short.js

'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;
};