in src/ecp2.js [22:914]
var ECP2 = function(ctx) {
"use strict";
/**
* Creates an instance of ECP2
*
* @constructor
* @this {ECP2}
*/
var ECP2 = function(input) {
if (input instanceof ECP2) {
// copy constructor
this.x = new ctx.FP2(input.x);
this.y = new ctx.FP2(input.y);
this.z = new ctx.FP2(input.z);
} else {
// default constructor (point at infinity)
this.x = new ctx.FP2(0);
this.y = new ctx.FP2(1);
this.z = new ctx.FP2(0);
}
};
ECP2.prototype = {
/**
* Tests for ECP2 point equal to infinity
*
* @this {ECP2}
* @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 ECP2 point to another ECP2 point
*
* @this {ECP2}
* @param P ECP2 instance
*/
copy: function(P) {
this.x.copy(P.x);
this.y.copy(P.y);
this.z.copy(P.z);
},
/**
* Set ECP2 to point-at-infinity
*
* @this {ECP2}
*/
inf: function() {
this.x.zero();
this.y.one();
this.z.zero();
},
/**
* conditional move of Q to P dependant on d
*
* @this {ECP2}
*/
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 {ECP2}
*/
select: function(W, b) {
var MP = new ECP2(),
m, babs;
m = b >> 31,
babs = (b ^ m) - m;
babs = (babs - 1) / 2;
this.cmove(W[0], ECP2.teq(babs, 0)); // conditional move
this.cmove(W[1], ECP2.teq(babs, 1));
this.cmove(W[2], ECP2.teq(babs, 2));
this.cmove(W[3], ECP2.teq(babs, 3));
this.cmove(W[4], ECP2.teq(babs, 4));
this.cmove(W[5], ECP2.teq(babs, 5));
this.cmove(W[6], ECP2.teq(babs, 6));
this.cmove(W[7], ECP2.teq(babs, 7));
MP.copy(this);
MP.neg();
this.cmove(MP, (m & 1));
},
/**
* Test P == Q
*
* @this {ECP2}
* @param Q ECP2 instance
*/
equals: function(Q) {
var a, b;
a = new ctx.FP2(0);
a.copy(this.x);
b = new ctx.FP2(0);
b.copy(Q.x);
a.copy(this.x);
a.mul(Q.z);
a.reduce();
b.copy(Q.x);
b.mul(this.z);
b.reduce();
if (!a.equals(b)) {
return false;
}
a.copy(this.y);
a.mul(Q.z);
a.reduce();
b.copy(Q.y);
b.mul(this.z);
b.reduce();
if (!a.equals(b)) {
return false;
}
return true;
},
/**
* set this=-this
*
* @this {ECP2}
*/
neg: function() {
this.y.norm();
this.y.neg();
this.y.norm();
return;
},
/**
* convert this to affine, from (x,y,z) to (x,y)
*
* @this {ECP2}
*/
affine: function() {
var one;
if (this.is_infinity()) {
return;
}
one = new ctx.FP2(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 {ECP2}
*/
getX: function() {
var W=new ECP2(); W.copy(this); W.affine();
return W.x;
},
/**
* extract affine y as ctx.FP2
*
* @this {ECP2}
*/
getY: function() {
var W=new ECP2(); W.copy(this); W.affine();
return W.y;
},
/**
* extract projective x
*
* @this {ECP2}
*/
getx: function() {
return this.x;
},
/**
* extract projective y
*
* @this {ECP2}
*/
gety: function() {
return this.y;
},
/**
* extract projective z
*
* @this {ECP2}
*/
getz: function() {
return this.z;
},
/**
* convert this to byte arrayextract projective x
*
* @this {ECP2}
* @param b byte array output
*/
toBytes: function(b) {
var t = [],
i;
var W=new ECP2(); W.copy(this);
W.affine();
W.x.getA().toBytes(t);
for (i = 0; i < ctx.BIG.MODBYTES; i++) {
b[i] = t[i];
}
W.x.getB().toBytes(t);
for (i = 0; i < ctx.BIG.MODBYTES; i++) {
b[i + ctx.BIG.MODBYTES] = t[i];
}
W.y.getA().toBytes(t);
for (i = 0; i < ctx.BIG.MODBYTES; i++) {
b[i + 2 * ctx.BIG.MODBYTES] = t[i];
}
W.y.getB().toBytes(t);
for (i = 0; i < ctx.BIG.MODBYTES; i++) {
b[i + 3 * ctx.BIG.MODBYTES] = t[i];
}
},
/**
* convert this to hex string
*
* @this {ECP2}
* @return hex string
*/
toString: function() {
var W=new ECP2(); W.copy(this);
if (W.is_infinity()) {
return "infinity";
}
W.affine();
return "(" + W.x.toString() + "," + W.y.toString() + ")";
},
/**
* set this=(x,y)
*
* @this {ECP2}
* @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 = ECP2.RHS(this.x);
y2 = new ctx.FP2(this.y);
y2.sqr();
if (!y2.equals(rhs)) {
this.inf();
}
},
/**
* set this=(x,.)
*
* @this {ECP2}
* @param ix x-value
*/
setx: function(ix) {
var rhs;
this.x.copy(ix);
this.z.one();
this.x.norm();
rhs = ECP2.RHS(this.x);
if (rhs.sqrt()) {
this.y.copy(rhs);
} else {
this.inf();
}
},
/**
* set this*=q, where q is Modulus, using Frobenius
*
* @this {ECP2}
*/
frob: function(X) {
var X2;
X2 = new ctx.FP2(X); //X2.copy(X);
X2.sqr();
this.x.conj();
this.y.conj();
this.z.conj();
this.z.reduce();
this.x.mul(X2);
this.y.mul(X2);
this.y.mul(X);
},
/**
* this+=this
*
* @this {ECP2}
*/
dbl: function() {
var iy, t0, t1, t2, x3, y3;
iy = new ctx.FP2(0);
iy.copy(this.y);
if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
iy.mul_ip();
iy.norm();
}
t0 = new ctx.FP2(0);
t0.copy(this.y);
t0.sqr();
if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
t0.mul_ip();
}
t1 = new ctx.FP2(0);
t1.copy(iy);
t1.mul(this.z);
t2 = new ctx.FP2(0);
t2.copy(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.mul_ip();
t2.norm();
}
x3 = new ctx.FP2(0);
x3.copy(t2);
x3.mul(this.z);
y3 = new ctx.FP2(0);
y3.copy(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 ECP2 instances
*
* param Q ECP2 instance
* @this {ECP2}
*/
add: function(Q) {
var b, t0, t1, t2, t3, t4, x3, y3, z3;
b = 3 * ctx.ROM_CURVE.CURVE_B_I;
t0 = new ctx.FP2(0);
t0.copy(this.x);
t0.mul(Q.x); // x.Q.x
t1 = new ctx.FP2(0);
t1.copy(this.y);
t1.mul(Q.y); // y.Q.y
t2 = new ctx.FP2(0);
t2.copy(this.z);
t2.mul(Q.z);
t3 = new ctx.FP2(0);
t3.copy(this.x);
t3.add(this.y);
t3.norm(); //t3=X1+Y1
t4 = new ctx.FP2(0);
t4.copy(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.mul_ip();
t3.norm(); //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.FP2(0);
x3.copy(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.mul_ip();
t4.norm(); //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.FP2(0);
y3.copy(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.mul_ip();
t0.norm(); // x.Q.x
t1.mul_ip();
t1.norm(); // y.Q.y
}
x3.copy(t0);
x3.add(t0);
t0.add(x3);
t0.norm();
t2.imul(b);
if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
t2.mul_ip(); t2.norm();
}
z3 = new ctx.FP2(0);
z3.copy(t1);
z3.add(t2);
z3.norm();
t1.sub(t2);
t1.norm();
y3.imul(b);
if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
y3.mul_ip();
y3.norm();
}
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 {ECP2}
* @param Q ECP2 instance
*/
sub: function(Q) {
var D;
var NQ=new ECP2(); NQ.copy(Q);
NQ.neg();
D = this.add(NQ);
return D;
},
/**
* Multiplies an ECP2 instance P by a BIG, side-channel resistant
*
* @this {ECP2}
* @param e BIG number multiplier
*/
mul: function(e) {
/* fixed size windows */
var mt = new ctx.BIG(),
t = new ctx.BIG(),
C = new ECP2(),
P = new ECP2(),
Q = new ECP2(),
W = [],
w = [],
i, nb, s, ns;
if (this.is_infinity()) {
return new ECP2();
}
// precompute table
Q.copy(this);
Q.dbl();
W[0] = new ECP2();
W[0].copy(this);
for (i = 1; i < 8; i++) {
W[i] = new ECP2();
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 {ECP2}
*/
ECP2.generator = function() {
var G=new ECP2(),
A = new ctx.BIG(0),
B = new ctx.BIG(0),
QX, QY;
A.rcopy(ctx.ROM_CURVE.CURVE_Pxa);
B.rcopy(ctx.ROM_CURVE.CURVE_Pxb);
QX = new ctx.FP2(0);
QX.bset(A, B);
A.rcopy(ctx.ROM_CURVE.CURVE_Pya);
B.rcopy(ctx.ROM_CURVE.CURVE_Pyb);
QY = new ctx.FP2(0);
QY.bset(A, B);
G.setxy(QX, QY);
return G;
};
/**
* convert from byte array to point
*
* @this {ECP2}
* @param b input byte array
*/
ECP2.fromBytes = function(b) {
var t = [],
ra, rb, 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);
rx = new ctx.FP2(ra, rb); //rx.bset(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);
ry = new ctx.FP2(ra, rb); //ry.bset(ra,rb);
P = new ECP2();
P.setxy(rx, ry);
return P;
};
/**
* Calculate RHS of curve equation x^3+B
*
* @this {ECP2}
* @param x x-value
*/
ECP2.RHS = function(x) {
var r, c, b;
//x.norm();
r = new ctx.FP2(x); //r.copy(x);
r.sqr();
c = new ctx.BIG(0);
c.rcopy(ctx.ROM_CURVE.CURVE_B);
b = new ctx.FP2(c); //b.bseta(c);
if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.D_TYPE) {
b.div_ip();
}
if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
b.norm();
b.mul_ip();
b.norm();
}
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 {ECP2}
*/
ECP2.mul4 = function(Q, u) {
var W = new ECP2(),
P = new ECP2(),
T = [],
mt = new ctx.BIG(),
t = [],
w = [],
s = [],
i, j, k, nb, bt, pb;
for (i = 0; i < 4; i++) {
t[i] = new ctx.BIG(u[i]); t[i].norm();
//Q[i].affine();
}
T[0] = new ECP2(); T[0].copy(Q[0]); // Q[0]
T[1] = new ECP2(); T[1].copy(T[0]); T[1].add(Q[1]); // Q[0]+Q[1]
T[2] = new ECP2(); T[2].copy(T[0]); T[2].add(Q[2]); // Q[0]+Q[2]
T[3] = new ECP2(); T[3].copy(T[1]); T[3].add(Q[2]); // Q[0]+Q[1]+Q[2]
T[4] = new ECP2(); T[4].copy(T[0]); T[4].add(Q[3]); // Q[0]+Q[3]
T[5] = new ECP2(); T[5].copy(T[1]); T[5].add(Q[3]); // Q[0]+Q[1]+Q[3]
T[6] = new ECP2(); T[6].copy(T[2]); T[6].add(Q[3]); // Q[0]+Q[2]+Q[3]
T[7] = new ECP2(); T[7].copy(T[3]); T[7].add(Q[3]); // Q[0]+Q[1]+Q[2]+Q[3]
// Make it odd
pb=1-t[0].parity();
t[0].inc(pb);
t[0].norm();
// Number of bits
mt.zero();
for (i=0;i<4;i++) {
mt.or(t[i]);
}
nb=1+mt.nbits();
// Sign pivot
s[nb-1]=1;
for (i=0;i<nb-1;i++) {
t[0].fshr(1);
s[i]=2*t[0].parity()-1;
}
// Recoded exponent
for (i=0; i<nb; i++) {
w[i]=0;
k=1;
for (j=1; j<4; j++) {
bt=s[i]*t[j].parity();
t[j].fshr(1);
t[j].dec(bt>>1);
t[j].norm();
w[i]+=bt*k;
k*=2;
}
}
// Main loop
P.select(T,2*w[nb-1]+1);
for (i=nb-2;i>=0;i--) {
P.dbl();
W.select(T,2*w[i]+s[i]);
P.add(W);
}
// apply correction
W.copy(P);
W.sub(Q[0]);
P.cmove(W,pb);
P.affine();
return P;
};
/* return 1 if b==c, no branching */
ECP2.teq = function(b, c) {
var x = b ^ c;
x -= 1; // if x=0, x now -1
return ((x >> 31) & 1);
};
/* needed for SOK */
ECP2.mapit = function(h) {
var fa = new ctx.BIG(0),
fb = new ctx.BIG(0),
q, x, one, Q, T, K, X, xQ, x2Q;
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 (;;) {
X = new ctx.FP2(one, x);
Q = new ECP2();
Q.setx(X);
if (!Q.is_infinity()) {
break;
}
x.inc(1);
x.norm();
}
/* Fast Hashing to G2 - Fuentes-Castaneda, Knapp and Rodriguez-Henriquez */
fa.rcopy(ctx.ROM_FIELD.Fra);
fb.rcopy(ctx.ROM_FIELD.Frb);
X = new ctx.FP2(fa, fb);
if (ctx.ECP.SEXTIC_TWIST == ctx.ECP.M_TYPE) {
X.inverse();
X.norm();
}
x = new ctx.BIG(0);
x.rcopy(ctx.ROM_CURVE.CURVE_Bnx);
if (ctx.ECP.CURVE_PAIRING_TYPE == ctx.ECP.BN) {
T = new ECP2();
T.copy(Q);
T = T.mul(x);
if (ctx.ECP.SIGN_OF_X == ctx.ECP.NEGATIVEX) {
T.neg();
}
K = new ECP2();
K.copy(T);
K.dbl();
K.add(T); //K.affine();
K.frob(X);
Q.frob(X);
Q.frob(X);
Q.frob(X);
Q.add(T);
Q.add(K);
T.frob(X);
T.frob(X);
Q.add(T);
}
if (ctx.ECP.CURVE_PAIRING_TYPE == ctx.ECP.BLS) {
xQ = Q.mul(x);
x2Q = xQ.mul(x);
if (ctx.ECP.SIGN_OF_X == ctx.ECP.NEGATIVEX) {
xQ.neg();
}
x2Q.sub(xQ);
x2Q.sub(Q);
xQ.sub(Q);
xQ.frob(X);
Q.dbl();
Q.frob(X);
Q.frob(X);
Q.add(x2Q);
Q.add(xQ);
}
Q.affine();
return Q;
};
return ECP2;
};