src/ecp8.js

/*
    Licensed to the Apache Software Foundation (ASF) under one
    or more contributor license agreements.  See the NOTICE file
    distributed with this work for additional information
    regarding copyright ownership.  The ASF licenses this file
    to you under the Apache License, Version 2.0 (the
    "License"); you may not use this file except in compliance
    with the License.  You may obtain a copy of the License at

    http://www.apache.org/licenses/LICENSE-2.0

    Unless required by applicable law or agreed to in writing,
    software distributed under the License is distributed on an
    "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
    KIND, either express or implied.  See the License for the
    specific language governing permissions and limitations
    under the License.
*/

/* AMCL Weierstrass elliptic curve functions over ctx.FP8 */

var ECP8 = function(ctx) {
    "use strict";

    /**
     * Creates an instance of ECP8
     *
     * @constructor
     * @this {ECP8}
    */    
    var ECP8 = function(input) {
        if (input instanceof ECP8) {
            // copy constructor
            this.x = new ctx.FP8(input.x);
            this.y = new ctx.FP8(input.y);
            this.z = new ctx.FP8(input.z);
        } else {
            // default constructor (point at infinity)
            this.x = new ctx.FP8(0);
            this.y = new ctx.FP8(1);
            this.z = new ctx.FP8(0);
        }
    };

    ECP8.prototype = {

	/**
         * Tests for ECP8 point equal to infinity
         *
         * @this {ECP8}
         * @param 1 if infinity, else returns 0
         */	
        is_infinity: function() {

            this.x.reduce();
            this.y.reduce();
            this.z.reduce();
            return (this.x.iszilch() && this.z.iszilch());
        },

	/**
         * Copy ECP8 point to another ECP8 point
         *
         * @this {ECP8}
         * @param P ECP8 instance
         */	
        copy: function(P) {
            this.x.copy(P.x);
            this.y.copy(P.y);
            this.z.copy(P.z);
        },

	/**
         * Set ECP8 to point-at-infinity
         *
         * @this {ECP8}
         */	
        inf: function() {
            this.x.zero();
            this.y.one();
            this.z.zero();
        },

	/**
         * conditional move of Q to P dependant on d 
         *
         * @this {ECP8}
         */	
        cmove: function(Q, d) {
            this.x.cmove(Q.x, d);
            this.y.cmove(Q.y, d);
            this.z.cmove(Q.z, d);
        },

	/**
         * Constant time select from pre-computed table 
         *
         * @this {ECP8}
         */	
        select: function(W, b) {
            var MP = new ECP8(),
                m = b >> 31,
                babs = (b ^ m) - m;

            babs = (babs - 1) / 2;

            this.cmove(W[0], ECP8.teq(babs, 0)); // conditional move
            this.cmove(W[1], ECP8.teq(babs, 1));
            this.cmove(W[2], ECP8.teq(babs, 2));
            this.cmove(W[3], ECP8.teq(babs, 3));
            this.cmove(W[4], ECP8.teq(babs, 4));
            this.cmove(W[5], ECP8.teq(babs, 5));
            this.cmove(W[6], ECP8.teq(babs, 6));
            this.cmove(W[7], ECP8.teq(babs, 7));

            MP.copy(this);
            MP.neg();
            this.cmove(MP, (m & 1));
        },

	/**
         * Test P == Q 
         *
         * @this {ECP8}
         * @param Q ECP8 instance 
         */	
        equals: function(Q) {
            var a, b;

            a = new ctx.FP8(this.x);
            b = new ctx.FP8(Q.x);

            a.mul(Q.z);
            b.mul(this.z);
            if (!a.equals(b)) {
                return false;
            }

            a.copy(this.y);
            a.mul(Q.z);
            b.copy(Q.y);
            b.mul(this.z);
            if (!a.equals(b)) {
                return false;
            }

            return true;
        },

	/**
         * set this=-this 
         *
         * @this {ECP8}
         */	
        neg: function() {
            this.y.norm();
            this.y.neg();
            this.y.norm();
            return;
        },

	/**
         * convert this to affine, from (x,y,z) to (x,y) 
         *
         * @this {ECP8}
         */	
        affine: function() {
            var one;

            if (this.is_infinity()) {
                return;
            }

            one = new ctx.FP8(1);

            if (this.z.equals(one)) {
                this.x.reduce();
                this.y.reduce();
                return;
            }

            this.z.inverse();
            this.x.mul(this.z);
            this.x.reduce();
            this.y.mul(this.z);
            this.y.reduce();
            this.z.copy(one);
        },

	/**
         * extract affine x as ctx.FP2 
         *
         * @this {ECP8}
         */	
        getX: function() {
			var W=new ECP8(); W.copy(this); W.affine();
            return W.x;
        },

	/**
         * extract affine y as ctx.FP2
         *
         * @this {ECP8}
         */	
        getY: function() {
			var W=new ECP8(); W.copy(this); W.affine();
            return W.y;
        },

	/**
         * extract projective x
         *
         * @this {ECP8}
         */	
        getx: function() {
            return this.x;
        },

	/**
         * extract projective y
         *
         * @this {ECP8}
         */	
        gety: function() {
            return this.y;
        },

	/**
         * extract projective z
         *
         * @this {ECP8}
         */	
        getz: function() {
            return this.z;
        },

	/**
         * convert this to byte arrayextract projective x
         *
         * @this {ECP8}
         * @param b byte array output
         */	
        toBytes: function(b) {
            var t = [],
                i;
			var W=new ECP8(); W.copy(this);
            W.affine();
            W.x.geta().geta().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i] = t[i];
            }
            W.x.geta().geta().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + ctx.BIG.MODBYTES] = t[i];
            }
            W.x.geta().getb().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 2*ctx.BIG.MODBYTES] = t[i];
            }
            W.x.geta().getb().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 3*ctx.BIG.MODBYTES] = t[i];
            }

            W.x.getb().geta().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 4*ctx.BIG.MODBYTES] = t[i];
            }
            W.x.getb().geta().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 5*ctx.BIG.MODBYTES] = t[i];
            }
            W.x.getb().getb().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 6*ctx.BIG.MODBYTES] = t[i];
            }
            W.x.getb().getb().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 7*ctx.BIG.MODBYTES] = t[i];
            }

            W.y.geta().geta().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 8 * ctx.BIG.MODBYTES] = t[i];
            }
            W.y.geta().geta().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 9 * ctx.BIG.MODBYTES] = t[i];
            }
            W.y.geta().getb().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 10 * ctx.BIG.MODBYTES] = t[i];
            }
            W.y.geta().getb().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 11 * ctx.BIG.MODBYTES] = t[i];
            }

            W.y.getb().geta().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 12 * ctx.BIG.MODBYTES] = t[i];
            }
            W.y.getb().geta().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 13 * ctx.BIG.MODBYTES] = t[i];
            }
            W.y.getb().getb().getA().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 14 * ctx.BIG.MODBYTES] = t[i];
            }
            W.y.getb().getb().getB().toBytes(t);
            for (i = 0; i < ctx.BIG.MODBYTES; i++) {
                b[i + 15 * ctx.BIG.MODBYTES] = t[i];
            }
        },

	/**
         * convert this to hex string 
         *
         * @this {ECP8}
         * @return hex string
         */	
        toString: function() {
			var W=new ECP8(); W.copy(this);
            if (W.is_infinity()) {
                return "infinity";
            }
            W.affine();
            return "(" + W.x.toString() + "," + W.y.toString() + ")";
        },

	/**
         * set this=(x,y)
         *
         * @this {ECP8}
         * @param ix x-value
         * @param iy y-value
         */	
        setxy: function(ix, iy) {
            var rhs, y2;

            this.x.copy(ix);
            this.y.copy(iy);
            this.z.one();
			this.x.norm();

            rhs = ECP8.RHS(this.x);

            y2 = new ctx.FP8(this.y); //y2.copy(this.y);
            y2.sqr();

            if (!y2.equals(rhs)) {
                this.inf();
            }
        },

	/**
         * set this=(x,.) 
         *
         * @this {ECP8}
         * @param ix x-value
         */	
        setx: function(ix) {
            var rhs;

            this.x.copy(ix);
            this.z.one();
			this.x.norm();
            rhs = ECP8.RHS(this.x);

            if (rhs.sqrt()) {
                this.y.copy(rhs);
            } else {
                this.inf();
            }
        },

	/**
         * set this*=q, where q is Modulus, using Frobenius 
         *
         * @this {ECP8}
         */	
        frob: function(F,n) {
            for (var i=0;i<n;i++) {
                this.x.frob(F[2]);
                this.x.qmul(F[0]);
                if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
                    this.x.div_i2();
                }
                if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
                    this.x.times_i2();
                }

                this.y.frob(F[2]);
                this.y.qmul(F[1]);

                if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
                    this.y.div_i();
                }
                if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
                    this.y.times_i2(); this.y.times_i2(); this.y.times_i();
                }
                this.z.frob(F[2]);
            }
        },

	/**
         * this+=this 
         *
         * @this {ECP8}
         */	
        dbl: function() {
            var iy, t0, t1, t2, x3, y3;

            iy = new ctx.FP8(this.y);
            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
                iy.times_i();
                iy.norm();
            }

            t0 = new ctx.FP8(this.y);
            t0.sqr();
            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
                t0.times_i();
            }
            t1 = new ctx.FP8(iy);
            t1.mul(this.z);
            t2 = new ctx.FP8(this.z);
            t2.sqr();

            this.z.copy(t0);
            this.z.add(t0);
            this.z.norm();
            this.z.add(this.z);
            this.z.add(this.z);
            this.z.norm();

            t2.imul(3 * ctx.ROM_CURVE.CURVE_B_I);
            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
                t2.times_i();
            }

            x3 = new ctx.FP8(t2);
            x3.mul(this.z);

            y3 = new ctx.FP8(t0);

            y3.add(t2);
            y3.norm();
            this.z.mul(t1);
            t1.copy(t2);
            t1.add(t2);
            t2.add(t1);
            t2.norm();
            t0.sub(t2);
            t0.norm(); //y^2-9bz^2
            y3.mul(t0);
            y3.add(x3); //(y^2+3z*2)(y^2-9z^2)+3b.z^2.8y^2
            t1.copy(this.x);
            t1.mul(iy); //
            this.x.copy(t0);
            this.x.norm();
            this.x.mul(t1);
            this.x.add(this.x); //(y^2-9bz^2)xy2

            this.x.norm();
            this.y.copy(y3);
            this.y.norm();

            return 1;
        },

	/**
         * Adds ECP8 instances
         *
         * param Q ECP8 instance
         * @this {ECP8}
         */		
        add: function(Q) {
            var b, t0, t1, t2, t3, t4, x3, y3, z3;

            b = 3 * ctx.ROM_CURVE.CURVE_B_I;
            t0 = new ctx.FP8(this.x);
            t0.mul(Q.x); // x.Q.x
            t1 = new ctx.FP8(this.y);
            t1.mul(Q.y); // y.Q.y

            t2 = new ctx.FP8(this.z);
            t2.mul(Q.z);
            t3 = new ctx.FP8(this.x);
            t3.add(this.y);
            t3.norm(); //t3=X1+Y1
            t4 = new ctx.FP8(Q.x);
            t4.add(Q.y);
            t4.norm(); //t4=X2+Y2
            t3.mul(t4); //t3=(X1+Y1)(X2+Y2)
            t4.copy(t0);
            t4.add(t1); //t4=X1.X2+Y1.Y2

            t3.sub(t4);
            t3.norm();
            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
                t3.times_i();  //t3=(X1+Y1)(X2+Y2)-(X1.X2+Y1.Y2) = X1.Y2+X2.Y1
            }

            t4.copy(this.y);
            t4.add(this.z);
            t4.norm(); //t4=Y1+Z1
            x3 = new ctx.FP8(Q.y);
            x3.add(Q.z);
            x3.norm(); //x3=Y2+Z2

            t4.mul(x3); //t4=(Y1+Z1)(Y2+Z2)
            x3.copy(t1); //
            x3.add(t2); //X3=Y1.Y2+Z1.Z2

            t4.sub(x3);
            t4.norm();
            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
                t4.times_i();  //t4=(Y1+Z1)(Y2+Z2) - (Y1.Y2+Z1.Z2) = Y1.Z2+Y2.Z1
            }

            x3.copy(this.x);
            x3.add(this.z);
            x3.norm(); // x3=X1+Z1
            y3 = new ctx.FP8(Q.x);
            y3.add(Q.z);
            y3.norm(); // y3=X2+Z2
            x3.mul(y3); // x3=(X1+Z1)(X2+Z2)
            y3.copy(t0);
            y3.add(t2); // y3=X1.X2+Z1+Z2
            y3.rsub(x3);
            y3.norm(); // y3=(X1+Z1)(X2+Z2) - (X1.X2+Z1.Z2) = X1.Z2+X2.Z1

            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
                t0.times_i();
                t1.times_i();
            }

            x3.copy(t0);
            x3.add(t0);
            t0.add(x3);
            t0.norm();
            t2.imul(b);
            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
                t2.times_i();
            }

            z3 = new ctx.FP8(t1);
            z3.add(t2);
            z3.norm();
            t1.sub(t2);
            t1.norm();
            y3.imul(b);
            if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
                y3.times_i();
            }

            x3.copy(y3);
            x3.mul(t4);
            t2.copy(t3);
            t2.mul(t1);
            x3.rsub(t2);
            y3.mul(t0);
            t1.mul(z3);
            y3.add(t1);
            t0.mul(t3);
            z3.mul(t4);
            z3.add(t0);

            this.x.copy(x3);
            this.x.norm();
            this.y.copy(y3);
            this.y.norm();
            this.z.copy(z3);
            this.z.norm();

            return 0;
        },

	/**
         * Subtracts ECP instance Q  from this
         *
         * @this {ECP8}
         * @param Q ECP8 instance
         */	
        sub: function(Q) {
            var D;
			var NQ=new ECP8(); NQ.copy(Q);
            NQ.neg();
            D = this.add(NQ);
            return D;
        },

	/**
         * Multiplies an ECP8 instance P by a BIG, side-channel resistant
         *
         * @this {ECP8}
         * @param e BIG number multiplier
         */		
        mul: function(e) {
            /* fixed size windows */
            var mt = new ctx.BIG(),
                t = new ctx.BIG(),
                C = new ECP8(),
                P = new ECP8(),
                Q = new ECP8(),
                W = [],
                w = [],
                i, nb, s, ns;

            if (this.is_infinity()) {
                return new ECP8();
            }

            // precompute table
            Q.copy(this);
            Q.dbl();
            W[0] = new ECP8();
            W[0].copy(this);

            for (i = 1; i < 8; i++) {
                W[i] = new ECP8();
                W[i].copy(W[i - 1]);
                W[i].add(Q);
            }

            // make exponent odd - add 2P if even, P if odd
            t.copy(e);
            s = t.parity();
            t.inc(1);
            t.norm();
            ns = t.parity();
            mt.copy(t);
            mt.inc(1);
            mt.norm();
            t.cmove(mt, s);
            Q.cmove(this, ns);
            C.copy(Q);

            nb = 1 + Math.floor((t.nbits() + 3) / 4);

            // convert exponent to signed 4-bit window
            for (i = 0; i < nb; i++) {
                w[i] = (t.lastbits(5) - 16);
                t.dec(w[i]);
                t.norm();
                t.fshr(4);
            }
            w[nb] = t.lastbits(5);

            P.copy(W[Math.floor((w[nb] - 1) / 2)]);
            for (i = nb - 1; i >= 0; i--) {
                Q.select(W, w[i]);
                P.dbl();
                P.dbl();
                P.dbl();
                P.dbl();
                P.add(Q);
            }
            P.sub(C);
            P.affine();

            return P;
        }
    };

    /**
      * Set group generator
      *
      * @this {ECP8}
      */	
    ECP8.generator = function() {
        var G=new ECP8(),
            A = new ctx.BIG(0),
            B = new ctx.BIG(0),
            XAA, XAB, XA, XBA, XBB, XB, X,
            YAA, YAB, YA, YBA, YBB, YB, Y;

        A.rcopy(ctx.ROM_CURVE.CURVE_Pxaaa);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pxaab);
        XAA= new ctx.FP2(A,B);

        A.rcopy(ctx.ROM_CURVE.CURVE_Pxaba);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pxabb);

        XAB= new ctx.FP2(A,B);
        XA=new ctx.FP4(XAA,XAB);

        A.rcopy(ctx.ROM_CURVE.CURVE_Pxbaa);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pxbab);
        XBA= new ctx.FP2(A,B);

        A.rcopy(ctx.ROM_CURVE.CURVE_Pxbba);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pxbbb);

        XBB= new ctx.FP2(A,B);
        XB=new ctx.FP4(XBA,XBB);

        X=new ctx.FP8(XA,XB);


        A.rcopy(ctx.ROM_CURVE.CURVE_Pyaaa);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pyaab);
        YAA= new ctx.FP2(A,B);

        A.rcopy(ctx.ROM_CURVE.CURVE_Pyaba);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pyabb);

        YAB= new ctx.FP2(A,B);
        YA=new ctx.FP4(YAA,YAB);

        A.rcopy(ctx.ROM_CURVE.CURVE_Pybaa);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pybab);
        YBA= new ctx.FP2(A,B);

        A.rcopy(ctx.ROM_CURVE.CURVE_Pybba);
        B.rcopy(ctx.ROM_CURVE.CURVE_Pybbb);

        YBB= new ctx.FP2(A,B);
        YB=new ctx.FP4(YBA,YBB);

        Y=new ctx.FP8(YA,YB);

        G.setxy(X,Y);

        return G;
    };

    /**
      * convert from byte array to point 
      *
      * @this {ECP8}
      * @param b input byte array
      */	    
    ECP8.fromBytes = function(b) {
        var t = [],
            ra, rb, ra4, rb4, ra8, rb8, i, rx, ry, P;

        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        ra4=new ctx.FP2(ra,rb);

        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i  + 2*ctx.BIG.MODBYTES];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 3*ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        rb4=new ctx.FP2(ra,rb);

        ra8=new ctx.FP4(ra4,rb4);

        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 4*ctx.BIG.MODBYTES];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 5*ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        ra4=new ctx.FP2(ra,rb);

        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i  + 6*ctx.BIG.MODBYTES];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 7*ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        rb4=new ctx.FP2(ra,rb);

        rb8=new ctx.FP4(ra4,rb4);

        rx = new ctx.FP8(ra8, rb8); //rx.bset(ra,rb);


        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 8 * ctx.BIG.MODBYTES];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 9 * ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        ra4=new ctx.FP2(ra,rb);

        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 10 * ctx.BIG.MODBYTES];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 11 * ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        rb4=new ctx.FP2(ra,rb);

        ra8=new ctx.FP4(ra4,rb4);

        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 12 * ctx.BIG.MODBYTES];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 13 * ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        ra4=new ctx.FP2(ra,rb);

        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 14 * ctx.BIG.MODBYTES];
        }
        ra = ctx.BIG.fromBytes(t);
        for (i = 0; i < ctx.BIG.MODBYTES; i++) {
            t[i] = b[i + 15 * ctx.BIG.MODBYTES];
        }
        rb = ctx.BIG.fromBytes(t);
        rb4=new ctx.FP2(ra,rb);

        rb8=new ctx.FP4(ra4,rb4);

        ry = new ctx.FP8(ra8, rb8); //ry.bset(ra,rb);

        P = new ECP8();
        P.setxy(rx, ry);

        return P;
    };

    /**
      * Calculate RHS of curve equation x^3+B
      *
      * @this {ECP8}
      * @param x x-value
      */	    
    ECP8.RHS = function(x) {
        var r, c, b;

        //x.norm();
        r = new ctx.FP8(x); //r.copy(x);
        r.sqr();

        c = new ctx.BIG(0);
        c.rcopy(ctx.ROM_CURVE.CURVE_B);
        b = new ctx.FP8(c); //b.bseta(c);

        if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
            b.div_i();
        }
        if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
            b.times_i();
        }

        r.mul(x);
        r.add(b);

        r.reduce();
        return r;
    };

    /* P=u0.Q0+u1*Q1+u2*Q2+u3*Q3... */
    // Bos & Costello https://eprint.iacr.org/2013/458.pdf
    // Faz-Hernandez & Longa & Sanchez  https://eprint.iacr.org/2013/158.pdf
    // Side channel attack secure

    /**
      * Calculate P=u0.Q0+u1*Q1+u2*Q2+u3*Q3... 
      *
      * @this {ECP8}
      */     
    ECP8.mul16 = function(Q, u) {
        var W = new ECP8(),
            P = new ECP8(),
            T1 = [],
            T2 = [],
            T3 = [],
            T4 = [],
            mt = new ctx.BIG(),
            t = [],
            w1 = [],
            s1 = [],
            w2 = [],
            s2 = [],
            w3 = [],
            s3 = [],
            w4 = [],
            s4 = [],
            F=ECP8.frob_constants(),
            i, j, k, nb, bt, pb1, pb2, pb3, pb4;

        for (i = 0; i < 16; i++) {
            t[i] = new ctx.BIG(u[i]); t[i].norm();
            //Q[i].affine();
        }

        T1[0] = new ECP8(); T1[0].copy(Q[0]); // Q[0]
        T1[1] = new ECP8(); T1[1].copy(T1[0]); T1[1].add(Q[1]); // Q[0]+Q[1]
        T1[2] = new ECP8(); T1[2].copy(T1[0]); T1[2].add(Q[2]); // Q[0]+Q[2]
        T1[3] = new ECP8(); T1[3].copy(T1[1]); T1[3].add(Q[2]); // Q[0]+Q[1]+Q[2]
        T1[4] = new ECP8(); T1[4].copy(T1[0]); T1[4].add(Q[3]); // Q[0]+Q[3]
        T1[5] = new ECP8(); T1[5].copy(T1[1]); T1[5].add(Q[3]); // Q[0]+Q[1]+Q[3]
        T1[6] = new ECP8(); T1[6].copy(T1[2]); T1[6].add(Q[3]); // Q[0]+Q[2]+Q[3]
        T1[7] = new ECP8(); T1[7].copy(T1[3]); T1[7].add(Q[3]); // Q[0]+Q[1]+Q[2]+Q[3]

        //  Use Frobenius
        for (i=0;i<8;i++) {
            T2[i] = new ECP8(); T2[i].copy(T1[i]);
            T2[i].frob(F,4);
            T3[i] = new ECP8(); T3[i].copy(T2[i]);
            T3[i].frob(F,4);
            T4[i] = new ECP8(); T4[i].copy(T3[i]);
            T4[i].frob(F,4);
        }

        // Make it odd
        pb1=1-t[0].parity();
        t[0].inc(pb1);
        t[0].norm();

        pb2=1-t[4].parity();
        t[4].inc(pb2);
        t[4].norm();

        pb3=1-t[8].parity();
        t[8].inc(pb3);
        t[8].norm();

        pb4=1-t[12].parity();
        t[12].inc(pb4);
        t[12].norm();

        // Number of bits
        mt.zero();
        for (i=0;i<16;i++) {
            mt.or(t[i]);
        }

        nb=1+mt.nbits();

        // Sign pivot
        s1[nb-1]=1;
        s2[nb-1]=1;
        s3[nb-1]=1;
        s4[nb-1]=1;
        for (i=0;i<nb-1;i++) {
            t[0].fshr(1);
            s1[i]=2*t[0].parity()-1;
            t[4].fshr(1);
            s2[i]=2*t[4].parity()-1;

            t[8].fshr(1);
            s3[i]=2*t[8].parity()-1;
            t[12].fshr(1);
            s4[i]=2*t[12].parity()-1;
        }

        // Recoded exponent
        for (i=0; i<nb; i++) {
            w1[i]=0;
            k=1;
            for (j=1; j<4; j++) {
                bt=s1[i]*t[j].parity();
                t[j].fshr(1);
                t[j].dec(bt>>1);
                t[j].norm();
                w1[i]+=bt*k;
                k*=2;
            }
            w2[i]=0;
            k=1;
            for (j=5; j<8; j++) {
                bt=s2[i]*t[j].parity();
                t[j].fshr(1);
                t[j].dec(bt>>1);
                t[j].norm();
                w2[i]+=bt*k;
                k*=2;
            }

            w3[i]=0;
            k=1;
            for (j=9; j<12; j++) {
                bt=s3[i]*t[j].parity();
                t[j].fshr(1);
                t[j].dec(bt>>1);
                t[j].norm();
                w3[i]+=bt*k;
                k*=2;
            }
            w4[i]=0;
            k=1;
            for (j=13; j<16; j++) {
                bt=s4[i]*t[j].parity();
                t[j].fshr(1);
                t[j].dec(bt>>1);
                t[j].norm();
                w4[i]+=bt*k;
                k*=2;
            }
        }

        // Main loop
        P.select(T1,2*w1[nb-1]+1);
        W.select(T2,2*w2[nb-1]+1);
        P.add(W);
        W.select(T3,2*w3[nb-1]+1);
        P.add(W);
        W.select(T4,2*w4[nb-1]+1);
        P.add(W);
        for (i=nb-2;i>=0;i--) {
            P.dbl();
            W.select(T1,2*w1[i]+s1[i]);
            P.add(W);
            W.select(T2,2*w2[i]+s2[i]);
            P.add(W);
            W.select(T3,2*w3[i]+s3[i]);
            P.add(W);
            W.select(T4,2*w4[i]+s4[i]);
            P.add(W);
        }

        // apply correction
        W.copy(P);
        W.sub(Q[0]);
        P.cmove(W,pb1);

        W.copy(P);
        W.sub(Q[4]);
        P.cmove(W,pb2);

        W.copy(P);
        W.sub(Q[8]);
        P.cmove(W,pb3);

        W.copy(P);
        W.sub(Q[12]);
        P.cmove(W,pb4);

        P.affine();
        return P;
    };

    /* return 1 if b==c, no branching */
    ECP8.teq = function(b, c) {
        var x = b ^ c;
        x -= 1; // if x=0, x now -1
        return ((x >> 31) & 1);
    };

    /* needed for SOK */
    ECP8.mapit = function(h) {
        var F=ECP8.frob_constants(),
            q, x, one, Q, X, X2, X4,
            xQ, x2Q, x3Q, x4Q, x5Q, x6Q, x7Q, x8Q;

        q = new ctx.BIG(0);
        q.rcopy(ctx.ROM_FIELD.Modulus);
        x = ctx.BIG.fromBytes(h);
        one = new ctx.BIG(1);
        x.mod(q);

        for (;;) {
            X2 = new ctx.FP2(one, x);
            X4 = new ctx.FP4(X2);
            X = new ctx.FP8(X4);
            Q = new ECP8();
            Q.setx(X);
            if (!Q.is_infinity()) {
                break;
            }
            x.inc(1);
            x.norm();
        }

        /* Fast Hashing to G2 - Fuentes-Castaneda, Knapp and Rodriguez-Henriquez */
        x = new ctx.BIG(0);
        x.rcopy(ctx.ROM_CURVE.CURVE_Bnx);


        xQ = Q.mul(x);
        x2Q = xQ.mul(x);
        x3Q = x2Q.mul(x);
        x4Q = x3Q.mul(x);
        x5Q = x4Q.mul(x);
        x6Q = x5Q.mul(x);
        x7Q = x6Q.mul(x);
        x8Q = x7Q.mul(x);

        if (ctx.ECP.SIGN_OF_X == ctx.ECP.NEGATIVEX) {
            xQ.neg();
            x3Q.neg();
            x5Q.neg();
            x7Q.neg();
        }

        x8Q.sub(x7Q);
        x8Q.sub(Q);

        x7Q.sub(x6Q);
        x7Q.frob(F,1);

        x6Q.sub(x5Q);
        x6Q.frob(F,2);

        x5Q.sub(x4Q);
        x5Q.frob(F,3);

        x4Q.sub(x3Q);
        x4Q.frob(F,4);

        x3Q.sub(x2Q);
        x3Q.frob(F,5);

        x2Q.sub(xQ);
        x2Q.frob(F,6);

        xQ.sub(Q);
        xQ.frob(F,7);

        Q.dbl();
        Q.frob(F,8);

        Q.add(x8Q);
        Q.add(x7Q);
        Q.add(x6Q);
        Q.add(x5Q);

        Q.add(x4Q);
        Q.add(x3Q);
        Q.add(x2Q);
        Q.add(xQ);

        Q.affine();
        return Q;
    };

    ECP8.frob_constants = function() {
        var fa = new ctx.BIG(0),
            fb = new ctx.BIG(0),
            F=[],
            X, F0, F1, F2;

        fa.rcopy(ctx.ROM_FIELD.Fra);
        fb.rcopy(ctx.ROM_FIELD.Frb);
        X = new ctx.FP2(fa, fb);

        F0=new ctx.FP2(X); F0.sqr();
        F2=new ctx.FP2(F0);
        F2.mul_ip(); F2.norm();
        F1=new ctx.FP2(F2); F1.sqr();
        F2.mul(F1);

        F2.mul_ip(); F2.norm();

        F1.copy(X);
        if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
            F1.mul_ip();
            F1.inverse();
            F0.copy(F1); F0.sqr();
        }
        F0.mul_ip(); F0.norm();
        F1.mul(F0);

        F[0]=new ctx.FP2(F0); F[1]=new ctx.FP2(F1); F[2]=new ctx.FP2(F2);
        return F;
    };

    return ECP8;
};

// CommonJS module exports
if (typeof module !== "undefined" && typeof module.exports !== "undefined") {
  module.exports.ECP8 = ECP8;
}