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/*
Copyright 2018 0kims association
This file is part of zksnark javascript library.
zksnark javascript library is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
zksnark javascript library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with zksnark javascript library. If not, see <https://www.gnu.org/licenses/>.
*/
const bigInt = require("./bigint.js");
const assert = require("assert");
const F1Field = require("./zqfield.js");
const F2Field = require("./f2field.js");
const F3Field = require("./f3field.js");
const GCurve = require("./gcurve.js");
class BN128 {
constructor() {
this.q = bigInt("21888242871839275222246405745257275088696311157297823662689037894645226208583");
this.r = bigInt("21888242871839275222246405745257275088548364400416034343698204186575808495617");
this.g1 = [ bigInt(1), bigInt(2) ];
this.g2 = [
[
bigInt("10857046999023057135944570762232829481370756359578518086990519993285655852781"),
bigInt("11559732032986387107991004021392285783925812861821192530917403151452391805634")
],
[
bigInt("8495653923123431417604973247489272438418190587263600148770280649306958101930"),
bigInt("4082367875863433681332203403145435568316851327593401208105741076214120093531")
]
];
this.nonResidueF2 = bigInt("21888242871839275222246405745257275088696311157297823662689037894645226208582");
this.nonResidueF6 = [ bigInt("9"), bigInt("1") ];
this.F1 = new F1Field(this.q);
this.F2 = new F2Field(this.F1, this.nonResidueF2);
this.G1 = new GCurve(this.F1, this.g1);
this.G2 = new GCurve(this.F2, this.g2);
this.F6 = new F3Field(this.F2, this.nonResidueF6);
this.F12 = new F2Field(this.F6, this.nonResidueF6);
const self = this;
this.F12._mulByNonResidue = function(a) {
return [self.F2.mul(this.nonResidue, a[2]), a[0], a[1]];
};
this._preparePairing();
}
_preparePairing() {
this.loopCount = bigInt("29793968203157093288");// CONSTANT
// Set loopCountNeg
if (this.loopCount.isNegative()) {
this.loopCount = this.neg();
this.loopCountNeg = true;
} else {
this.loopCountNeg = false;
}
// Set loop_count_bits
let lc = this.loopCount;
this.loop_count_bits = []; // Constant
while (!lc.isZero()) {
this.loop_count_bits.push( lc.isOdd() );
lc = lc.shr(1);
}
this.two_inv = this.F1.inverse(bigInt(2));
this.coef_b = bigInt(3);
this.twist = [bigInt(9) , bigInt(1)];
this.twist_coeff_b = this.F2.mulScalar( this.F2.inverse(this.twist), this.coef_b );
this.frobenius_coeffs_c1_1 = bigInt("21888242871839275222246405745257275088696311157297823662689037894645226208582");
this.twist_mul_by_q_X =
[
bigInt("21575463638280843010398324269430826099269044274347216827212613867836435027261"),
bigInt("10307601595873709700152284273816112264069230130616436755625194854815875713954")
];
this.twist_mul_by_q_Y =
[
bigInt("2821565182194536844548159561693502659359617185244120367078079554186484126554"),
bigInt("3505843767911556378687030309984248845540243509899259641013678093033130930403")
];
this.final_exponent = bigInt("552484233613224096312617126783173147097382103762957654188882734314196910839907541213974502761540629817009608548654680343627701153829446747810907373256841551006201639677726139946029199968412598804882391702273019083653272047566316584365559776493027495458238373902875937659943504873220554161550525926302303331747463515644711876653177129578303191095900909191624817826566688241804408081892785725967931714097716709526092261278071952560171111444072049229123565057483750161460024353346284167282452756217662335528813519139808291170539072125381230815729071544861602750936964829313608137325426383735122175229541155376346436093930287402089517426973178917569713384748081827255472576937471496195752727188261435633271238710131736096299798168852925540549342330775279877006784354801422249722573783561685179618816480037695005515426162362431072245638324744480");
}
pairing(p1, p2) {
const pre1 = this.precomputeG1(p1);
const pre2 = this.precomputeG2(p2);
const r1 = this.millerLoop(pre1, pre2);
const res = this.finalExponentiation(r1);
return res;
}
precomputeG1(p) {
const Pcopy = this.G1.affine(p);
const res = {};
res.PX = Pcopy[0];
res.PY = Pcopy[1];
return res;
}
precomputeG2(p) {
const Qcopy = this.G2.affine(p);
const res = {
QX: Qcopy[0],
QY: Qcopy[1],
coeffs: []
};
const R = {
X: Qcopy[0],
Y: Qcopy[1],
Z: this.F2.one
};
let c;
for (let i = this.loop_count_bits.length-2; i >= 0; --i)
{
const bit = this.loop_count_bits[i];
c = this._doubleStep(R);
res.coeffs.push(c);
if (bit)
{
c = this._addStep(Qcopy, R);
res.coeffs.push(c);
}
}
const Q1 = this.G2.affine(this._g2MulByQ(Qcopy));
assert(this.F2.equals(Q1[2], this.F2.one));
const Q2 = this.G2.affine(this._g2MulByQ(Q1));
assert(this.F2.equals(Q2[2], this.F2.one));
if (this.loopCountNef)
{
R.Y = this.F2.neg(R.Y);
}
Q2[1] = this.F2.neg(Q2[1]);
c = this._addStep(Q1, R);
res.coeffs.push(c);
c = this._addStep(Q2, R);
res.coeffs.push(c);
return res;
}
millerLoop(pre1, pre2) {
let f = this.F12.one;
let idx = 0;
let c;
for (let i = this.loop_count_bits.length-2; i >= 0; --i)
{
const bit = this.loop_count_bits[i];
/* code below gets executed for all bits (EXCEPT the MSB itself) of
alt_bn128_param_p (skipping leading zeros) in MSB to LSB
order */
c = pre2.coeffs[idx++];
f = this.F12.square(f);
f = this._mul_by_024(
f,
c.ell_0,
this.F2.mulScalar(c.ell_VW , pre1.PY),
this.F2.mulScalar(c.ell_VV , pre1.PX, ));
if (bit)
{
c = pre2.coeffs[idx++];
f = this._mul_by_024(
f,
c.ell_0,
this.F2.mulScalar(c.ell_VW, pre1.PY, ),
this.F2.mulScalar(c.ell_VV, pre1.PX, ));
}
}
if (this.loopCountNef)
{
f = this.F12.inverse(f);
}
c = pre2.coeffs[idx++];
f = this._mul_by_024(
f,
c.ell_0,
this.F2.mulScalar(c.ell_VW, pre1.PY),
this.F2.mulScalar(c.ell_VV, pre1.PX));
c = pre2.coeffs[idx++];
f = this._mul_by_024(
f,
c.ell_0,
this.F2.mulScalar(c.ell_VW, pre1.PY, ),
this.F2.mulScalar(c.ell_VV, pre1.PX));
return f;
}
finalExponentiation(elt) {
// TODO: There is an optimization in FF
const res = this.F12.exp(elt,this.final_exponent);
return res;
}
_doubleStep(current) {
const X = current.X;
const Y = current.Y;
const Z = current.Z;
const A = this.F2.mulScalar(this.F2.mul(X,Y), this.two_inv); // A = X1 * Y1 / 2
const B = this.F2.square(Y); // B = Y1^2
const C = this.F2.square(Z); // C = Z1^2
const D = this.F2.add(C, this.F2.add(C,C)); // D = 3 * C
const E = this.F2.mul(this.twist_coeff_b, D); // E = twist_b * D
const F = this.F2.add(E, this.F2.add(E,E)); // F = 3 * E
const G =
this.F2.mulScalar(
this.F2.add( B , F ),
this.two_inv); // G = (B+F)/2
const H =
this.F2.sub(
this.F2.square( this.F2.add(Y,Z) ),
this.F2.add( B , C)); // H = (Y1+Z1)^2-(B+C)
const I = this.F2.sub(E, B); // I = E-B
const J = this.F2.square(X); // J = X1^2
const E_squared = this.F2.square(E); // E_squared = E^2
current.X = this.F2.mul( A, this.F2.sub(B,F) ); // X3 = A * (B-F)
current.Y =
this.F2.sub(
this.F2.sub( this.F2.square(G) , E_squared ),
this.F2.add( E_squared , E_squared )); // Y3 = G^2 - 3*E^2
current.Z = this.F2.mul( B, H ); // Z3 = B * H
const c = {
ell_0 : this.F2.mul( I, this.twist), // ell_0 = xi * I
ell_VW: this.F2.neg( H ), // ell_VW = - H (later: * yP)
ell_VV: this.F2.add( J , this.F2.add(J,J) ) // ell_VV = 3*J (later: * xP)
};
return c;
}
_addStep(base, current) {
const X1 = current.X;
const Y1 = current.Y;
const Z1 = current.Z;
const x2 = base[0];
const y2 = base[1];
const D = this.F2.sub( X1, this.F2.mul(x2,Z1) ); // D = X1 - X2*Z1
const E = this.F2.sub( Y1, this.F2.mul(y2,Z1) ); // E = Y1 - Y2*Z1
const F = this.F2.square(D); // F = D^2
const G = this.F2.square(E); // G = E^2
const H = this.F2.mul(D,F); // H = D*F
const I = this.F2.mul(X1,F); // I = X1 * F
const J =
this.F2.sub(
this.F2.add( H, this.F2.mul(Z1,G) ),
this.F2.add( I, I )); // J = H + Z1*G - (I+I)
current.X = this.F2.mul( D , J ); // X3 = D*J
current.Y =
this.F2.sub(
this.F2.mul( E , this.F2.sub(I,J) ),
this.F2.mul( H , Y1)); // Y3 = E*(I-J)-(H*Y1)
current.Z = this.F2.mul(Z1,H);
const c = {
ell_0 :
this.F2.mul(
this.twist,
this.F2.sub(
this.F2.mul(E , x2),
this.F2.mul(D , y2))), // ell_0 = xi * (E * X2 - D * Y2)
ell_VV : this.F2.neg(E), // ell_VV = - E (later: * xP)
ell_VW : D // ell_VW = D (later: * yP )
};
return c;
}
_mul_by_024(a, ell_0, ell_VW, ell_VV) {
// Old implementation
const b = [
[ell_0, this.F2.zero, ell_VV],
[this.F2.zero, ell_VW, this.F2.zero]
];
return this.F12.mul(a,b);
/*
// This is a new implementation,
// But it does not look worthy
// at least in javascript.
let z0 = a[0][0];
let z1 = a[0][1];
let z2 = a[0][2];
let z3 = a[1][0];
let z4 = a[1][1];
let z5 = a[1][2];
const x0 = ell_0;
const x2 = ell_VV;
const x4 = ell_VW;
const D0 = this.F2.mul(z0, x0);
const D2 = this.F2.mul(z2, x2);
const D4 = this.F2.mul(z4, x4);
const t2 = this.F2.add(z0, z4);
let t1 = this.F2.add(z0, z2);
const s0 = this.F2.add(this.F2.add(z1,z3),z5);
// For z.a_.a_ = z0.
let S1 = this.F2.mul(z1, x2);
let T3 = this.F2.add(S1, D4);
let T4 = this.F2.add( this.F2.mul(this.nonResidueF6, T3),D0);
z0 = T4;
// For z.a_.b_ = z1
T3 = this.F2.mul(z5, x4);
S1 = this.F2.add(S1, T3);
T3 = this.F2.add(T3, D2);
T4 = this.F2.mul(this.nonResidueF6, T3);
T3 = this.F2.mul(z1, x0);
S1 = this.F2.add(S1, T3);
T4 = this.F2.add(T4, T3);
z1 = T4;
// For z.a_.c_ = z2
let t0 = this.F2.add(x0, x2);
T3 = this.F2.sub(
this.F2.mul(t1, t0),
this.F2.add(D0, D2));
T4 = this.F2.mul(z3, x4);
S1 = this.F2.add(S1, T4);
T3 = this.F2.add(T3, T4);
// For z.b_.a_ = z3 (z3 needs z2)
t0 = this.F2.add(z2, z4);
z2 = T3;
t1 = this.F2.add(x2, x4);
T3 = this.F2.sub(
this.F2.mul(t0,t1),
this.F2.add(D2, D4));
T4 = this.F2.mul(this.nonResidueF6, T3);
T3 = this.F2.mul(z3, x0);
S1 = this.F2.add(S1, T3);
T4 = this.F2.add(T4, T3);
z3 = T4;
// For z.b_.b_ = z4
T3 = this.F2.mul(z5, x2);
S1 = this.F2.add(S1, T3);
T4 = this.F2.mul(this.nonResidueF6, T3);
t0 = this.F2.add(x0, x4);
T3 = this.F2.sub(
this.F2.mul(t2,t0),
this.F2.add(D0, D4));
T4 = this.F2.add(T4, T3);
z4 = T4;
// For z.b_.c_ = z5.
t0 = this.F2.add(this.F2.add(x0, x2), x4);
T3 = this.F2.sub(this.F2.mul(s0, t0), S1);
z5 = T3;
return [
[z0, z1, z2],
[z3, z4, z5]
];
*/
}
_g2MulByQ(p) {
const fmx = [p[0][0], this.F1.mul(p[0][1], this.frobenius_coeffs_c1_1 )];
const fmy = [p[1][0], this.F1.mul(p[1][1], this.frobenius_coeffs_c1_1 )];
const fmz = [p[2][0], this.F1.mul(p[2][1], this.frobenius_coeffs_c1_1 )];
return [
this.F2.mul(this.twist_mul_by_q_X , fmx),
this.F2.mul(this.twist_mul_by_q_Y , fmy),
fmz
];
}
}
module.exports = BN128;