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) */

/* general purpose constructor */
var FP2 =function(c,d)
{
	if (c instanceof FP2)
	{
		this.a=new FP(c.a);
		this.b=new FP(c.b);
	}
	else
	{
		this.a=new FP(c);
		this.b=new 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 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 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 FP(this.a);
		var t=new 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+=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);
	},
/* 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 FP(this.a);
		var w3=new FP(this.a);
		var mb=new FP(this.b);

		w3.mul(this.b);
		w1.add(this.b);
		mb.neg();
		this.a.add(mb);
		this.a.mul(w1);
		this.b.copy(w3); this.b.add(w3);
		this.norm();
	},
/* this*=y */
	mul: function(y)
	{
		this.norm();  // This is needed here as {a,b} is not normed before additions

		var w1=new FP(this.a);
		var w2=new FP(this.b);
		var w5=new FP(this.a);
		var mw=new FP(0);

		w1.mul(y.a);  // w1=a*y.a  - this norms w1 and y.a, NOT a
		w2.mul(y.b);  // w2=b*y.b  - this norms w2 and y.b, NOT b
		w5.add(this.b);    // w5=a+b
		this.b.copy(y.a); this.b.add(y.b); // b=y.a+y.b

		this.b.mul(w5);
		mw.copy(w1); mw.add(w2); mw.neg();

		this.b.add(mw); mw.add(w1);
		this.a.copy(w1); this.a.add(mw);

		this.norm();
	},

/* 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()
	{
		if (this.iszilch()) return true;
		var w1=new FP(this.b);
		var w2=new 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.div2();
		if (w2.jacobi()!=1)
		{
			w2.copy(this.a); w2.sub(w1); 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()
	{
		this.norm();
		var w1=new FP(this.a);
		var w2=new FP(this.b);
		w1.sqr();
		w2.sqr();
		w1.add(w2);
		w1.inverse();
		this.a.mul(w1);
		w1.neg();
		this.b.mul(w1);
	},
/* this/=2 */
	div2: function()
	{
		this.a.div2();
		this.b.div2();
	},
/* this*=sqrt(-1) */
	times_i: function()
	{
		var z=new 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);
		var z=new FP(this.a); //z.copy(this.a);
		this.a.copy(this.b);
		this.a.neg();
		this.b.copy(z);
		this.add(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.div2();
	},
/* this=this^e */
	pow: function(e)
	{
		var bt;
		var r=new FP2(1);
		this.norm();
		var x=new FP2(this); //x.copy(this);
		e.norm();
		while (true)
		{
			bt=e.parity();
			e.fshr(1);
			if (bt==1) r.mul(x);
			if (e.iszilch()) break;
			x.sqr();
		}

		r.reduce();
		return r;
	}

};