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pair.c
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pair.c
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/*
* Copyright (c) 2012-2020 MIRACL UK Ltd.
*
* This file is part of MIRACL Core
* (see https://github.com/miracl/core).
*
* Licensed 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.
*/
/* CORE BN Curve pairing functions */
//#define HAS_MAIN
#include "pair_ZZZ.h"
// Point doubling for pairings
static void PAIR_ZZZ_double(ECP2_ZZZ *A, FP2_YYY *AA, FP2_YYY *BB, FP2_YYY *CC)
{
FP2_YYY YY;
FP2_YYY_copy(CC, &(A->x)); //FP2 XX=new FP2(A.getx()); //X
FP2_YYY_copy(&YY, &(A->y)); //FP2 YY=new FP2(A.gety()); //Y
FP2_YYY_copy(BB, &(A->z)); //FP2 ZZ=new FP2(A.getz()); //Z
FP2_YYY_copy(AA, &YY); //FP2 YZ=new FP2(YY); //Y
FP2_YYY_mul(AA, AA, BB); //YZ.mul(ZZ); //YZ
FP2_YYY_sqr(CC, CC); //XX.sqr(); //X^2
FP2_YYY_sqr(&YY, &YY); //YY.sqr(); //Y^2
FP2_YYY_sqr(BB, BB); //ZZ.sqr(); //Z^2
FP2_YYY_add(AA, AA, AA);
FP2_YYY_neg(AA, AA);
FP2_YYY_norm(AA); // -2YZ
FP2_YYY_mul_ip(AA);
FP2_YYY_norm(AA); // -2YZi
FP2_YYY_imul(BB, BB, 3 * CURVE_B_I_ZZZ); //3Bz^2
FP2_YYY_imul(CC, CC, 3); //3X^2
#if SEXTIC_TWIST_ZZZ==D_TYPE
FP2_YYY_mul_ip(&YY); // Y^2.i
FP2_YYY_norm(&YY);
FP2_YYY_mul_ip(CC); // 3X^2.i
FP2_YYY_norm(CC);
#endif
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP2_YYY_mul_ip(BB); // 3Bz^2.i
FP2_YYY_norm(BB);
#endif
FP2_YYY_sub(BB, BB, &YY);
FP2_YYY_norm(BB);
ECP2_ZZZ_dbl(A); //A.dbl();
}
// Point addition for pairings
static void PAIR_ZZZ_add(ECP2_ZZZ *A, ECP2_ZZZ *B, FP2_YYY *AA, FP2_YYY *BB, FP2_YYY *CC)
{
FP2_YYY T1;
FP2_YYY_copy(AA, &(A->x)); //FP2 X1=new FP2(A.getx()); // X1
FP2_YYY_copy(CC, &(A->y)); //FP2 Y1=new FP2(A.gety()); // Y1
FP2_YYY_copy(&T1, &(A->z)); //FP2 T1=new FP2(A.getz()); // Z1
FP2_YYY_copy(BB, &T1); //FP2 T2=new FP2(A.getz()); // Z1
FP2_YYY_mul(&T1, &T1, &(B->y)); //T1.mul(B.gety()); // T1=Z1.Y2
FP2_YYY_mul(BB, BB, &(B->x)); //T2.mul(B.getx()); // T2=Z1.X2
FP2_YYY_sub(AA, AA, BB); //X1.sub(T2);
FP2_YYY_norm(AA); //X1.norm(); // X1=X1-Z1.X2
FP2_YYY_sub(CC, CC, &T1); //Y1.sub(T1);
FP2_YYY_norm(CC); //Y1.norm(); // Y1=Y1-Z1.Y2
FP2_YYY_copy(&T1, AA); //T1.copy(X1); // T1=X1-Z1.X2
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP2_YYY_mul_ip(AA);
FP2_YYY_norm(AA);
#endif
FP2_YYY_mul(&T1, &T1, &(B->y)); //T1.mul(B.gety()); // T1=(X1-Z1.X2).Y2
FP2_YYY_copy(BB, CC); //T2.copy(Y1); // T2=Y1-Z1.Y2
FP2_YYY_mul(BB, BB, &(B->x)); //T2.mul(B.getx()); // T2=(Y1-Z1.Y2).X2
FP2_YYY_sub(BB, BB, &T1); //T2.sub(T1);
FP2_YYY_norm(BB); //T2.norm(); // T2=(Y1-Z1.Y2).X2 - (X1-Z1.X2).Y2
FP2_YYY_neg(CC, CC); //Y1.neg();
FP2_YYY_norm(CC); //Y1.norm(); // Y1=-(Y1-Z1.Y2).Xs - ***
ECP2_ZZZ_add(A, B); //A.add(B);
}
/* Line function */
static void PAIR_ZZZ_line(FP12_YYY *v, ECP2_ZZZ *A, ECP2_ZZZ *B, FP_YYY *Qx, FP_YYY *Qy)
{
FP2_YYY AA, BB, CC;
FP4_YYY a, b, c;
if (A == B)
PAIR_ZZZ_double(A, &AA, &BB, &CC);
else
PAIR_ZZZ_add(A, B, &AA, &BB, &CC);
FP2_YYY_pmul(&CC, &CC, Qx);
FP2_YYY_pmul(&AA, &AA, Qy);
FP4_YYY_from_FP2s(&a, &AA, &BB);
#if SEXTIC_TWIST_ZZZ==D_TYPE
FP4_YYY_from_FP2(&b, &CC);
FP4_YYY_zero(&c);
#endif
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP4_YYY_zero(&b);
FP4_YYY_from_FP2H(&c, &CC);
#endif
FP12_YYY_from_FP4s(v, &a, &b, &c);
v->type = FP_SPARSER;
}
/* prepare ate parameter, n=6u+2 (BN) or n=u (BLS), n3=3*n */
int PAIR_ZZZ_nbits(BIG_XXX n3, BIG_XXX n)
{
BIG_XXX x;
BIG_XXX_rcopy(x, CURVE_Bnx_ZZZ);
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
BIG_XXX_pmul(n, x, 6);
#if SIGN_OF_X_ZZZ==POSITIVEX
BIG_XXX_inc(n, 2);
#else
BIG_XXX_dec(n, 2);
#endif
#else
BIG_XXX_copy(n, x);
#endif
BIG_XXX_norm(n);
BIG_XXX_pmul(n3, n, 3);
BIG_XXX_norm(n3);
return BIG_XXX_nbits(n3);
}
/*
For multi-pairing, product of n pairings
1. Declare FP12 array of length number of bits in Ate parameter
2. Initialise this array by calling PAIR_initmp()
3. Accumulate each pairing by calling PAIR_another() n times
4. Call PAIR_miller()
5. Call final exponentiation PAIR_fexp()
*/
/* prepare for multi-pairing */
void PAIR_ZZZ_initmp(FP12_YYY r[])
{
int i;
for (i = ATE_BITS_ZZZ - 1; i >= 0; i--)
FP12_YYY_one(&r[i]);
return;
}
// Store precomputed line details in an FP4
static void PAIR_ZZZ_pack(FP4_YYY *T, FP2_YYY* AA, FP2_YYY* BB, FP2_YYY *CC)
{
FP2_YYY I, A, B;
FP2_YYY_inv(&I, CC);
FP2_YYY_mul(&A, AA, &I);
FP2_YYY_mul(&B, BB, &I);
FP4_YYY_from_FP2s(T, &A, &B);
}
// Unpack G2 line function details and include G1
static void PAIR_ZZZ_unpack(FP12_YYY *v, FP4_YYY* T, FP_YYY *Qx, FP_YYY *Qy)
{
FP4_YYY a, b, c;
FP2_YYY t;
FP4_YYY_copy(&a, T);
FP2_YYY_pmul(&a.a, &a.a, Qy);
FP2_YYY_from_FP(&t, Qx);
#if SEXTIC_TWIST_ZZZ==D_TYPE
FP4_YYY_from_FP2(&b, &t);
FP4_YYY_zero(&c);
#endif
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP4_YYY_zero(&b);
FP4_YYY_from_FP2H(&c, &t);
#endif
FP12_YYY_from_FP4s(v, &a, &b, &c);
v->type = FP_SPARSEST;
}
/* basic Miller loop */
void PAIR_ZZZ_miller(FP12_YYY *res, FP12_YYY r[])
{
int i;
FP12_YYY_one(res);
for (i = ATE_BITS_ZZZ - 1; i >= 1; i--)
{
FP12_YYY_sqr(res, res);
FP12_YYY_ssmul(res, &r[i]);
FP12_YYY_zero(&r[i]);
}
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(res, res);
#endif
FP12_YYY_ssmul(res, &r[0]);
FP12_YYY_zero(&r[0]);
return;
}
// Precompute table of line functions for fixed G2 value
void PAIR_ZZZ_precomp(FP4_YYY T[], ECP2_ZZZ* GV)
{
int i, j, nb, bt;
BIG_XXX n, n3;
FP2_YYY AA, BB, CC;
ECP2_ZZZ A, G, NG;
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
ECP2_ZZZ K;
FP2_YYY X;
FP_YYY Qx, Qy;
FP_YYY_rcopy(&Qx, Fra_YYY);
FP_YYY_rcopy(&Qy, Frb_YYY);
FP2_YYY_from_FPs(&X, &Qx, &Qy);
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP2_YYY_inv(&X, &X);
FP2_YYY_norm(&X);
#endif
#endif
ECP2_ZZZ_copy(&A, GV);
ECP2_ZZZ_copy(&G, GV);
ECP2_ZZZ_copy(&NG, GV);
ECP2_ZZZ_neg(&NG);
nb = PAIR_ZZZ_nbits(n3, n);
j = 0;
for (i = nb - 2; i >= 1; i--)
{
PAIR_ZZZ_double(&A, &AA, &BB, &CC);
PAIR_ZZZ_pack(&T[j++], &AA, &BB, &CC);
bt = BIG_XXX_bit(n3, i) - BIG_XXX_bit(n, i); // bt=BIG_bit(n,i);
if (bt == 1)
{
PAIR_ZZZ_add(&A, &G, &AA, &BB, &CC);
PAIR_ZZZ_pack(&T[j++], &AA, &BB, &CC);
}
if (bt == -1)
{
PAIR_ZZZ_add(&A, &NG, &AA, &BB, &CC);
PAIR_ZZZ_pack(&T[j++], &AA, &BB, &CC);
}
}
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
#if SIGN_OF_X_ZZZ==NEGATIVEX
ECP2_ZZZ_neg(&A);
#endif
ECP2_ZZZ_copy(&K, &G);
ECP2_ZZZ_frob(&K, &X);
PAIR_ZZZ_add(&A, &K, &AA, &BB, &CC);
PAIR_ZZZ_pack(&T[j++], &AA, &BB, &CC);
ECP2_ZZZ_frob(&K, &X);
ECP2_ZZZ_neg(&K);
PAIR_ZZZ_add(&A, &K, &AA, &BB, &CC);
PAIR_ZZZ_pack(&T[j++], &AA, &BB, &CC);
#endif
}
/* Accumulate another set of line functions for n-pairing, assuming precomputation on G2 */
void PAIR_ZZZ_another_pc(FP12_YYY r[], FP4_YYY T[], ECP_ZZZ *QV)
{
int i, j, nb, bt;
BIG_XXX x, n, n3;
FP12_YYY lv, lv2;
ECP_ZZZ Q;
FP_YYY Qx, Qy;
if (ECP_ZZZ_isinf(QV)) return;
nb = PAIR_ZZZ_nbits(n3, n);
ECP_ZZZ_copy(&Q, QV);
ECP_ZZZ_affine(&Q);
FP_YYY_copy(&Qx, &(Q.x));
FP_YYY_copy(&Qy, &(Q.y));
j = 0;
for (i = nb - 2; i >= 1; i--)
{
PAIR_ZZZ_unpack(&lv, &T[j++], &Qx, &Qy);
bt = BIG_XXX_bit(n3, i) - BIG_XXX_bit(n, i); // bt=BIG_bit(n,i);
if (bt == 1)
{
PAIR_ZZZ_unpack(&lv2, &T[j++], &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
}
if (bt == -1)
{
PAIR_ZZZ_unpack(&lv2, &T[j++], &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
}
FP12_YYY_ssmul(&r[i], &lv);
}
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
PAIR_ZZZ_unpack(&lv, &T[j++], &Qx, &Qy);
PAIR_ZZZ_unpack(&lv2, &T[j++], &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(&r[0], &lv);
#endif
}
/* Accumulate another set of line functions for n-pairing */
void PAIR_ZZZ_another(FP12_YYY r[], ECP2_ZZZ* PV, ECP_ZZZ* QV)
{
int i, j, nb, bt;
BIG_XXX x, n, n3;
FP12_YYY lv, lv2;
ECP2_ZZZ A, NP, P;
ECP_ZZZ Q;
FP_YYY Qx, Qy;
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
ECP2_ZZZ K;
FP2_YYY X;
FP_YYY_rcopy(&Qx, Fra_YYY);
FP_YYY_rcopy(&Qy, Frb_YYY);
FP2_YYY_from_FPs(&X, &Qx, &Qy);
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP2_YYY_inv(&X, &X);
FP2_YYY_norm(&X);
#endif
#endif
if (ECP_ZZZ_isinf(QV)) return;
nb = PAIR_ZZZ_nbits(n3, n);
ECP2_ZZZ_copy(&P, PV);
ECP_ZZZ_copy(&Q, QV);
ECP2_ZZZ_affine(&P);
ECP_ZZZ_affine(&Q);
FP_YYY_copy(&Qx, &(Q.x));
FP_YYY_copy(&Qy, &(Q.y));
ECP2_ZZZ_copy(&A, &P);
ECP2_ZZZ_copy(&NP, &P); ECP2_ZZZ_neg(&NP);
for (i = nb - 2; i >= 1; i--)
{
PAIR_ZZZ_line(&lv, &A, &A, &Qx, &Qy);
bt = BIG_XXX_bit(n3, i) - BIG_XXX_bit(n, i); // bt=BIG_bit(n,i);
if (bt == 1)
{
PAIR_ZZZ_line(&lv2, &A, &P, &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
}
if (bt == -1)
{
PAIR_ZZZ_line(&lv2, &A, &NP, &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
}
FP12_YYY_ssmul(&r[i], &lv);
}
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
#if SIGN_OF_X_ZZZ==NEGATIVEX
ECP2_ZZZ_neg(&A);
#endif
ECP2_ZZZ_copy(&K, &P);
ECP2_ZZZ_frob(&K, &X);
PAIR_ZZZ_line(&lv, &A, &K, &Qx, &Qy);
ECP2_ZZZ_frob(&K, &X);
ECP2_ZZZ_neg(&K);
PAIR_ZZZ_line(&lv2, &A, &K, &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(&r[0], &lv);
#endif
}
/* Optimal R-ate pairing r=e(P,Q) */
void PAIR_ZZZ_ate(FP12_YYY *r, ECP2_ZZZ *P1, ECP_ZZZ *Q1)
{
BIG_XXX x, n, n3;
FP_YYY Qx, Qy;
int i, nb, bt;
ECP2_ZZZ A, NP, P;
ECP_ZZZ Q;
FP12_YYY lv, lv2;
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
ECP2_ZZZ KA;
FP2_YYY X;
FP_YYY_rcopy(&Qx, Fra_YYY);
FP_YYY_rcopy(&Qy, Frb_YYY);
FP2_YYY_from_FPs(&X, &Qx, &Qy);
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP2_YYY_inv(&X, &X);
FP2_YYY_norm(&X);
#endif
#endif
FP12_YYY_one(r);
if (ECP_ZZZ_isinf(Q1)) return;
nb = PAIR_ZZZ_nbits(n3, n);
ECP2_ZZZ_copy(&P, P1);
ECP_ZZZ_copy(&Q, Q1);
ECP2_ZZZ_affine(&P);
ECP_ZZZ_affine(&Q);
FP_YYY_copy(&Qx, &(Q.x));
FP_YYY_copy(&Qy, &(Q.y));
ECP2_ZZZ_copy(&A, &P);
ECP2_ZZZ_copy(&NP, &P); ECP2_ZZZ_neg(&NP);
/* Main Miller Loop */
for (i = nb - 2; i >= 1; i--) //0
{
FP12_YYY_sqr(r, r);
PAIR_ZZZ_line(&lv, &A, &A, &Qx, &Qy);
bt = BIG_XXX_bit(n3, i) - BIG_XXX_bit(n, i); // bt=BIG_bit(n,i);
if (bt == 1)
{
PAIR_ZZZ_line(&lv2, &A, &P, &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
}
if (bt == -1)
{
PAIR_ZZZ_line(&lv2, &A, &NP, &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
}
FP12_YYY_ssmul(r, &lv);
}
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(r, r);
#endif
/* R-ate fixup required for BN curves */
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
#if SIGN_OF_X_ZZZ==NEGATIVEX
ECP2_ZZZ_neg(&A);
#endif
ECP2_ZZZ_copy(&KA, &P);
ECP2_ZZZ_frob(&KA, &X);
PAIR_ZZZ_line(&lv, &A, &KA, &Qx, &Qy);
ECP2_ZZZ_frob(&KA, &X);
ECP2_ZZZ_neg(&KA);
PAIR_ZZZ_line(&lv2, &A, &KA, &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(r, &lv);
#endif
}
/* Optimal R-ate double pairing e(P,Q).e(R,S) */
void PAIR_ZZZ_double_ate(FP12_YYY *r, ECP2_ZZZ *P1, ECP_ZZZ *Q1, ECP2_ZZZ *R1, ECP_ZZZ *S1)
{
BIG_XXX x, n, n3;
FP_YYY Qx, Qy, Sx, Sy;
int i, nb, bt;
ECP2_ZZZ A, B, NP, NR, P, R;
ECP_ZZZ Q, S;
FP12_YYY lv, lv2;
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
FP2_YYY X;
ECP2_ZZZ K;
FP_YYY_rcopy(&Qx, Fra_YYY);
FP_YYY_rcopy(&Qy, Frb_YYY);
FP2_YYY_from_FPs(&X, &Qx, &Qy);
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP2_YYY_inv(&X, &X);
FP2_YYY_norm(&X);
#endif
#endif
if (ECP_ZZZ_isinf(Q1))
{
PAIR_ZZZ_ate(r, R1, S1);
return;
}
if (ECP_ZZZ_isinf(S1))
{
PAIR_ZZZ_ate(r, P1, Q1);
return;
}
nb = PAIR_ZZZ_nbits(n3, n);
ECP2_ZZZ_copy(&P, P1);
ECP_ZZZ_copy(&Q, Q1);
ECP2_ZZZ_affine(&P);
ECP_ZZZ_affine(&Q);
ECP2_ZZZ_copy(&R, R1);
ECP_ZZZ_copy(&S, S1);
ECP2_ZZZ_affine(&R);
ECP_ZZZ_affine(&S);
FP_YYY_copy(&Qx, &(Q.x));
FP_YYY_copy(&Qy, &(Q.y));
FP_YYY_copy(&Sx, &(S.x));
FP_YYY_copy(&Sy, &(S.y));
ECP2_ZZZ_copy(&A, &P);
ECP2_ZZZ_copy(&B, &R);
ECP2_ZZZ_copy(&NP, &P); ECP2_ZZZ_neg(&NP);
ECP2_ZZZ_copy(&NR, &R); ECP2_ZZZ_neg(&NR);
FP12_YYY_one(r);
/* Main Miller Loop */
for (i = nb - 2; i >= 1; i--)
{
FP12_YYY_sqr(r, r);
PAIR_ZZZ_line(&lv, &A, &A, &Qx, &Qy);
PAIR_ZZZ_line(&lv2, &B, &B, &Sx, &Sy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(r, &lv);
bt = BIG_XXX_bit(n3, i) - BIG_XXX_bit(n, i); // bt=BIG_bit(n,i);
if (bt == 1)
{
PAIR_ZZZ_line(&lv, &A, &P, &Qx, &Qy);
PAIR_ZZZ_line(&lv2, &B, &R, &Sx, &Sy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(r, &lv);
}
if (bt == -1)
{
PAIR_ZZZ_line(&lv, &A, &NP, &Qx, &Qy);
PAIR_ZZZ_line(&lv2, &B, &NR, &Sx, &Sy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(r, &lv);
}
}
/* R-ate fixup required for BN curves */
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(r, r);
#endif
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
#if SIGN_OF_X_ZZZ==NEGATIVEX
ECP2_ZZZ_neg(&A);
ECP2_ZZZ_neg(&B);
#endif
ECP2_ZZZ_copy(&K, &P);
ECP2_ZZZ_frob(&K, &X);
PAIR_ZZZ_line(&lv, &A, &K, &Qx, &Qy);
ECP2_ZZZ_frob(&K, &X);
ECP2_ZZZ_neg(&K);
PAIR_ZZZ_line(&lv2, &A, &K, &Qx, &Qy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(r, &lv);
ECP2_ZZZ_copy(&K, &R);
ECP2_ZZZ_frob(&K, &X);
PAIR_ZZZ_line(&lv, &B, &K, &Sx, &Sy);
ECP2_ZZZ_frob(&K, &X);
ECP2_ZZZ_neg(&K);
PAIR_ZZZ_line(&lv2, &B, &K, &Sx, &Sy);
FP12_YYY_smul(&lv, &lv2);
FP12_YYY_ssmul(r, &lv);
#endif
}
/* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */
void PAIR_ZZZ_fexp(FP12_YYY *r)
{
FP2_YYY X;
BIG_XXX x;
FP_YYY a, b;
FP12_YYY t0, y0, y1, y2, y3;
BIG_XXX_rcopy(x, CURVE_Bnx_ZZZ);
FP_YYY_rcopy(&a, Fra_YYY);
FP_YYY_rcopy(&b, Frb_YYY);
FP2_YYY_from_FPs(&X, &a, &b);
/* Easy part of final exp */
FP12_YYY_inv(&t0, r);
FP12_YYY_conj(r, r);
FP12_YYY_mul(r, &t0);
FP12_YYY_copy(&t0, r);
FP12_YYY_frob(r, &X);
FP12_YYY_frob(r, &X);
FP12_YYY_mul(r, &t0);
/* Hard part of final exp - see Duquesne & Ghamman eprint 2015/192.pdf */
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
FP12_YYY_pow(&t0, r, x); // t0=f^-u
#if SIGN_OF_X_ZZZ==POSITIVEX
FP12_YYY_conj(&t0, &t0);
#endif
FP12_YYY_usqr(&y3, &t0); // y3=t0^2
FP12_YYY_copy(&y0, &t0);
FP12_YYY_mul(&y0, &y3); // y0=t0*y3
FP12_YYY_copy(&y2, &y3);
FP12_YYY_frob(&y2, &X); // y2=y3^p
FP12_YYY_mul(&y2, &y3); //y2=y2*y3
FP12_YYY_usqr(&y2, &y2); //y2=y2^2
FP12_YYY_mul(&y2, &y3); // y2=y2*y3
FP12_YYY_pow(&t0, &y0, x); //t0=y0^-u
#if SIGN_OF_X_ZZZ==POSITIVEX
FP12_YYY_conj(&t0, &t0);
#endif
FP12_YYY_conj(&y0, r); //y0=~r
FP12_YYY_copy(&y1, &t0);
FP12_YYY_frob(&y1, &X);
FP12_YYY_frob(&y1, &X); //y1=t0^p^2
FP12_YYY_mul(&y1, &y0); // y1=y0*y1
FP12_YYY_conj(&t0, &t0); // t0=~t0
FP12_YYY_copy(&y3, &t0);
FP12_YYY_frob(&y3, &X); //y3=t0^p
FP12_YYY_mul(&y3, &t0); // y3=t0*y3
FP12_YYY_usqr(&t0, &t0); // t0=t0^2
FP12_YYY_mul(&y1, &t0); // y1=t0*y1
FP12_YYY_pow(&t0, &y3, x); // t0=y3^-u
#if SIGN_OF_X_ZZZ==POSITIVEX
FP12_YYY_conj(&t0, &t0);
#endif
FP12_YYY_usqr(&t0, &t0); //t0=t0^2
FP12_YYY_conj(&t0, &t0); //t0=~t0
FP12_YYY_mul(&y3, &t0); // y3=t0*y3
FP12_YYY_frob(r, &X);
FP12_YYY_copy(&y0, r);
FP12_YYY_frob(r, &X);
FP12_YYY_mul(&y0, r);
FP12_YYY_frob(r, &X);
FP12_YYY_mul(&y0, r);
FP12_YYY_usqr(r, &y3); //r=y3^2
FP12_YYY_mul(r, &y2); //r=y2*r
FP12_YYY_copy(&y3, r);
FP12_YYY_mul(&y3, &y0); // y3=r*y0
FP12_YYY_mul(r, &y1); // r=r*y1
FP12_YYY_usqr(r, r); // r=r^2
FP12_YYY_mul(r, &y3); // r=r*y3
FP12_YYY_reduce(r);
#else
// Ghamman & Fouotsa Method
FP12_YYY_usqr(&y0, r);
FP12_YYY_pow(&y1, &y0, x);
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(&y1, &y1);
#endif
BIG_XXX_fshr(x, 1);
FP12_YYY_pow(&y2, &y1, x);
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(&y2, &y2);
#endif
BIG_XXX_fshl(x, 1); // x must be even
FP12_YYY_conj(&y3, r);
FP12_YYY_mul(&y1, &y3);
FP12_YYY_conj(&y1, &y1);
FP12_YYY_mul(&y1, &y2);
FP12_YYY_pow(&y2, &y1, x);
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(&y2, &y2);
#endif
FP12_YYY_pow(&y3, &y2, x);
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(&y3, &y3);
#endif
FP12_YYY_conj(&y1, &y1);
FP12_YYY_mul(&y3, &y1);
FP12_YYY_conj(&y1, &y1);
FP12_YYY_frob(&y1, &X);
FP12_YYY_frob(&y1, &X);
FP12_YYY_frob(&y1, &X);
FP12_YYY_frob(&y2, &X);
FP12_YYY_frob(&y2, &X);
FP12_YYY_mul(&y1, &y2);
FP12_YYY_pow(&y2, &y3, x);
#if SIGN_OF_X_ZZZ==NEGATIVEX
FP12_YYY_conj(&y2, &y2);
#endif
FP12_YYY_mul(&y2, &y0);
FP12_YYY_mul(&y2, r);
FP12_YYY_mul(&y1, &y2);
FP12_YYY_copy(&y2, &y3);
FP12_YYY_frob(&y2, &X);
FP12_YYY_mul(&y1, &y2);
FP12_YYY_copy(r, &y1);
FP12_YYY_reduce(r);
#endif
}
#ifdef USE_GLV_ZZZ
/* GLV method */
static void glv(BIG_XXX u[2], BIG_XXX e)
{
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
int i, j;
BIG_XXX v[2], t, q;
DBIG_XXX d;
BIG_XXX_rcopy(q, CURVE_Order_ZZZ);
for (i = 0; i < 2; i++)
{
BIG_XXX_rcopy(t, CURVE_W_ZZZ[i]);
BIG_XXX_mul(d, t, e);
BIG_XXX_ddiv(v[i], d, q);
BIG_XXX_zero(u[i]);
}
BIG_XXX_copy(u[0], e);
for (i = 0; i < 2; i++)
for (j = 0; j < 2; j++)
{
BIG_XXX_rcopy(t, CURVE_SB_ZZZ[j][i]);
BIG_XXX_modmul(t, v[j], t, q);
BIG_XXX_add(u[i], u[i], q);
BIG_XXX_sub(u[i], u[i], t);
BIG_XXX_mod(u[i], q);
}
#else
// -(x^2).P = (Beta.x,y)
BIG_XXX x, x2, q;
BIG_XXX_rcopy(x, CURVE_Bnx_ZZZ);
BIG_XXX_smul(x2, x, x);
BIG_XXX_copy(u[0], e);
BIG_XXX_mod(u[0], x2);
BIG_XXX_copy(u[1], e);
BIG_XXX_sdiv(u[1], x2);
BIG_XXX_rcopy(q, CURVE_Order_ZZZ);
BIG_XXX_sub(u[1], q, u[1]);
#endif
return;
}
#endif // USE_GLV
/* Galbraith & Scott Method */
static void gs(BIG_XXX u[4], BIG_XXX e)
{
int i;
#if PAIRING_FRIENDLY_ZZZ==BN_CURVE
int j;
BIG_XXX v[4], t, q;
DBIG_XXX d;
BIG_XXX_rcopy(q, CURVE_Order_ZZZ);
for (i = 0; i < 4; i++)
{
BIG_XXX_rcopy(t, CURVE_WB_ZZZ[i]);
BIG_XXX_mul(d, t, e);
BIG_XXX_ddiv(v[i], d, q);
BIG_XXX_zero(u[i]);
}
BIG_XXX_copy(u[0], e);
for (i = 0; i < 4; i++)
for (j = 0; j < 4; j++)
{
BIG_XXX_rcopy(t, CURVE_BB_ZZZ[j][i]);
BIG_XXX_modmul(t, v[j], t, q);
BIG_XXX_add(u[i], u[i], q);
BIG_XXX_sub(u[i], u[i], t);
BIG_XXX_mod(u[i], q);
}
#else
BIG_XXX x, w, q;
BIG_XXX_rcopy(q, CURVE_Order_ZZZ);
BIG_XXX_rcopy(x, CURVE_Bnx_ZZZ);
BIG_XXX_copy(w, e);
for (i = 0; i < 3; i++)
{
BIG_XXX_copy(u[i], w);
BIG_XXX_mod(u[i], x);
BIG_XXX_sdiv(w, x);
}
BIG_XXX_copy(u[3], w);
/* */
#if SIGN_OF_X_ZZZ==NEGATIVEX
BIG_XXX_modneg(u[1], u[1], q);
BIG_XXX_modneg(u[3], u[3], q);
#endif
#endif
return;
}
/* Multiply P by e in group G1 */
void PAIR_ZZZ_G1mul(ECP_ZZZ *P, BIG_XXX e)
{
#ifdef USE_GLV_ZZZ /* Note this method is patented */
int np, nn;
ECP_ZZZ Q;
FP_YYY cru;
BIG_XXX t, q;
BIG_XXX u[2];
BIG_XXX_rcopy(q, CURVE_Order_ZZZ);
glv(u, e);
ECP_ZZZ_copy(&Q, P); ECP_ZZZ_affine(&Q);
FP_YYY_rcopy(&cru, CRu_YYY);
FP_YYY_mul(&(Q.x), &(Q.x), &cru);
/* note that -a.B = a.(-B). Use a or -a depending on which is smaller */
np = BIG_XXX_nbits(u[0]);
BIG_XXX_modneg(t, u[0], q);
nn = BIG_XXX_nbits(t);
if (nn < np)
{
BIG_XXX_copy(u[0], t);
ECP_ZZZ_neg(P);
}
np = BIG_XXX_nbits(u[1]);
BIG_XXX_modneg(t, u[1], q);
nn = BIG_XXX_nbits(t);
if (nn < np)
{
BIG_XXX_copy(u[1], t);
ECP_ZZZ_neg(&Q);
}
BIG_XXX_norm(u[0]);
BIG_XXX_norm(u[1]);
ECP_ZZZ_mul2(P, &Q, u[0], u[1]);
#else
ECP_ZZZ_mul(P, e);
#endif
}
/* Multiply P by e in group G2 */
void PAIR_ZZZ_G2mul(ECP2_ZZZ *P, BIG_XXX e)
{
#ifdef USE_GS_G2_ZZZ /* Well I didn't patent it :) */
int i, np, nn;
ECP2_ZZZ Q[4];
FP2_YYY X;
FP_YYY fx, fy;
BIG_XXX x, y, u[4];
FP_YYY_rcopy(&fx, Fra_YYY);
FP_YYY_rcopy(&fy, Frb_YYY);
FP2_YYY_from_FPs(&X, &fx, &fy);
#if SEXTIC_TWIST_ZZZ==M_TYPE
FP2_YYY_inv(&X, &X);
FP2_YYY_norm(&X);
#endif
BIG_XXX_rcopy(y, CURVE_Order_ZZZ);
gs(u, e);
ECP2_ZZZ_copy(&Q[0], P);
for (i = 1; i < 4; i++)
{
ECP2_ZZZ_copy(&Q[i], &Q[i - 1]);
ECP2_ZZZ_frob(&Q[i], &X);
}
for (i = 0; i < 4; i++)
{
np = BIG_XXX_nbits(u[i]);
BIG_XXX_modneg(x, u[i], y);
nn = BIG_XXX_nbits(x);
if (nn < np)
{
BIG_XXX_copy(u[i], x);
ECP2_ZZZ_neg(&Q[i]);
}
BIG_XXX_norm(u[i]);
}
ECP2_ZZZ_mul4(P, Q, u);
#else
ECP2_ZZZ_mul(P, e);
#endif
}
/* f=f^e */
void PAIR_ZZZ_GTpow(FP12_YYY *f, BIG_XXX e)
{
#ifdef USE_GS_GT_ZZZ /* Note that this option requires a lot of RAM! Maybe better to use compressed XTR method, see fp4.c */
int i, np, nn;
FP12_YYY g[4];
FP2_YYY X;
BIG_XXX t, q;
FP_YYY fx, fy;
BIG_XXX u[4];
FP_YYY_rcopy(&fx, Fra_YYY);
FP_YYY_rcopy(&fy, Frb_YYY);
FP2_YYY_from_FPs(&X, &fx, &fy);
BIG_XXX_rcopy(q, CURVE_Order_ZZZ);
gs(u, e);
FP12_YYY_copy(&g[0], f);
for (i = 1; i < 4; i++)
{
FP12_YYY_copy(&g[i], &g[i - 1]);
FP12_YYY_frob(&g[i], &X);
}
for (i = 0; i < 4; i++)
{
np = BIG_XXX_nbits(u[i]);
BIG_XXX_modneg(t, u[i], q);
nn = BIG_XXX_nbits(t);
if (nn < np)
{
BIG_XXX_copy(u[i], t);
FP12_YYY_conj(&g[i], &g[i]);
}
BIG_XXX_norm(u[i]);
}
FP12_YYY_pow4(f, g, u);
#else
FP12_YYY_pow(f, f, e);
#endif
}