sjcl.ecc = {}; /** * Represents a point on a curve in affine coordinates. * @constructor * @param {sjcl.ecc.curve} curve The curve that this point lies on. * @param {bigInt} x The x coordinate. * @param {bigInt} y The y coordinate. */ sjcl.ecc.point = function(curve,x,y) { if (x === undefined) { this.isIdentity = true; } else { this.x = x; this.y = y; this.isIdentity = false; } this.curve = curve; }; sjcl.ecc.point.prototype = { toJac: function() { return new sjcl.ecc.pointJac(this.curve, this.x, this.y, new this.curve.field(1)); }, mult: function(k) { return this.toJac().mult(k, this).toAffine(); }, /** * Multiply this point by k, added to affine2*k2, and return the answer in Jacobian coordinates. * @param {bigInt} k The coefficient to multiply this by. * @param {bigInt} k2 The coefficient to multiply affine2 this by. * @param {sjcl.ecc.point} affine The other point in affine coordinates. * @return {sjcl.ecc.pointJac} The result of the multiplication and addition, in Jacobian coordinates. */ mult2: function(k, k2, affine2) { return this.toJac().mult2(k, this, k2, affine2).toAffine(); }, multiples: function() { var m, i, j; if (this._multiples === undefined) { j = this.toJac().doubl(); m = this._multiples = [new sjcl.ecc.point(this.curve), this, j.toAffine()]; for (i=3; i<16; i++) { j = j.add(this); m.push(j.toAffine()); } } return this._multiples; }, isValid: function() { return this.y.square().equals(this.curve.b.add(this.x.mul(this.curve.a.add(this.x.square())))); }, toBits: function() { return sjcl.bitArray.concat(this.x.toBits(), this.y.toBits()); } }; /** * Represents a point on a curve in Jacobian coordinates. Coordinates can be specified as bigInts or strings (which * will be converted to bigInts). * * @constructor * @param {bigInt/string} x The x coordinate. * @param {bigInt/string} y The y coordinate. * @param {bigInt/string} z The z coordinate. * @param {sjcl.ecc.curve} curve The curve that this point lies on. */ sjcl.ecc.pointJac = function(curve, x, y, z) { if (x === undefined) { this.isIdentity = true; } else { this.x = x; this.y = y; this.z = z; this.isIdentity = false; } this.curve = curve; }; sjcl.ecc.pointJac.prototype = { /** * Adds S and T and returns the result in Jacobian coordinates. Note that S must be in Jacobian coordinates and T must be in affine coordinates. * @param {sjcl.ecc.pointJac} S One of the points to add, in Jacobian coordinates. * @param {sjcl.ecc.point} T The other point to add, in affine coordinates. * @return {sjcl.ecc.pointJac} The sum of the two points, in Jacobian coordinates. */ add: function(T) { var S = this, sz2, c, d, c2, x1, x2, x, y1, y2, y, z; if (S.curve !== T.curve) { throw("sjcl.ecc.add(): Points must be on the same curve to add them!"); } if (S.isIdentity) { return T.toJac(); } else if (T.isIdentity) { return S; } sz2 = S.z.square(); c = T.x.mul(sz2).subM(S.x); if (c.equals(0)) { if (S.y.equals(T.y.mul(sz2.mul(S.z)))) { // same point return S.doubl(); } else { // inverses return new sjcl.ecc.pointJac(S.curve); } } d = T.y.mul(sz2.mul(S.z)).subM(S.y); c2 = c.square(); x1 = d.square(); x2 = c.square().mul(c).addM( S.x.add(S.x).mul(c2) ); x = x1.subM(x2); y1 = S.x.mul(c2).subM(x).mul(d); y2 = S.y.mul(c.square().mul(c)); y = y1.subM(y2); z = S.z.mul(c); return new sjcl.ecc.pointJac(this.curve,x,y,z); }, /** * doubles this point. * @return {sjcl.ecc.pointJac} The doubled point. */ doubl: function() { if (this.isIdentity) { return this; } var y2 = this.y.square(), a = y2.mul(this.x.mul(4)), b = y2.square().mul(8), z2 = this.z.square(), c = this.x.sub(z2).mul(3).mul(this.x.add(z2)), x = c.square().subM(a).subM(a), y = a.sub(x).mul(c).subM(b), z = this.y.add(this.y).mul(this.z); return new sjcl.ecc.pointJac(this.curve, x, y, z); }, /** * Returns a copy of this point converted to affine coordinates. * @return {sjcl.ecc.point} The converted point. */ toAffine: function() { if (this.isIdentity || this.z.equals(0)) { return new sjcl.ecc.point(this.curve); } var zi = this.z.inverse(), zi2 = zi.square(); return new sjcl.ecc.point(this.curve, this.x.mul(zi2).fullReduce(), this.y.mul(zi2.mul(zi)).fullReduce()); }, /** * Multiply this point by k and return the answer in Jacobian coordinates. * @param {bigInt} k The coefficient to multiply by. * @param {sjcl.ecc.point} affine This point in affine coordinates. * @return {sjcl.ecc.pointJac} The result of the multiplication, in Jacobian coordinates. */ mult: function(k, affine) { if (typeof(k) === "number") { k = [k]; } else if (k.limbs !== undefined) { k = k.normalize().limbs; } var i, j, out = new sjcl.ecc.point(this.curve).toJac(), multiples = affine.multiples(); for (i=k.length-1; i>=0; i--) { for (j=sjcl.bn.prototype.radix-4; j>=0; j-=4) { out = out.doubl().doubl().doubl().doubl().add(multiples[k[i]>>j & 0xF]); } } return out; }, /** * Multiply this point by k, added to affine2*k2, and return the answer in Jacobian coordinates. * @param {bigInt} k The coefficient to multiply this by. * @param {sjcl.ecc.point} affine This point in affine coordinates. * @param {bigInt} k2 The coefficient to multiply affine2 this by. * @param {sjcl.ecc.point} affine The other point in affine coordinates. * @return {sjcl.ecc.pointJac} The result of the multiplication and addition, in Jacobian coordinates. */ mult2: function(k1, affine, k2, affine2) { if (typeof(k1) === "number") { k1 = [k1]; } else if (k1.limbs !== undefined) { k1 = k1.normalize().limbs; } if (typeof(k2) === "number") { k2 = [k2]; } else if (k2.limbs !== undefined) { k2 = k2.normalize().limbs; } var i, j, out = new sjcl.ecc.point(this.curve).toJac(), m1 = affine.multiples(), m2 = affine2.multiples(), l1, l2; for (i=Math.max(k1.length,k2.length)-1; i>=0; i--) { l1 = k1[i] | 0; l2 = k2[i] | 0; for (j=sjcl.bn.prototype.radix-4; j>=0; j-=4) { out = out.doubl().doubl().doubl().doubl().add(m1[l1>>j & 0xF]).add(m2[l2>>j & 0xF]); } } return out; }, isValid: function() { var z2 = this.z.square(), z4 = z2.square(), z6 = z4.mul(z2); return this.y.square().equals( this.curve.b.mul(z6).add(this.x.mul( this.curve.a.mul(z4).add(this.x.square())))); } }; /** * Construct an elliptic curve. Most users will not use this and instead start with one of the NIST curves defined below. * * @constructor * @param {bigInt} p The prime modulus. * @param {bigInt} r The prime order of the curve. * @param {bigInt} a The constant a in the equation of the curve y^2 = x^3 + ax + b (for NIST curves, a is always -3). * @param {bigInt} x The x coordinate of a base point of the curve. * @param {bigInt} y The y coordinate of a base point of the curve. */ sjcl.ecc.curve = function(Field, r, a, b, x, y) { this.field = Field; this.r = Field.prototype.modulus.sub(r); this.a = new Field(a); this.b = new Field(b); this.G = new sjcl.ecc.point(this, new Field(x), new Field(y)); }; sjcl.ecc.curve.prototype.fromBits = function (bits) { var w = sjcl.bitArray, l = this.field.prototype.exponent + 7 & -8, p = new sjcl.ecc.point(this, this.field.fromBits(w.bitSlice(bits, 0, l)), this.field.fromBits(w.bitSlice(bits, l, 2*l))); if (!p.isValid()) { throw new sjcl.exception.corrupt("not on the curve!"); } return p; }; sjcl.ecc.curves = { c192: new sjcl.ecc.curve( sjcl.bn.prime.p192, "0x662107c8eb94364e4b2dd7ce", -3, "0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1", "0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012", "0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811"), c224: new sjcl.ecc.curve( sjcl.bn.prime.p224, "0xe95c1f470fc1ec22d6baa3a3d5c4", -3, "0xb4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4", "0xb70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21", "0xbd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34"), c256: new sjcl.ecc.curve( sjcl.bn.prime.p256, "0x4319055358e8617b0c46353d039cdaae", -3, "0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b", "0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296", "0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5"), c384: new sjcl.ecc.curve( sjcl.bn.prime.p384, "0x389cb27e0bc8d21fa7e5f24cb74f58851313e696333ad68c", -3, "0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef", "0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7", "0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f") }; /* Diffie-Hellman-like public-key system */ sjcl.ecc._dh = function(cn) { sjcl.ecc[cn] = { publicKey: function(curve, point) { this._curve = curve; if (point instanceof Array) { this._point = curve.fromBits(point); } else { this._point = point; } }, secretKey: function(curve, exponent) { this._curve = curve; this._exponent = exponent; }, generateKeys: function(curve, paranoia) { if (curve === undefined) { curve = 256; } if (typeof curve === "number") { curve = sjcl.ecc.curves['c'+curve]; if (curve === undefined) { throw new sjcl.exception.invalid("no such curve"); } } var sec = sjcl.bn.random(curve.r, paranoia), pub = curve.G.mult(sec); return { pub: new sjcl.ecc[cn].publicKey(curve, pub), sec: new sjcl.ecc[cn].secretKey(curve, sec) }; } }; }; sjcl.ecc._dh("elGamal"); sjcl.ecc.elGamal.publicKey.prototype = { kem: function(paranoia) { var sec = sjcl.bn.random(this._curve.r, paranoia), tag = this._curve.G.mult(sec).toBits(), key = sjcl.hash.sha256.hash(this._point.mult(sec).toBits()); return { key: key, tag: tag }; } }; sjcl.ecc.elGamal.secretKey.prototype = { unkem: function(tag) { return sjcl.hash.sha256.hash(this._curve.fromBits(tag).mult(this._exponent).toBits()); }, dh: function(pk) { return sjcl.hash.sha256.hash(pk._point.mult(this._exponent).toBits()); } }; sjcl.ecc._dh("ecdsa"); sjcl.ecc.ecdsa.secretKey.prototype = { sign: function(hash, paranoia) { var R = this._curve.r, l = R.bitLength(), k = sjcl.bn.random(R.sub(1), paranoia).add(1), r = this._curve.G.mult(k).x.mod(R), s = sjcl.bn.fromBits(hash).add(r.mul(this._exponent)).inverseMod(R).mul(k).mod(R); return sjcl.bitArray.concat(r.toBits(l), s.toBits(l)); } }; sjcl.ecc.ecdsa.publicKey.prototype = { verify: function(hash, rs) { var w = sjcl.bitArray, R = this._curve.r, l = R.bitLength(), r = sjcl.bn.fromBits(w.bitSlice(rs,0,l)), s = sjcl.bn.fromBits(w.bitSlice(rs,l,2*l)), hG = sjcl.bn.fromBits(hash).mul(s).mod(R), hA = r.mul(s).mod(R), r2 = this._curve.G.mult2(hG, hA, this._point).x; if (r.equals(0) || s.equals(0) || r.greaterEquals(R) || s.greaterEquals(R) || !r2.equals(r)) { throw (new sjcl.exception.corrupt("signature didn't check out")); } return true; } };