version3/js/fp2.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.
*/

/* Finite Field arithmetic  Fp^2 functions */

/* FP2 elements are of the form a+ib, where i is sqrt(-1) */

var FP2 = function(ctx) {
    "use strict";

    /* general purpose constructor */
    var FP2 = function(c, d) {
        if (c instanceof FP2) {
            this.a = new ctx.FP(c.a);
            this.b = new ctx.FP(c.b);
        } else {
            this.a = new ctx.FP(c);
            this.b = new ctx.FP(d);
        }
    };

    FP2.prototype = {
        /* reduce components mod Modulus */
        reduce: function() {
            this.a.reduce();
            this.b.reduce();
        },

        /* normalise components of w */
        norm: function() {
            this.a.norm();
            this.b.norm();
        },

        /* test this=0 ? */
        iszilch: function() {
            //this.reduce();
            return (this.a.iszilch() && this.b.iszilch());
        },

        /* test this=1 ? */
        isunity: function() {
            var one = new ctx.FP(1);
            return (this.a.equals(one) && this.b.iszilch());
        },

        /* conditional copy of g to this depending on d */
        cmove: function(g, d) {
            this.a.cmove(g.a, d);
            this.b.cmove(g.b, d);
        },

        /* test this=x */
        equals: function(x) {
            return (this.a.equals(x.a) && this.b.equals(x.b));
        },

        /* extract a */
        getA: function() {
            return this.a.redc();
        },

        /* extract b */
        getB: function() {
            return this.b.redc();
        },

        /* set from pair of FPs */
        set: function(c, d) {
            this.a.copy(c);
            this.b.copy(d);
        },

        /* set a */
        seta: function(c) {
            this.a.copy(c);
            this.b.zero();
        },

        /* set from two BIGs */
        bset: function(c, d) {
            this.a.bcopy(c);
            this.b.bcopy(d);
        },

        /* set from one ctx.BIG */
        bseta: function(c) {
            this.a.bcopy(c);
            this.b.zero();
        },

        /* copy this=x */
        copy: function(x) {
            this.a.copy(x.a);
            this.b.copy(x.b);
        },

        /* set this=0 */
        zero: function() {
            this.a.zero();
            this.b.zero();
        },

        /* set this=1 */
        one: function() {
            this.a.one();
            this.b.zero();
        },

        /* negate this */
        neg: function() {
            //      this.norm();
            var m = new ctx.FP(this.a),
                t = new ctx.FP(0);

            m.add(this.b);
            m.neg();
            //      m.norm();
            t.copy(m);
            t.add(this.b);
            this.b.copy(m);
            this.b.add(this.a);
            this.a.copy(t);
            //this.norm();
        },

        /* conjugate this */
        conj: function() {
            this.b.neg();
            this.b.norm();
        },

        /* this+=a */
        add: function(x) {
            this.a.add(x.a);
            this.b.add(x.b);
        },

        /* this-=x */
        sub: function(x) {
            var m = new FP2(x); //var m=new FP2(0); m.copy(x);
            m.neg();
            this.add(m);
        },

        rsub: function(x) {
            this.neg();
            this.add(x);
        },

        /* this*=s, where s is FP */
        pmul: function(s) {
            this.a.mul(s);
            this.b.mul(s);
        },

        /* this*=c, where s is int */
        imul: function(c) {
            this.a.imul(c);
            this.b.imul(c);
        },

        /* this*=this */
        sqr: function() {
            //      this.norm();

            var w1 = new ctx.FP(this.a),
                w3 = new ctx.FP(this.a),
                mb = new ctx.FP(this.b);

            //      w3.mul(this.b);
            w1.add(this.b);


            w3.add(this.a);
            w3.norm();
            this.b.mul(w3);

            mb.neg();
            this.a.add(mb);

            this.a.norm();
            w1.norm();

            this.a.mul(w1);
            //      this.b.copy(w3); this.b.add(w3);
            //      this.b.norm();
        },

        /* this*=y */
        /* Now using Lazy reduction - inputs must be normed */
        mul: function(y) {
            var p = new ctx.BIG(0),
                pR = new ctx.DBIG(0),
                A, B, C, D, E, F;

            p.rcopy(ctx.ROM_FIELD.Modulus);
            pR.ucopy(p);

            if ((this.a.XES + this.b.XES) * (y.a.XES + y.b.XES) > ctx.FP.FEXCESS) {
                if (this.a.XES > 1) {
                    this.a.reduce();
                }

                if (this.b.XES > 1) {
                    this.b.reduce();
                }
            }

            A = ctx.BIG.mul(this.a.f, y.a.f);
            B = ctx.BIG.mul(this.b.f, y.b.f);

            C = new ctx.BIG(this.a.f);
            D = new ctx.BIG(y.a.f);

            C.add(this.b.f);
            C.norm();
            D.add(y.b.f);
            D.norm();

            E = ctx.BIG.mul(C, D);
            F = new ctx.DBIG(0);
            F.copy(A);
            F.add(B);
            B.rsub(pR);

            A.add(B);
            A.norm();
            E.sub(F);
            E.norm();

            this.a.f.copy(ctx.FP.mod(A));
            this.a.XES = 3;
            this.b.f.copy(ctx.FP.mod(E));
            this.b.XES = 2;
        },

        /* sqrt(a+ib) = sqrt(a+sqrt(a*a-n*b*b)/2)+ib/(2*sqrt(a+sqrt(a*a-n*b*b)/2)) */
        /* returns true if this is QR */
        sqrt: function() {
            var w1, w2;

            if (this.iszilch()) {
                return true;
            }

            w1 = new ctx.FP(this.b);
            w2 = new ctx.FP(this.a);

            w1.sqr();
            w2.sqr();
            w1.add(w2);
            if (w1.jacobi() != 1) {
                this.zero();
                return false;
            }
            w1 = w1.sqrt();
            w2.copy(this.a);
            w2.add(w1);
            w2.norm();
            w2.div2();
            if (w2.jacobi() != 1) {
                w2.copy(this.a);
                w2.sub(w1);
                w2.norm();
                w2.div2();
                if (w2.jacobi() != 1) {
                    this.zero();
                    return false;
                }
            }
            w2 = w2.sqrt();
            this.a.copy(w2);
            w2.add(w2);
            w2.inverse();
            this.b.mul(w2);

            return true;
        },

        /* convert this to hex string */
        toString: function() {
            return ("[" + this.a.toString() + "," + this.b.toString() + "]");
        },

        /* this=1/this */
        inverse: function() {
            var w1, w2;

            this.norm();

            w1 = new ctx.FP(this.a);
            w2 = new ctx.FP(this.b);

            w1.sqr();
            w2.sqr();
            w1.add(w2);
            w1.inverse();
            this.a.mul(w1);
            w1.neg();
            w1.norm();
            this.b.mul(w1);
        },

        /* this/=2 */
        div2: function() {
            this.a.div2();
            this.b.div2();
        },

        /* this*=sqrt(-1) */
        times_i: function() {
            var z = new ctx.FP(this.a); //z.copy(this.a);
            this.a.copy(this.b);
            this.a.neg();
            this.b.copy(z);
        },

        /* w*=(1+sqrt(-1)) */
        /* where X*2-(1+sqrt(-1)) is irreducible for FP4, assumes p=3 mod 8 */
        mul_ip: function() {
            //      this.norm();
            var t = new FP2(this), // t.copy(this);
                z = new ctx.FP(this.a); //z.copy(this.a);

            this.a.copy(this.b);
            this.a.neg();
            this.b.copy(z);
            this.add(t);
            //      this.norm();
        },

        div_ip2: function() {
            var t = new FP2(0);
            this.norm();
            t.a.copy(this.a);
            t.a.add(this.b);
            t.b.copy(this.b);
            t.b.sub(this.a);
            this.copy(t);
            this.norm();
        },

        /* w/=(1+sqrt(-1)) */
        div_ip: function() {
            var t = new FP2(0);
            this.norm();
            t.a.copy(this.a);
            t.a.add(this.b);
            t.b.copy(this.b);
            t.b.sub(this.a);
            this.copy(t);
            this.norm();
            this.div2();
        },

        /* this=this^e */
        pow: function(e) {
            this.norm();

            var r = new FP2(1),
                x = new FP2(this), //x.copy(this);
                bt;

            e.norm();

            for (;;) {
                bt = e.parity();
                e.fshr(1);

                if (bt == 1) {
                    r.mul(x);
                }

                if (e.iszilch()) {
                    break;
                }
                x.sqr();
            }

            r.reduce();

            return r;
        }

    };

    return FP2;
};