>(j + 1) == 0) { break; }
pow = pow.mulmod(pow, N);
}
}
return out;
},
/** this ^ x mod N with Montomery reduction */
montpowermod: function(x, N) {
x = new sjcl.bn(x).normalize().trim();
N = new sjcl.bn(N);
var i, j,
radix = this.radix,
out = new this._class(1),
pow = this.copy();
// Generate R as a cap of N.
var R, s, wind, bitsize = x.bitLength();
R = new sjcl.bn({
limbs: N.copy().normalize().trim().limbs.map(function() { return 0; })
});
for (s = this.radix; s > 0; s--) {
if (((N.limbs[N.limbs.length - 1] >> s) & 1) == 1) {
R.limbs[R.limbs.length - 1] = 1 << s;
break;
}
}
// Calculate window size as a function of the exponent's size.
if (bitsize == 0) {
return this;
} else if (bitsize < 18) {
wind = 1;
} else if (bitsize < 48) {
wind = 3;
} else if (bitsize < 144) {
wind = 4;
} else if (bitsize < 768) {
wind = 5;
} else {
wind = 6;
}
// Find R' and N' such that R * R' - N * N' = 1.
var RR = R.copy(), NN = N.copy(), RP = new sjcl.bn(1), NP = new sjcl.bn(0), RT = R.copy();
while (RT.greaterEquals(1)) {
RT.halveM();
if ((RP.limbs[0] & 1) == 0) {
RP.halveM();
NP.halveM();
} else {
RP.addM(NN);
RP.halveM();
NP.halveM();
NP.addM(RR);
}
}
RP = RP.normalize();
NP = NP.normalize();
RR.doubleM();
var R2 = RR.mulmod(RR, N);
// Check whether the invariant holds.
// If it doesn't, we can't use Montgomery reduction on this modulus.
if (!RR.mul(RP).sub(N.mul(NP)).equals(1)) {
return false;
}
var montIn = function(c) { return montMul(c, R2); },
montMul = function(a, b) {
// Standard Montgomery reduction
var k, carry, ab, right, abBar, mask = (1 << (s + 1)) - 1;
ab = a.mul(b);
right = ab.mul(NP);
right.limbs = right.limbs.slice(0, R.limbs.length);
if (right.limbs.length == R.limbs.length) {
right.limbs[R.limbs.length - 1] &= mask;
}
right = right.mul(N);
abBar = ab.add(right).normalize().trim();
abBar.limbs = abBar.limbs.slice(R.limbs.length - 1);
// Division. Equivelent to calling *.halveM() s times.
for (k=0; k < abBar.limbs.length; k++) {
if (k > 0) {
abBar.limbs[k - 1] |= (abBar.limbs[k] & mask) << (radix - s - 1);
}
abBar.limbs[k] = abBar.limbs[k] >> (s + 1);
}
if (abBar.greaterEquals(N)) {
abBar.subM(N);
}
return abBar;
},
montOut = function(c) { return montMul(c, 1); };
pow = montIn(pow);
out = montIn(out);
// Sliding-Window Exponentiation (HAC 14.85)
var h, precomp = {}, cap = (1 << (wind - 1)) - 1;
precomp[1] = pow.copy();
precomp[2] = montMul(pow, pow);
for (h=1; h<=cap; h++) {
precomp[(2 * h) + 1] = montMul(precomp[(2 * h) - 1], precomp[2]);
}
var getBit = function(exp, i) { // Gets ith bit of exp.
var off = i % exp.radix;
return (exp.limbs[Math.floor(i / exp.radix)] & (1 << off)) >> off;
};
for (i = x.bitLength() - 1; i >= 0; ) {
if (getBit(x, i) == 0) {
// If the next bit is zero:
// Square, move forward one bit.
out = montMul(out, out);
i = i - 1;
} else {
// If the next bit is one:
// Find the longest sequence of bits after this one, less than `wind`
// bits long, that ends with a 1. Convert the sequence into an
// integer and look up the pre-computed value to add.
var l = i - wind + 1;
while (getBit(x, l) == 0) {
l++;
}
var indx = 0;
for (j = l; j <= i; j++) {
indx += getBit(x, j) << (j - l);
out = montMul(out, out);
}
out = montMul(out, precomp[indx]);
i = l - 1;
}
}
return montOut(out);
},
trim: function() {
var l = this.limbs, p;
do {
p = l.pop();
} while (l.length && p === 0);
l.push(p);
return this;
},
/** Reduce mod a modulus. Stubbed for subclassing. */
reduce: function() {
return this;
},
/** Reduce and normalize. */
fullReduce: function() {
return this.normalize();
},
/** Propagate carries. */
normalize: function() {
var carry=0, i, pv = this.placeVal, ipv = this.ipv, l, m, limbs = this.limbs, ll = limbs.length, mask = this.radixMask;
for (i=0; i < ll || (carry !== 0 && carry !== -1); i++) {
l = (limbs[i]||0) + carry;
m = limbs[i] = l & mask;
carry = (l-m)*ipv;
}
if (carry === -1) {
limbs[i-1] -= pv;
}
this.trim();
return this;
},
/** Constant-time normalize. Does not allocate additional space. */
cnormalize: function() {
var carry=0, i, ipv = this.ipv, l, m, limbs = this.limbs, ll = limbs.length, mask = this.radixMask;
for (i=0; i < ll-1; i++) {
l = limbs[i] + carry;
m = limbs[i] = l & mask;
carry = (l-m)*ipv;
}
limbs[i] += carry;
return this;
},
/** Serialize to a bit array */
toBits: function(len) {
this.fullReduce();
len = len || this.exponent || this.bitLength();
var i = Math.floor((len-1)/24), w=sjcl.bitArray, e = (len + 7 & -8) % this.radix || this.radix,
out = [w.partial(e, this.getLimb(i))];
for (i--; i >= 0; i--) {
out = w.concat(out, [w.partial(Math.min(this.radix,len), this.getLimb(i))]);
len -= this.radix;
}
return out;
},
/** Return the length in bits, rounded up to the nearest byte. */
bitLength: function() {
this.fullReduce();
var out = this.radix * (this.limbs.length - 1),
b = this.limbs[this.limbs.length - 1];
for (; b; b >>>= 1) {
out ++;
}
return out+7 & -8;
}
};
/** @memberOf sjcl.bn
* @this { sjcl.bn }
*/
sjcl.bn.fromBits = function(bits) {
var Class = this, out = new Class(), words=[], w=sjcl.bitArray, t = this.prototype,
l = Math.min(this.bitLength || 0x100000000, w.bitLength(bits)), e = l % t.radix || t.radix;
words[0] = w.extract(bits, 0, e);
for (; e < l; e += t.radix) {
words.unshift(w.extract(bits, e, t.radix));
}
out.limbs = words;
return out;
};
sjcl.bn.prototype.ipv = 1 / (sjcl.bn.prototype.placeVal = Math.pow(2,sjcl.bn.prototype.radix));
sjcl.bn.prototype.radixMask = (1 << sjcl.bn.prototype.radix) - 1;
/**
* Creates a new subclass of bn, based on reduction modulo a pseudo-Mersenne prime,
* i.e. a prime of the form 2^e + sum(a * 2^b),where the sum is negative and sparse.
*/
sjcl.bn.pseudoMersennePrime = function(exponent, coeff) {
/** @constructor
* @private
*/
function p(it) {
this.initWith(it);
/*if (this.limbs[this.modOffset]) {
this.reduce();
}*/
}
var ppr = p.prototype = new sjcl.bn(), i, tmp, mo;
mo = ppr.modOffset = Math.ceil(tmp = exponent / ppr.radix);
ppr.exponent = exponent;
ppr.offset = [];
ppr.factor = [];
ppr.minOffset = mo;
ppr.fullMask = 0;
ppr.fullOffset = [];
ppr.fullFactor = [];
ppr.modulus = p.modulus = new sjcl.bn(Math.pow(2,exponent));
ppr.fullMask = 0|-Math.pow(2, exponent % ppr.radix);
for (i=0; i<coeff.length; i++) {
ppr.offset[i] = Math.floor(coeff[i][0] / ppr.radix - tmp);
ppr.fullOffset[i] = Math.ceil(coeff[i][0] / ppr.radix - tmp);
ppr.factor[i] = coeff[i][1] * Math.pow(1/2, exponent - coeff[i][0] + ppr.offset[i] * ppr.radix);
ppr.fullFactor[i] = coeff[i][1] * Math.pow(1/2, exponent - coeff[i][0] + ppr.fullOffset[i] * ppr.radix);
ppr.modulus.addM(new sjcl.bn(Math.pow(2,coeff[i][0])*coeff[i][1]));
ppr.minOffset = Math.min(ppr.minOffset, -ppr.offset[i]); // conservative
}
ppr._class = p;
ppr.modulus.cnormalize();
/** Approximate reduction mod p. May leave a number which is negative or slightly larger than p.
* @memberof sjcl.bn
* @this { sjcl.bn }
*/
ppr.reduce = function() {
var i, k, l, mo = this.modOffset, limbs = this.limbs, off = this.offset, ol = this.offset.length, fac = this.factor, ll;
i = this.minOffset;
while (limbs.length > mo) {
l = limbs.pop();
ll = limbs.length;
for (k=0; k<ol; k++) {
limbs[ll+off[k]] -= fac[k] * l;
}
i--;
if (!i) {
limbs.push(0);
this.cnormalize();
i = this.minOffset;
}
}
this.cnormalize();
return this;
};
/** @memberof sjcl.bn
* @this { sjcl.bn }
*/
ppr._strongReduce = (ppr.fullMask === -1) ? ppr.reduce : function() {
var limbs = this.limbs, i = limbs.length - 1, k, l;
this.reduce();
if (i === this.modOffset - 1) {
l = limbs[i] & this.fullMask;
limbs[i] -= l;
for (k=0; k<this.fullOffset.length; k++) {
limbs[i+this.fullOffset[k]] -= this.fullFactor[k] * l;
}
this.normalize();
}
};
/** mostly constant-time, very expensive full reduction.
* @memberof sjcl.bn
* @this { sjcl.bn }
*/
ppr.fullReduce = function() {
var greater, i;
// massively above the modulus, may be negative
this._strongReduce();
// less than twice the modulus, may be negative
this.addM(this.modulus);
this.addM(this.modulus);
this.normalize();
// probably 2-3x the modulus
this._strongReduce();
// less than the power of 2. still may be more than
// the modulus
// HACK: pad out to this length
for (i=this.limbs.length; i<this.modOffset; i++) {
this.limbs[i] = 0;
}
// constant-time subtract modulus
greater = this.greaterEquals(this.modulus);
for (i=0; i<this.limbs.length; i++) {
this.limbs[i] -= this.modulus.limbs[i] * greater;
}
this.cnormalize();
return this;
};
/** @memberof sjcl.bn
* @this { sjcl.bn }
*/
ppr.inverse = function() {
return (this.power(this.modulus.sub(2)));
};
p.fromBits = sjcl.bn.fromBits;
return p;
};
// a small Mersenne prime
var sbp = sjcl.bn.pseudoMersennePrime;
sjcl.bn.prime = {
p127: sbp(127, [[0,-1]]),
// Bernstein's prime for Curve25519
p25519: sbp(255, [[0,-19]]),
// Koblitz primes
p192k: sbp(192, [[32,-1],[12,-1],[8,-1],[7,-1],[6,-1],[3,-1],[0,-1]]),
p224k: sbp(224, [[32,-1],[12,-1],[11,-1],[9,-1],[7,-1],[4,-1],[1,-1],[0,-1]]),
p256k: sbp(256, [[32,-1],[9,-1],[8,-1],[7,-1],[6,-1],[4,-1],[0,-1]]),
// NIST primes
p192: sbp(192, [[0,-1],[64,-1]]),
p224: sbp(224, [[0,1],[96,-1]]),
p256: sbp(256, [[0,-1],[96,1],[192,1],[224,-1]]),
p384: sbp(384, [[0,-1],[32,1],[96,-1],[128,-1]]),
p521: sbp(521, [[0,-1]])
};
sjcl.bn.random = function(modulus, paranoia) {
if (typeof modulus !== "object") { modulus = new sjcl.bn(modulus); }
var words, i, l = modulus.limbs.length, m = modulus.limbs[l-1]+1, out = new sjcl.bn();
while (true) {
// get a sequence whose first digits make sense
do {
words = sjcl.random.randomWords(l, paranoia);
if (words[l-1] < 0) { words[l-1] += 0x100000000; }
} while (Math.floor(words[l-1] / m) === Math.floor(0x100000000 / m));
words[l-1] %= m;
// mask off all the limbs
for (i=0; i<l-1; i++) {
words[i] &= modulus.radixMask;
}
// check the rest of the digitssj
out.limbs = words;
if (!out.greaterEquals(modulus)) {
return out;
}
}
};