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'use strict';
var curve = require('../curve');
var elliptic = require('../../elliptic');
var BN = require('bn.js');
var inherits = require('inherits');
var Base = curve.base;
var assert = elliptic.utils.assert;
function ShortCurve(conf) {
Base.call(this, 'short', conf);
this.a = new BN(conf.a, 16).toRed(this.red);
this.b = new BN(conf.b, 16).toRed(this.red);
this.tinv = this.two.redInvm();
this.zeroA = this.a.fromRed().cmpn(0) === 0;
this.threeA = this.a.fromRed().sub(this.p).cmpn(-3) === 0;
// If the curve is endomorphic, precalculate beta and lambda
this.endo = this._getEndomorphism(conf);
this._endoWnafT1 = new Array(4);
this._endoWnafT2 = new Array(4);
}
inherits(ShortCurve, Base);
module.exports = ShortCurve;
ShortCurve.prototype._getEndomorphism = function _getEndomorphism(conf) {
// No efficient endomorphism
if (!this.zeroA || !this.g || !this.n || this.p.modn(3) !== 1)
return;
// Compute beta and lambda, that lambda * P = (beta * Px; Py)
var beta;
var lambda;
if (conf.beta) {
beta = new BN(conf.beta, 16).toRed(this.red);
} else {
var betas = this._getEndoRoots(this.p);
// Choose the smallest beta
beta = betas[0].cmp(betas[1]) < 0 ? betas[0] : betas[1];
beta = beta.toRed(this.red);
}
if (conf.lambda) {
lambda = new BN(conf.lambda, 16);
} else {
// Choose the lambda that is matching selected beta
var lambdas = this._getEndoRoots(this.n);
if (this.g.mul(lambdas[0]).x.cmp(this.g.x.redMul(beta)) === 0) {
lambda = lambdas[0];
} else {
lambda = lambdas[1];
assert(this.g.mul(lambda).x.cmp(this.g.x.redMul(beta)) === 0);
}
}
// Get basis vectors, used for balanced length-two representation
var basis;
if (conf.basis) {
basis = conf.basis.map(function(vec) {
return {
a: new BN(vec.a, 16),
b: new BN(vec.b, 16)
};
});
} else {
basis = this._getEndoBasis(lambda);
}
return {
beta: beta,
lambda: lambda,
basis: basis
};
};
ShortCurve.prototype._getEndoRoots = function _getEndoRoots(num) {
// Find roots of for x^2 + x + 1 in F
// Root = (-1 +- Sqrt(-3)) / 2
//
var red = num === this.p ? this.red : BN.mont(num);
var tinv = new BN(2).toRed(red).redInvm();
var ntinv = tinv.redNeg();
var s = new BN(3).toRed(red).redNeg().redSqrt().redMul(tinv);
var l1 = ntinv.redAdd(s).fromRed();
var l2 = ntinv.redSub(s).fromRed();
return [ l1, l2 ];
};
ShortCurve.prototype._getEndoBasis = function _getEndoBasis(lambda) {
// aprxSqrt >= sqrt(this.n)
var aprxSqrt = this.n.ushrn(Math.floor(this.n.bitLength() / 2));
// 3.74
// Run EGCD, until r(L + 1) < aprxSqrt
var u = lambda;
var v = this.n.clone();
var x1 = new BN(1);
var y1 = new BN(0);
var x2 = new BN(0);
var y2 = new BN(1);
// NOTE: all vectors are roots of: a + b * lambda = 0 (mod n)
var a0;
var b0;
// First vector
var a1;
var b1;
// Second vector
var a2;
var b2;
var prevR;
var i = 0;
var r;
var x;
while (u.cmpn(0) !== 0) {
var q = v.div(u);
r = v.sub(q.mul(u));
x = x2.sub(q.mul(x1));
var y = y2.sub(q.mul(y1));
if (!a1 && r.cmp(aprxSqrt) < 0) {
a0 = prevR.neg();
b0 = x1;
a1 = r.neg();
b1 = x;
} else if (a1 && ++i === 2) {
break;
}
prevR = r;
v = u;
u = r;
x2 = x1;
x1 = x;
y2 = y1;
y1 = y;
}
a2 = r.neg();
b2 = x;
var len1 = a1.sqr().add(b1.sqr());
var len2 = a2.sqr().add(b2.sqr());
if (len2.cmp(len1) >= 0) {
a2 = a0;
b2 = b0;
}
// Normalize signs
if (a1.negative) {
a1 = a1.neg();
b1 = b1.neg();
}
if (a2.negative) {
a2 = a2.neg();
b2 = b2.neg();
}
return [
{ a: a1, b: b1 },
{ a: a2, b: b2 }
];
};
ShortCurve.prototype._endoSplit = function _endoSplit(k) {
var basis = this.endo.basis;
var v1 = basis[0];
var v2 = basis[1];
var c1 = v2.b.mul(k).divRound(this.n);
var c2 = v1.b.neg().mul(k).divRound(this.n);
var p1 = c1.mul(v1.a);
var p2 = c2.mul(v2.a);
var q1 = c1.mul(v1.b);
var q2 = c2.mul(v2.b);
// Calculate answer
var k1 = k.sub(p1).sub(p2);
var k2 = q1.add(q2).neg();
return { k1: k1, k2: k2 };
};
ShortCurve.prototype.pointFromX = function pointFromX(x, odd) {
x = new BN(x, 16);
if (!x.red)
x = x.toRed(this.red);
var y2 = x.redSqr().redMul(x).redIAdd(x.redMul(this.a)).redIAdd(this.b);
var y = y2.redSqrt();
if (y.redSqr().redSub(y2).cmp(this.zero) !== 0)
throw new Error('invalid point');
// XXX Is there any way to tell if the number is odd without converting it
// to non-red form?
var isOdd = y.fromRed().isOdd();
if (odd && !isOdd || !odd && isOdd)
y = y.redNeg();
return this.point(x, y);
};
ShortCurve.prototype.validate = function validate(point) {
if (point.inf)
return true;
var x = point.x;
var y = point.y;
var ax = this.a.redMul(x);
var rhs = x.redSqr().redMul(x).redIAdd(ax).redIAdd(this.b);
return y.redSqr().redISub(rhs).cmpn(0) === 0;
};
ShortCurve.prototype._endoWnafMulAdd =
function _endoWnafMulAdd(points, coeffs, jacobianResult) {
var npoints = this._endoWnafT1;
var ncoeffs = this._endoWnafT2;
for (var i = 0; i < points.length; i++) {
var split = this._endoSplit(coeffs[i]);
var p = points[i];
var beta = p._getBeta();
if (split.k1.negative) {
split.k1.ineg();
p = p.neg(true);
}
if (split.k2.negative) {
split.k2.ineg();
beta = beta.neg(true);
}
npoints[i * 2] = p;
npoints[i * 2 + 1] = beta;
ncoeffs[i * 2] = split.k1;
ncoeffs[i * 2 + 1] = split.k2;
}
var res = this._wnafMulAdd(1, npoints, ncoeffs, i * 2, jacobianResult);
// Clean-up references to points and coefficients
for (var j = 0; j < i * 2; j++) {
npoints[j] = null;
ncoeffs[j] = null;
}
return res;
};
function Point(curve, x, y, isRed) {
Base.BasePoint.call(this, curve, 'affine');
if (x === null && y === null) {
this.x = null;
this.y = null;
this.inf = true;
} else {
this.x = new BN(x, 16);
this.y = new BN(y, 16);
// Force redgomery representation when loading from JSON
if (isRed) {
this.x.forceRed(this.curve.red);
this.y.forceRed(this.curve.red);
}
if (!this.x.red)
this.x = this.x.toRed(this.curve.red);
if (!this.y.red)
this.y = this.y.toRed(this.curve.red);
this.inf = false;
}
}
inherits(Point, Base.BasePoint);
ShortCurve.prototype.point = function point(x, y, isRed) {
return new Point(this, x, y, isRed);
};
ShortCurve.prototype.pointFromJSON = function pointFromJSON(obj, red) {
return Point.fromJSON(this, obj, red);
};
Point.prototype._getBeta = function _getBeta() {
if (!this.curve.endo)
return;
var pre = this.precomputed;
if (pre && pre.beta)
return pre.beta;
var beta = this.curve.point(this.x.redMul(this.curve.endo.beta), this.y);
if (pre) {
var curve = this.curve;
var endoMul = function(p) {
return curve.point(p.x.redMul(curve.endo.beta), p.y);
};
pre.beta = beta;
beta.precomputed = {
beta: null,
naf: pre.naf && {
wnd: pre.naf.wnd,
points: pre.naf.points.map(endoMul)
},
doubles: pre.doubles && {
step: pre.doubles.step,
points: pre.doubles.points.map(endoMul)
}
};
}
return beta;
};
Point.prototype.toJSON = function toJSON() {
if (!this.precomputed)
return [ this.x, this.y ];
return [ this.x, this.y, this.precomputed && {
doubles: this.precomputed.doubles && {
step: this.precomputed.doubles.step,
points: this.precomputed.doubles.points.slice(1)
},
naf: this.precomputed.naf && {
wnd: this.precomputed.naf.wnd,
points: this.precomputed.naf.points.slice(1)
}
} ];
};
Point.fromJSON = function fromJSON(curve, obj, red) {
if (typeof obj === 'string')
obj = JSON.parse(obj);
var res = curve.point(obj[0], obj[1], red);
if (!obj[2])
return res;
function obj2point(obj) {
return curve.point(obj[0], obj[1], red);
}
var pre = obj[2];
res.precomputed = {
beta: null,
doubles: pre.doubles && {
step: pre.doubles.step,
points: [ res ].concat(pre.doubles.points.map(obj2point))
},
naf: pre.naf && {
wnd: pre.naf.wnd,
points: [ res ].concat(pre.naf.points.map(obj2point))
}
};
return res;
};
Point.prototype.inspect = function inspect() {
if (this.isInfinity())
return '<EC Point Infinity>';
return '<EC Point x: ' + this.x.fromRed().toString(16, 2) +
' y: ' + this.y.fromRed().toString(16, 2) + '>';
};
Point.prototype.isInfinity = function isInfinity() {
return this.inf;
};
Point.prototype.add = function add(p) {
// O + P = P
if (this.inf)
return p;
// P + O = P
if (p.inf)
return this;
// P + P = 2P
if (this.eq(p))
return this.dbl();
// P + (-P) = O
if (this.neg().eq(p))
return this.curve.point(null, null);
// P + Q = O
if (this.x.cmp(p.x) === 0)
return this.curve.point(null, null);
var c = this.y.redSub(p.y);
if (c.cmpn(0) !== 0)
c = c.redMul(this.x.redSub(p.x).redInvm());
var nx = c.redSqr().redISub(this.x).redISub(p.x);
var ny = c.redMul(this.x.redSub(nx)).redISub(this.y);
return this.curve.point(nx, ny);
};
Point.prototype.dbl = function dbl() {
if (this.inf)
return this;
// 2P = O
var ys1 = this.y.redAdd(this.y);
if (ys1.cmpn(0) === 0)
return this.curve.point(null, null);
var a = this.curve.a;
var x2 = this.x.redSqr();
var dyinv = ys1.redInvm();
var c = x2.redAdd(x2).redIAdd(x2).redIAdd(a).redMul(dyinv);
var nx = c.redSqr().redISub(this.x.redAdd(this.x));
var ny = c.redMul(this.x.redSub(nx)).redISub(this.y);
return this.curve.point(nx, ny);
};
Point.prototype.getX = function getX() {
return this.x.fromRed();
};
Point.prototype.getY = function getY() {
return this.y.fromRed();
};
Point.prototype.mul = function mul(k) {
k = new BN(k, 16);
if (this._hasDoubles(k))
return this.curve._fixedNafMul(this, k);
else if (this.curve.endo)
return this.curve._endoWnafMulAdd([ this ], [ k ]);
else
return this.curve._wnafMul(this, k);
};
Point.prototype.mulAdd = function mulAdd(k1, p2, k2) {
var points = [ this, p2 ];
var coeffs = [ k1, k2 ];
if (this.curve.endo)
return this.curve._endoWnafMulAdd(points, coeffs);
else
return this.curve._wnafMulAdd(1, points, coeffs, 2);
};
Point.prototype.jmulAdd = function jmulAdd(k1, p2, k2) {
var points = [ this, p2 ];
var coeffs = [ k1, k2 ];
if (this.curve.endo)
return this.curve._endoWnafMulAdd(points, coeffs, true);
else
return this.curve._wnafMulAdd(1, points, coeffs, 2, true);
};
Point.prototype.eq = function eq(p) {
return this === p ||
this.inf === p.inf &&
(this.inf || this.x.cmp(p.x) === 0 && this.y.cmp(p.y) === 0);
};
Point.prototype.neg = function neg(_precompute) {
if (this.inf)
return this;
var res = this.curve.point(this.x, this.y.redNeg());
if (_precompute && this.precomputed) {
var pre = this.precomputed;
var negate = function(p) {
return p.neg();
};
res.precomputed = {
naf: pre.naf && {
wnd: pre.naf.wnd,
points: pre.naf.points.map(negate)
},
doubles: pre.doubles && {
step: pre.doubles.step,
points: pre.doubles.points.map(negate)
}
};
}
return res;
};
Point.prototype.toJ = function toJ() {
if (this.inf)
return this.curve.jpoint(null, null, null);
var res = this.curve.jpoint(this.x, this.y, this.curve.one);
return res;
};
function JPoint(curve, x, y, z) {
Base.BasePoint.call(this, curve, 'jacobian');
if (x === null && y === null && z === null) {
this.x = this.curve.one;
this.y = this.curve.one;
this.z = new BN(0);
} else {
this.x = new BN(x, 16);
this.y = new BN(y, 16);
this.z = new BN(z, 16);
}
if (!this.x.red)
this.x = this.x.toRed(this.curve.red);
if (!this.y.red)
this.y = this.y.toRed(this.curve.red);
if (!this.z.red)
this.z = this.z.toRed(this.curve.red);
this.zOne = this.z === this.curve.one;
}
inherits(JPoint, Base.BasePoint);
ShortCurve.prototype.jpoint = function jpoint(x, y, z) {
return new JPoint(this, x, y, z);
};
JPoint.prototype.toP = function toP() {
if (this.isInfinity())
return this.curve.point(null, null);
var zinv = this.z.redInvm();
var zinv2 = zinv.redSqr();
var ax = this.x.redMul(zinv2);
var ay = this.y.redMul(zinv2).redMul(zinv);
return this.curve.point(ax, ay);
};
JPoint.prototype.neg = function neg() {
return this.curve.jpoint(this.x, this.y.redNeg(), this.z);
};
JPoint.prototype.add = function add(p) {
// O + P = P
if (this.isInfinity())
return p;
// P + O = P
if (p.isInfinity())
return this;
// 12M + 4S + 7A
var pz2 = p.z.redSqr();
var z2 = this.z.redSqr();
var u1 = this.x.redMul(pz2);
var u2 = p.x.redMul(z2);
var s1 = this.y.redMul(pz2.redMul(p.z));
var s2 = p.y.redMul(z2.redMul(this.z));
var h = u1.redSub(u2);
var r = s1.redSub(s2);
if (h.cmpn(0) === 0) {
if (r.cmpn(0) !== 0)
return this.curve.jpoint(null, null, null);
else
return this.dbl();
}
var h2 = h.redSqr();
var h3 = h2.redMul(h);
var v = u1.redMul(h2);
var nx = r.redSqr().redIAdd(h3).redISub(v).redISub(v);
var ny = r.redMul(v.redISub(nx)).redISub(s1.redMul(h3));
var nz = this.z.redMul(p.z).redMul(h);
return this.curve.jpoint(nx, ny, nz);
};
JPoint.prototype.mixedAdd = function mixedAdd(p) {
// O + P = P
if (this.isInfinity())
return p.toJ();
// P + O = P
if (p.isInfinity())
return this;
// 8M + 3S + 7A
var z2 = this.z.redSqr();
var u1 = this.x;
var u2 = p.x.redMul(z2);
var s1 = this.y;
var s2 = p.y.redMul(z2).redMul(this.z);
var h = u1.redSub(u2);
var r = s1.redSub(s2);
if (h.cmpn(0) === 0) {
if (r.cmpn(0) !== 0)
return this.curve.jpoint(null, null, null);
else
return this.dbl();
}
var h2 = h.redSqr();
var h3 = h2.redMul(h);
var v = u1.redMul(h2);
var nx = r.redSqr().redIAdd(h3).redISub(v).redISub(v);
var ny = r.redMul(v.redISub(nx)).redISub(s1.redMul(h3));
var nz = this.z.redMul(h);
return this.curve.jpoint(nx, ny, nz);
};
JPoint.prototype.dblp = function dblp(pow) {
if (pow === 0)
return this;
if (this.isInfinity())
return this;
if (!pow)
return this.dbl();
if (this.curve.zeroA || this.curve.threeA) {
var r = this;
for (var i = 0; i < pow; i++)
r = r.dbl();
return r;
}
// 1M + 2S + 1A + N * (4S + 5M + 8A)
// N = 1 => 6M + 6S + 9A
var a = this.curve.a;
var tinv = this.curve.tinv;
var jx = this.x;
var jy = this.y;
var jz = this.z;
var jz4 = jz.redSqr().redSqr();
// Reuse results
var jyd = jy.redAdd(jy);
for (var i = 0; i < pow; i++) {
var jx2 = jx.redSqr();
var jyd2 = jyd.redSqr();
var jyd4 = jyd2.redSqr();
var c = jx2.redAdd(jx2).redIAdd(jx2).redIAdd(a.redMul(jz4));
var t1 = jx.redMul(jyd2);
var nx = c.redSqr().redISub(t1.redAdd(t1));
var t2 = t1.redISub(nx);
var dny = c.redMul(t2);
dny = dny.redIAdd(dny).redISub(jyd4);
var nz = jyd.redMul(jz);
if (i + 1 < pow)
jz4 = jz4.redMul(jyd4);
jx = nx;
jz = nz;
jyd = dny;
}
return this.curve.jpoint(jx, jyd.redMul(tinv), jz);
};
JPoint.prototype.dbl = function dbl() {
if (this.isInfinity())
return this;
if (this.curve.zeroA)
return this._zeroDbl();
else if (this.curve.threeA)
return this._threeDbl();
else
return this._dbl();
};
JPoint.prototype._zeroDbl = function _zeroDbl() {
var nx;
var ny;
var nz;
// Z = 1
if (this.zOne) {
// hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html
// #doubling-mdbl-2007-bl
// 1M + 5S + 14A
// XX = X1^2
var xx = this.x.redSqr();
// YY = Y1^2
var yy = this.y.redSqr();
// YYYY = YY^2
var yyyy = yy.redSqr();
// S = 2 * ((X1 + YY)^2 - XX - YYYY)
var s = this.x.redAdd(yy).redSqr().redISub(xx).redISub(yyyy);
s = s.redIAdd(s);
// M = 3 * XX + a; a = 0
var m = xx.redAdd(xx).redIAdd(xx);
// T = M ^ 2 - 2*S
var t = m.redSqr().redISub(s).redISub(s);
// 8 * YYYY
var yyyy8 = yyyy.redIAdd(yyyy);
yyyy8 = yyyy8.redIAdd(yyyy8);
yyyy8 = yyyy8.redIAdd(yyyy8);
// X3 = T
nx = t;
// Y3 = M * (S - T) - 8 * YYYY
ny = m.redMul(s.redISub(t)).redISub(yyyy8);
// Z3 = 2*Y1
nz = this.y.redAdd(this.y);
} else {
// hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html
// #doubling-dbl-2009-l
// 2M + 5S + 13A
// A = X1^2
var a = this.x.redSqr();
// B = Y1^2
var b = this.y.redSqr();
// C = B^2
var c = b.redSqr();
// D = 2 * ((X1 + B)^2 - A - C)
var d = this.x.redAdd(b).redSqr().redISub(a).redISub(c);
d = d.redIAdd(d);
// E = 3 * A
var e = a.redAdd(a).redIAdd(a);
// F = E^2
var f = e.redSqr();
// 8 * C
var c8 = c.redIAdd(c);
c8 = c8.redIAdd(c8);
c8 = c8.redIAdd(c8);
// X3 = F - 2 * D
nx = f.redISub(d).redISub(d);
// Y3 = E * (D - X3) - 8 * C
ny = e.redMul(d.redISub(nx)).redISub(c8);
// Z3 = 2 * Y1 * Z1
nz = this.y.redMul(this.z);
nz = nz.redIAdd(nz);
}
return this.curve.jpoint(nx, ny, nz);
};
JPoint.prototype._threeDbl = function _threeDbl() {
var nx;
var ny;
var nz;
// Z = 1
if (this.zOne) {
// hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html
// #doubling-mdbl-2007-bl
// 1M + 5S + 15A
// XX = X1^2
var xx = this.x.redSqr();
// YY = Y1^2
var yy = this.y.redSqr();
// YYYY = YY^2
var yyyy = yy.redSqr();
// S = 2 * ((X1 + YY)^2 - XX - YYYY)
var s = this.x.redAdd(yy).redSqr().redISub(xx).redISub(yyyy);
s = s.redIAdd(s);
// M = 3 * XX + a
var m = xx.redAdd(xx).redIAdd(xx).redIAdd(this.curve.a);
// T = M^2 - 2 * S
var t = m.redSqr().redISub(s).redISub(s);
// X3 = T
nx = t;
// Y3 = M * (S - T) - 8 * YYYY
var yyyy8 = yyyy.redIAdd(yyyy);
yyyy8 = yyyy8.redIAdd(yyyy8);
yyyy8 = yyyy8.redIAdd(yyyy8);
ny = m.redMul(s.redISub(t)).redISub(yyyy8);
// Z3 = 2 * Y1
nz = this.y.redAdd(this.y);
} else {
// hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
// 3M + 5S
// delta = Z1^2
var delta = this.z.redSqr();
// gamma = Y1^2
var gamma = this.y.redSqr();
// beta = X1 * gamma
var beta = this.x.redMul(gamma);
// alpha = 3 * (X1 - delta) * (X1 + delta)
var alpha = this.x.redSub(delta).redMul(this.x.redAdd(delta));
alpha = alpha.redAdd(alpha).redIAdd(alpha);
// X3 = alpha^2 - 8 * beta
var beta4 = beta.redIAdd(beta);
beta4 = beta4.redIAdd(beta4);
var beta8 = beta4.redAdd(beta4);
nx = alpha.redSqr().redISub(beta8);
// Z3 = (Y1 + Z1)^2 - gamma - delta
nz = this.y.redAdd(this.z).redSqr().redISub(gamma).redISub(delta);
// Y3 = alpha * (4 * beta - X3) - 8 * gamma^2
var ggamma8 = gamma.redSqr();
ggamma8 = ggamma8.redIAdd(ggamma8);
ggamma8 = ggamma8.redIAdd(ggamma8);
ggamma8 = ggamma8.redIAdd(ggamma8);
ny = alpha.redMul(beta4.redISub(nx)).redISub(ggamma8);
}
return this.curve.jpoint(nx, ny, nz);
};
JPoint.prototype._dbl = function _dbl() {
var a = this.curve.a;
// 4M + 6S + 10A
var jx = this.x;
var jy = this.y;
var jz = this.z;
var jz4 = jz.redSqr().redSqr();
var jx2 = jx.redSqr();
var jy2 = jy.redSqr();
var c = jx2.redAdd(jx2).redIAdd(jx2).redIAdd(a.redMul(jz4));
var jxd4 = jx.redAdd(jx);
jxd4 = jxd4.redIAdd(jxd4);
var t1 = jxd4.redMul(jy2);
var nx = c.redSqr().redISub(t1.redAdd(t1));
var t2 = t1.redISub(nx);
var jyd8 = jy2.redSqr();
jyd8 = jyd8.redIAdd(jyd8);
jyd8 = jyd8.redIAdd(jyd8);
jyd8 = jyd8.redIAdd(jyd8);
var ny = c.redMul(t2).redISub(jyd8);
var nz = jy.redAdd(jy).redMul(jz);
return this.curve.jpoint(nx, ny, nz);
};
JPoint.prototype.trpl = function trpl() {
if (!this.curve.zeroA)
return this.dbl().add(this);
// hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#tripling-tpl-2007-bl
// 5M + 10S + ...
// XX = X1^2
var xx = this.x.redSqr();
// YY = Y1^2
var yy = this.y.redSqr();
// ZZ = Z1^2
var zz = this.z.redSqr();
// YYYY = YY^2
var yyyy = yy.redSqr();
// M = 3 * XX + a * ZZ2; a = 0
var m = xx.redAdd(xx).redIAdd(xx);
// MM = M^2
var mm = m.redSqr();
// E = 6 * ((X1 + YY)^2 - XX - YYYY) - MM
var e = this.x.redAdd(yy).redSqr().redISub(xx).redISub(yyyy);
e = e.redIAdd(e);
e = e.redAdd(e).redIAdd(e);
e = e.redISub(mm);
// EE = E^2
var ee = e.redSqr();
// T = 16*YYYY
var t = yyyy.redIAdd(yyyy);
t = t.redIAdd(t);
t = t.redIAdd(t);
t = t.redIAdd(t);
// U = (M + E)^2 - MM - EE - T
var u = m.redIAdd(e).redSqr().redISub(mm).redISub(ee).redISub(t);
// X3 = 4 * (X1 * EE - 4 * YY * U)
var yyu4 = yy.redMul(u);
yyu4 = yyu4.redIAdd(yyu4);
yyu4 = yyu4.redIAdd(yyu4);
var nx = this.x.redMul(ee).redISub(yyu4);
nx = nx.redIAdd(nx);
nx = nx.redIAdd(nx);
// Y3 = 8 * Y1 * (U * (T - U) - E * EE)
var ny = this.y.redMul(u.redMul(t.redISub(u)).redISub(e.redMul(ee)));
ny = ny.redIAdd(ny);
ny = ny.redIAdd(ny);
ny = ny.redIAdd(ny);
// Z3 = (Z1 + E)^2 - ZZ - EE
var nz = this.z.redAdd(e).redSqr().redISub(zz).redISub(ee);
return this.curve.jpoint(nx, ny, nz);
};
JPoint.prototype.mul = function mul(k, kbase) {
k = new BN(k, kbase);
return this.curve._wnafMul(this, k);
};
JPoint.prototype.eq = function eq(p) {
if (p.type === 'affine')
return this.eq(p.toJ());
if (this === p)
return true;
// x1 * z2^2 == x2 * z1^2
var z2 = this.z.redSqr();
var pz2 = p.z.redSqr();
if (this.x.redMul(pz2).redISub(p.x.redMul(z2)).cmpn(0) !== 0)
return false;
// y1 * z2^3 == y2 * z1^3
var z3 = z2.redMul(this.z);
var pz3 = pz2.redMul(p.z);
return this.y.redMul(pz3).redISub(p.y.redMul(z3)).cmpn(0) === 0;
};
JPoint.prototype.eqXToP = function eqXToP(x) {
var zs = this.z.redSqr();
var rx = x.toRed(this.curve.red).redMul(zs);
if (this.x.cmp(rx) === 0)
return true;
var xc = x.clone();
var t = this.curve.redN.redMul(zs);
for (;;) {
xc.iadd(this.curve.n);
if (xc.cmp(this.curve.p) >= 0)
return false;
rx.redIAdd(t);
if (this.x.cmp(rx) === 0)
return true;
}
return false;
};
JPoint.prototype.inspect = function inspect() {
if (this.isInfinity())
return '<EC JPoint Infinity>';
return '<EC JPoint x: ' + this.x.toString(16, 2) +
' y: ' + this.y.toString(16, 2) +
' z: ' + this.z.toString(16, 2) + '>';
};
JPoint.prototype.isInfinity = function isInfinity() {
// XXX This code assumes that zero is always zero in red
return this.z.cmpn(0) === 0;
};