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ec_isogeny.c
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/********************************************************************************************
* SIDH: an efficient supersingular isogeny-based cryptography library for Diffie-Hellman
* key exchange.
*
* Copyright (c) Microsoft Corporation. All rights reserved.
*
*
* Abstract: elliptic curve and isogeny functions
*
*********************************************************************************************
* Update: New functions for improving the operation P+[k]Q
*
* This is derived work authored by:
* Faz-Hernández, Lopez, Ochoa-Jiménez and Rodríquez-Henríquez.
*
* Contributions:
* - Right2Left Ladder replacing 3-point ladder of De Feo et al.
* - Precomputation algorithm using a Right2Left Ladder.
* - New point tripling formulas. Improves by 1M-1S-1A in Fp2
*
* Copyright (c) Sept,2017 by Armando Faz <armfazh@ic.unicamp.br>.
* Institute of Computing.
* University of Campinas, Brazil.
*********************************************************************************************/
#include "SIDH_internal.h"
#include "SIDH.h"
#include <math.h>
#include <stdio.h>
#ifndef __NO_R2L__
#include "look_up_tables.h"
#endif
extern const uint64_t LIST[22][NWORDS64_FIELD];
void j_inv(const f2elm_t A, const f2elm_t C, f2elm_t jinv)
{ // Computes the j-invariant of a Montgomery curve with projective constant.
// Input: A,C in GF(p^2).
// Output: j=256*(A^2-3*C^2)^3/(C^4*(A^2-4*C^2)), which is the j-invariant of the Montgomery curve B*y^2=x^3+(A/C)*x^2+x or (equivalently) j-invariant of B'*y^2=C*x^3+A*x^2+C*x.
f2elm_t t0, t1;
fp2sqr751_mont(A, jinv); // jinv = A^2
fp2sqr751_mont(C, t1); // t1 = C^2
fp2add751(t1, t1, t0); // t0 = t1+t1
fp2sub751(jinv, t0, t0); // t0 = jinv-t0
fp2sub751(t0, t1, t0); // t0 = t0-t1
fp2sub751(t0, t1, jinv); // jinv = t0-t1
fp2sqr751_mont(t1, t1); // t1 = t1^2
fp2mul751_mont(jinv, t1, jinv); // jinv = jinv*t1
fp2add751(t0, t0, t0); // t0 = t0+t0
fp2add751(t0, t0, t0); // t0 = t0+t0
fp2sqr751_mont(t0, t1); // t1 = t0^2
fp2mul751_mont(t0, t1, t0); // t0 = t0*t1
fp2add751(t0, t0, t0); // t0 = t0+t0
fp2add751(t0, t0, t0); // t0 = t0+t0
fp2inv751_mont(jinv); // jinv = 1/jinv
fp2mul751_mont(jinv, t0, jinv); // jinv = t0*jinv
}
void xDBLADD_full(point_proj_t P, point_proj_t Q, const point_proj_t PQ, const f2elm_t A24)
{
// Simultaneous doubling and differential addition.
// Input:
// projective Montgomery points P=(XP:ZP) and Q=(XQ:ZQ) such that xP=XP/ZP and xQ=XQ/ZQ,
// projective difference xPQ=x(P-Q)/z(P-Q)
// and Montgomery curve constant A24=(A+2)/4.
// Output:
// projective Montgomery points P <- 2*P = (X2P:Z2P) such that x(2P)=X2P/Z2P, and Q <- P+Q = (XQP:ZQP) such that = x(Q+P)=XQP/ZQP.
f2elm_t t0, t1, t2;
fp2add751(P->X, P->Z, t0); // t0 = XP+ZP
fp2sub751(P->X, P->Z, t1); // t1 = XP-ZP
fp2sqr751_mont(t0, P->X); // XP = (XP+ZP)^2
fp2sub751(Q->X, Q->Z, t2); // t2 = XQ-ZQ
fp2add751(Q->X, Q->Z, Q->X); // XQ = XQ+ZQ
fp2mul751_mont(t0, t2, t0); // t0 = (XP+ZP)*(XQ-ZQ)
fp2sqr751_mont(t1, P->Z); // ZP = (XP-ZP)^2
fp2mul751_mont(t1, Q->X, t1); // t1 = (XP-ZP)*(XQ+ZQ)
fp2sub751(P->X, P->Z, t2); // t2 = (XP+ZP)^2-(XP-ZP)^2
fp2mul751_mont(P->X, P->Z, P->X); // XP = (XP+ZP)^2*(XP-ZP)^2
fp2mul751_mont(t2, A24, Q->X); // XQ = A24*[(XP+ZP)^2-(XP-ZP)^2]
fp2sub751(t0, t1, Q->Z); // ZQ = (XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)
fp2add751(Q->X, P->Z, P->Z); // ZP = A24*[(XP+ZP)^2-(XP-ZP)^2]+(XP-ZP)^2
fp2add751(t0, t1, Q->X); // XQ = (XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)
fp2mul751_mont(P->Z, t2, P->Z); // ZP = [A24*[(XP+ZP)^2-(XP-ZP)^2]+(XP-ZP)^2]*[(XP+ZP)^2-(XP-ZP)^2]
fp2sqr751_mont(Q->Z, Q->Z); // ZQ = [(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
fp2sqr751_mont(Q->X, Q->X); // XQ = [(XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)]^2
fp2mul751_mont(Q->X, PQ->Z, Q->X); // XQ = zPQ*[(XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)]^2
fp2mul751_mont(Q->Z, PQ->X, Q->Z); // ZQ = xPQ*[(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
}
void xDBLADD(point_proj_t P, point_proj_t Q, const f2elm_t xPQ, const f2elm_t A24)
{ // Simultaneous doubling and differential addition.
// Input: projective Montgomery points P=(XP:ZP) and Q=(XQ:ZQ) such that xP=XP/ZP and xQ=XQ/ZQ, affine difference xPQ=x(P-Q) and Montgomery curve constant A24=(A+2)/4.
// Output: projective Montgomery points P <- 2*P = (X2P:Z2P) such that x(2P)=X2P/Z2P, and Q <- P+Q = (XQP:ZQP) such that = x(Q+P)=XQP/ZQP.
f2elm_t t0, t1, t2;
fp2add751(P->X, P->Z, t0); // t0 = XP+ZP
fp2sub751(P->X, P->Z, t1); // t1 = XP-ZP
fp2sqr751_mont(t0, P->X); // XP = (XP+ZP)^2
fp2sub751(Q->X, Q->Z, t2); // t2 = XQ-ZQ
fp2add751(Q->X, Q->Z, Q->X); // XQ = XQ+ZQ
fp2mul751_mont(t0, t2, t0); // t0 = (XP+ZP)*(XQ-ZQ)
fp2sqr751_mont(t1, P->Z); // ZP = (XP-ZP)^2
fp2mul751_mont(t1, Q->X, t1); // t1 = (XP-ZP)*(XQ+ZQ)
fp2sub751(P->X, P->Z, t2); // t2 = (XP+ZP)^2-(XP-ZP)^2
fp2mul751_mont(P->X, P->Z, P->X); // XP = (XP+ZP)^2*(XP-ZP)^2
fp2mul751_mont(t2, A24, Q->X); // XQ = A24*[(XP+ZP)^2-(XP-ZP)^2]
fp2sub751(t0, t1, Q->Z); // ZQ = (XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)
fp2add751(Q->X, P->Z, P->Z); // ZP = A24*[(XP+ZP)^2-(XP-ZP)^2]+(XP-ZP)^2
fp2add751(t0, t1, Q->X); // XQ = (XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)
fp2mul751_mont(P->Z, t2, P->Z); // ZP = [A24*[(XP+ZP)^2-(XP-ZP)^2]+(XP-ZP)^2]*[(XP+ZP)^2-(XP-ZP)^2]
fp2sqr751_mont(Q->Z, Q->Z); // ZQ = [(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
fp2sqr751_mont(Q->X, Q->X); // XQ = [(XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)]^2
fp2mul751_mont(Q->Z, xPQ, Q->Z); // ZQ = xPQ*[(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
}
void xDBL(const point_proj_t P, point_proj_t Q, const f2elm_t A24, const f2elm_t C24)
{ // Doubling of a Montgomery point in projective coordinates (X:Z).
// Input: projective Montgomery x-coordinates P = (X1:Z1), where x1=X1/Z1 and Montgomery curve constant A24/C24=(A/C+2)/4.
// Output: projective Montgomery x-coordinates Q = 2*P = (X2:Z2).
f2elm_t t0, t1;
fp2sub751(P->X, P->Z, t0); // t0 = X1-Z1
fp2add751(P->X, P->Z, t1); // t1 = X1+Z1
fp2sqr751_mont(t0, t0); // t0 = (X1-Z1)^2
fp2sqr751_mont(t1, t1); // t1 = (X1+Z1)^2
fp2mul751_mont(C24, t0, Q->Z); // Z2 = C24*(X1-Z1)^2
fp2mul751_mont(t1, Q->Z, Q->X); // X2 = C24*(X1-Z1)^2*(X1+Z1)^2
fp2sub751(t1, t0, t1); // t1 = (X1+Z1)^2-(X1-Z1)^2
fp2mul751_mont(A24, t1, t0); // t0 = A24*[(X1+Z1)^2-(X1-Z1)^2]
fp2add751(Q->Z, t0, Q->Z); // Z2 = A24*[(X1+Z1)^2-(X1-Z1)^2] + C24*(X1-Z1)^2
fp2mul751_mont(Q->Z, t1, Q->Z); // Z2 = [A24*[(X1+Z1)^2-(X1-Z1)^2] + C24*(X1-Z1)^2]*[(X1+Z1)^2-(X1-Z1)^2]
}
void xDBLe(const point_proj_t P, point_proj_t Q, const f2elm_t A, const f2elm_t C, const int e)
{ // Computes [2^e](X:Z) on Montgomery curve with projective constant via e repeated doublings.
// Input: projective Montgomery x-coordinates P = (XP:ZP), such that xP=XP/ZP and Montgomery curve constant A/C.
// Output: projective Montgomery x-coordinates Q <- (2^e)*P.
f2elm_t A24num, A24den;
int i;
fp2add751(C, C, A24num);
fp2add751(A24num, A24num, A24den);
fp2add751(A24num, A, A24num);
copy_words((digit_t*)P, (digit_t*)Q, 2*2*NWORDS_FIELD);
for (i = 0; i < e; i++) {
xDBL(Q, Q, A24num, A24den);
}
}
void xADD(point_proj_t P, const point_proj_t Q, const f2elm_t xPQ)
{ // Differential addition.
// Input: projective Montgomery points P=(XP:ZP) and Q=(XQ:ZQ) such that xP=XP/ZP and xQ=XQ/ZQ, and affine difference xPQ=x(P-Q).
// Output: projective Montgomery point P <- P+Q = (XQP:ZQP) such that = x(Q+P)=XQP/ZQP.
f2elm_t t0, t1;
fp2add751(P->X, P->Z, t0); // t0 = XP+ZP
fp2sub751(P->X, P->Z, t1); // t1 = XP-ZP
fp2sub751(Q->X, Q->Z, P->X); // XP = XQ-ZQ
fp2add751(Q->X, Q->Z, P->Z); // ZP = XQ+ZQ
fp2mul751_mont(t0, P->X, t0); // t0 = (XP+ZP)*(XQ-ZQ)
fp2mul751_mont(t1, P->Z, t1); // t1 = (XP-ZP)*(XQ+ZQ)
fp2sub751(t0, t1, P->Z); // ZP = (XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)
fp2add751(t0, t1, P->X); // XP = (XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)
fp2sqr751_mont(P->Z, P->Z); // ZP = [(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
fp2sqr751_mont(P->X, P->X); // XP = [(XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)]^2
fp2mul751_mont(P->Z, xPQ, P->Z); // ZP = xPQ*[(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
}
void DADD_basefield(point_basefield_proj_t P, const point_basefield_proj_t Q_P, const felm_t X)
{ // Differential addition.
// P <= P+(x,1) dif=Q_P
// Input: projective Montgomery points P=(XP:ZP) and Q-P=(XQP:ZQP) such that xP=XP/ZP and xQ=X/1,
// Output: projective Montgomery point P <= P+Q
// X = (x+1)/(x-1) pre-computed
felm_t t0, t1, t2;
fpadd751(P->X, P->Z, t0); // t0 = XP+ZP
fpsub751(P->X, P->Z, t1); // t1 = XP-ZP
fpmul751_mont(t1,X,t2); // t2 = (XP-ZP)*u
fpadd751(t0, t2, P->X); // XP = (XP+ZP)+u*(XP-ZP)
fpsub751(t0, t2, P->Z); // ZP = (XP+ZP)-u*(XP-ZP)
fpsqr751_mont(P->X, P->X); // XP = [(XP+ZP)+u*(XP-ZP)]^2
fpsqr751_mont(P->Z, P->Z); // ZP = [(XP+ZP)-u*(XP-ZP)]^2
fpmul751_mont(P->X, Q_P->Z, P->X); // XP = zPQ*[(XP+ZP)+u*(XP-ZP)]^2
fpmul751_mont(P->Z, Q_P->X, P->Z); // ZP = xPQ*[(XP+ZP)-u*(XP-ZP)]^2
}
void xDBL_basefield(const point_basefield_proj_t P, point_basefield_proj_t Q)
{ // Doubling of a Montgomery point in projective coordinates (X:Z) over the base field.
// Input: projective Montgomery x-coordinates P = (X1:Z1), where x1=X1/Z1 and Montgomery curve constant A24/C24=(A/C+2)/4.
// Output: projective Montgomery x-coordinates Q = 2*P = (X2:Z2).
felm_t t0, t1;
// NOTE: this function is fixed for A24=1, C24=2
fpsub751(P->X, P->Z, t0); // t0 = X1-Z1
fpadd751(P->X, P->Z, t1); // t1 = X1+Z1
fpsqr751_mont(t0, t0); // t0 = (X1-Z1)^2
fpsqr751_mont(t1, t1); // t1 = (X1+Z1)^2
fpadd751(t0, t0, Q->Z); // Z2 = C24*(X1-Z1)^2
fpmul751_mont(t1, Q->Z, Q->X); // X2 = C24*(X1-Z1)^2*(X1+Z1)^2
fpsub751(t1, t0, t1); // t1 = (X1+Z1)^2-(X1-Z1)^2
fpadd751(Q->Z, t1, Q->Z); // Z2 = A24*[(X1+Z1)^2-(X1-Z1)^2] + C24*(X1-Z1)^2
fpmul751_mont(Q->Z, t1, Q->Z); // Z2 = [A24*[(X1+Z1)^2-(X1-Z1)^2] + C24*(X1-Z1)^2]*[(X1+Z1)^2-(X1-Z1)^2]
}
void xDBLADD_basefield(point_basefield_proj_t P, point_basefield_proj_t Q, const felm_t xPQ, const felm_t A24)
{ // Simultaneous doubling and differential addition over the base field.
// Input: projective Montgomery points P=(XP:ZP) and Q=(XQ:ZQ) such that xP=XP/ZP and xQ=XQ/ZQ, affine difference xPQ=x(P-Q) and Montgomery curve constant A24=(A+2)/4.
// Output: projective Montgomery points P <- 2*P = (X2P:Z2P) such that x(2P)=X2P/Z2P, and Q <- P+Q = (XQP:ZQP) such that = x(Q+P)=XQP/ZQP.
felm_t t0, t1, t2;
// NOTE: this function is fixed for C24=2
fpadd751(P->X, P->Z, t0); // t0 = XP+ZP
fpsub751(P->X, P->Z, t1); // t1 = XP-ZP
fpsqr751_mont(t0, P->X); // XP = (XP+ZP)^2
fpsub751(Q->X, Q->Z, t2); // t2 = XQ-ZQ
fpadd751(Q->X, Q->Z, Q->X); // XQ = XQ+ZQ
fpmul751_mont(t0, t2, t0); // t0 = (XP+ZP)*(XQ-ZQ)
fpsqr751_mont(t1, P->Z); // ZP = (XP-ZP)^2
fpmul751_mont(t1, Q->X, t1); // t1 = (XP-ZP)*(XQ+ZQ)
fpsub751(P->X, P->Z, t2); // t2 = (XP+ZP)^2-(XP-ZP)^2
if (A24[0] == 1) {
fpadd751(P->Z, P->Z, P->Z); // ZP = C24*(XP-ZP)^2
fpmul751_mont(P->X, P->Z, P->X); // XP = C24*(XP+ZP)^2*(XP-ZP)^2
fpadd751(t2, P->Z, P->Z); // ZP = A24*[(XP+ZP)^2-(XP-ZP)^2]+C24*(XP-ZP)^2
} else {
fpmul751_mont(P->X, P->Z, P->X); // XP = (XP+ZP)^2*(XP-ZP)^2
fpmul751_mont(A24, t2, Q->X); // XQ = A24*[(XP+ZP)^2-(XP-ZP)^2]
fpadd751(P->Z, Q->X, P->Z); // ZP = A24*[(XP+ZP)^2-(XP-ZP)^2]+C24*(XP-ZP)^2
}
fpsub751(t0, t1, Q->Z); // ZQ = (XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)
fpadd751(t0, t1, Q->X); // XQ = (XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)
fpmul751_mont(P->Z, t2, P->Z); // ZP = [A24*[(XP+ZP)^2-(XP-ZP)^2]+C24*(XP-ZP)^2]*[(XP+ZP)^2-(XP-ZP)^2]
fpsqr751_mont(Q->Z, Q->Z); // ZQ = [(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
fpsqr751_mont(Q->X, Q->X); // XQ = [(XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)]^2
fpmul751_mont(Q->Z, xPQ, Q->Z); // ZQ = xPQ*[(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2
}
void ladder(const felm_t x, digit_t* m, point_basefield_proj_t P, point_basefield_proj_t Q, const felm_t A24, const unsigned int order_bits, const unsigned int order_fullbits, PCurveIsogenyStruct CurveIsogeny)
{ // The Montgomery ladder
// Inputs: the affine x-coordinate of a point P on E: B*y^2=x^3+A*x^2+x,
// scalar m
// curve constant A24 = (A+2)/4
// order_bits = subgroup order bitlength
// order_fullbits = smallest multiple of 32 larger than the order bitlength
// Output: Q = m*(x:1)
// CurveIsogeny must be set up in advance using SIDH_curve_initialize().
unsigned int bit = 0, owords = NBITS_TO_NWORDS(order_fullbits);
digit_t mask;
int i;
// Initializing with the points (1:0) and (x:1)
fpcopy751(CurveIsogeny->Montgomery_one, (digit_t*)P->X);
fpzero751(P->Z);
fpcopy751(x, Q->X);
fpcopy751(CurveIsogeny->Montgomery_one, (digit_t*)Q->Z);
for (i = order_fullbits-order_bits; i > 0; i--) {
mp_shiftl1(m, owords);
}
for (i = order_bits; i > 0; i--) {
bit = (unsigned int)(m[owords-1] >> (RADIX-1));
mp_shiftl1(m, owords);
mask = 0-(digit_t)bit;
swap_points_basefield(P, Q, mask);
xDBLADD_basefield(P, Q, x, A24); // If bit=0 then P <- 2*P and Q <- P+Q,
swap_points_basefield(P, Q, mask); // else if bit=1 then Q <- 2*Q and P <- P+Q
}
}
CRYPTO_STATUS BigMont_ladder(unsigned char* x, digit_t* m, unsigned char* xout, PCurveIsogenyStruct CurveIsogeny)
{ // BigMont's scalar multiplication using the Montgomery ladder
// Inputs: x, the affine x-coordinate of a point P on BigMont: y^2=x^3+A*x^2+x,
// scalar m.
// Output: xout, the affine x-coordinate of m*(x:1)
// CurveIsogeny must be set up in advance using SIDH_curve_initialize().
point_basefield_proj_t P1, P2;
digit_t scalar[BIGMONT_NWORDS_ORDER];
felm_t X, A24 = {0};
A24[0] = (digit_t)CurveIsogeny->BigMont_A24;
to_mont(A24, A24); // Conversion to Montgomery representation
to_mont((digit_t*)x, X);
copy_words(m, scalar, BIGMONT_NWORDS_ORDER);
ladder(X, scalar, P1, P2, A24, BIGMONT_NBITS_ORDER, BIGMONT_MAXBITS_ORDER, CurveIsogeny);
fpinv751_mont(P1->Z);
fpmul751_mont(P1->X, P1->Z, (digit_t*)xout);
from_mont((digit_t*)xout, (digit_t*)xout); // Conversion to standard representation
return CRYPTO_SUCCESS;
}
CRYPTO_STATUS precmp_secret_pt_Fp2(const point_basefield_t P, const digit_t* m, const unsigned int AliceOrBob, point_proj_t R, PCurveIsogenyStruct CurveIsogeny)
{ // Computes key generation entirely in the base field by exploiting a 1-dimensional Montgomery ladder in the trace zero subgroup and
// recovering the y-coordinate for the addition. All operations in the base field GF(p).
// Input: The scalar m, point P = (x,y) on E in the base field subgroup and Q = (x1,y1*i) on E in the trace-zero subgroup.
// x,y,x1,y1 are all in the base field.
// Output: R = (RX0+RX1*i)/RZ0 (the x-coordinate of P+[m]Q).
felm_t S,Q_S;
f2elm_t _2iQ_mu;
point_affine *_2iQ;
f2elm_t dif;
point_proj R1,R2,R3;
point_full_proj kQ,full_P,full_R;
digit_t scalar[NWORDS_ORDER];
digit_t mask,prev;
const uint64_t * Tab;
unsigned int bit = 0, nbits;
unsigned int i;
const digit_t _3inv_oA[NWORDS_ORDER] = { /* 1/3 mod oA */
0xAAAAAAAAAAAAAAAB,0xAAAAAAAAAAAAAAAA,0xAAAAAAAAAAAAAAAA,
0xAAAAAAAAAAAAAAAA,0xAAAAAAAAAAAAAAAA,0x000AAAAAAAAAAAAA
};
const digit_t _4inv_oB[NWORDS_ORDER] = { /* (1/4 mod oB)*2^384 mod (3^239) */
0xEB5506643C20F9BD,0xD8C154C49CFD1FCA,0xCA45971B4D3D250B,
0x7A2D8BDBAEE77B53,0x8EA63AA6E40654F8,0x10EB80AE5003815
};
const felm_t SA_X = { /* (x,y) = SA; SA_X = to_mont(x) */
0xFFFFFFFFFFFDB652,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0x6B9FFFFFFFFFFFFF,
0x8EC4E4A1129C0C42,0x711E035AD4A8A7A6,0x3FB0E0B52A8F9A48,0x38BE00CA8AAAEDF1,0xFD1AFE3322A8147D,0x428AB0851139
};
const felm_t SB_X = { /* (x,y) = SB; SB_X = to_mont(x) */
0x84485B5F84F87723,0x23C95C099F84DE4F,0xB8AF1E7118B30D0F,0xB9B1CE62F48547F9,0x3DA55BB3FB08A700,0xB5817ADE5B91F597,
0xE79E1FCF0AF940C9,0xBB60344181114E9D,0x0F7BAD5863C28A11,0x3FE7A11898419A7C,0x1070C40098C51E33,0x122697AF2B19
};
const felm_t QA_SB_X = { /* (x,y) = QA-SB; QA-SB_X = to_mont(x) */
0xBDB500AF28BF19C6,0xA9DBA0DC9C500777,0x555587C5242091F8,0xB2F79AE2BAD40CDF,0x9A53DFEFCCA87778,0xCB63F90D3B29E592,
0xFDA7FE26D6BE7CE8,0xB4C80AA87FF27B01,0xBA2873AF800A8E11,0x734ACDAA1430C012,0x6F4563B6927C6BA,0x4FCACEFE6BA2
};
const felm_t QB_SA_X = { /* (x,y) = QB-SA; QB-SA_X = to_mont(x) */
0xBD0C0516F26F441B,0x400260A336ADFA44,0x5C8493B76AD372C7,0xC5C0C661C9D57C0D,0xD4545A50CEC58A98,0xCDE3DFC6D1739EF5,
0x3B00EF1493497D3F,0x3B9823CA7356AEF4,0x4E8B29ED0EE9DE2D,0xA0355CA6B1B38187,0x690010CA832C9F01,0x1F72CAA86A34
};
/* Alice
* xx,yy:= ((3*2^370)*QA)[1];
* to_mont(xx);
* to_mont(yy);
*/
const point_affine _2iQ_Alice = {
.x = {
{
0xFFFFFFFFFFFDB652,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0x6B9FFFFFFFFFFFFF,
0x8EC4E4A1129C0C42,0x711E035AD4A8A7A6,0x3FB0E0B52A8F9A48,0x38BE00CA8AAAEDF1,0xFD1AFE3322A8147D,0x428AB0851139
},
{0}
},
.y = {
{0},
{
0x957EBAD489F67C6F,0x948C7DD037C464D5,0x525626E3E5A93782,0x168CF49C8D303980,0x3D3CF159A048DAA,0x478404EE5A178BC9,
0x4E61C8483CD8F624,0x49DF712544173CDE,0x566370793BAF611D,0x7E290975577CF148,0x97E421BC9BE88A62,0x5370762DBFA4
}
}
};
/* Bob
* xx,yy:= ((2^381)*QB)[1];
* to_mont(xx);
* to_mont(yy);
*/
const point_affine _2iQ_Bob = {
.x = {
{
0xE7D83DC64F12E2B9,0x6EE99F795F1A9233,0x414D2A3584E41776,0x40045C1254A2FAF5,0x762325D4B9940E0B,0x85A9D3293308D273,
0x8BBDDE6675B0738,0x2AFB36F660844EFC,0xF126084E14464CB2,0xBEA6574730D48163,0xA01A811F3E8320,0x214AB2679FC4
},
{0}
},
.y = {
{0},
{
0x649C4220DC572954,0x7AF81067892DEAD6,0x9A4A55D7D9B9FC47,0x1088D8BA9EEBB1D8,0x7CEFD297E1571129,0xA062161EA25B0712,
0x163F00B337478492,0x7E7ECB08F7AAFB3A,0xB45BD73348434A07,0xC22E8036A1EE60C2,0xE433D634E26AB3B9,0x3E1E09D93302
}
}
};
/* Alice
* xx:= (3*(2^370)*QA)[1];
* mu:=(xx+1)/(xx-1)
* to_mont(mu);
*/
const felm_t _2iQ_mu_Alice = {0};
/* Bob
* xx:= ((2^381)*QB)[1];
* mu:=(xx+1)/(xx-1)
* to_mont(mu);
*/
const felm_t _2iQ_mu_Bob = {
0x7A2EF704BA86438A,0x65745351CB267E9A,0x365EBFE7AB67F2A6,0x716A6B3EB9686110,0xCD12A4C8FC52419,0x741300535188FA50,
0x47AFE3AE69F2FEED,0x51D646896C5D4D44,0x15A655580B962731,0x3D76FEE0903A845A,0x9BC37064137AF0EC,0x1001531B0EED
};
if (AliceOrBob == ALICE)
{
digit_t tmp[2*NWORDS_ORDER];
nbits = CurveIsogeny->oAbits-2;
fpcopy751(SB_X,S);
fpcopy751(QA_SB_X,Q_S);
fpcopy751(_2iQ_mu_Alice,_2iQ_mu[0]); fpzero751(_2iQ_mu[1]);
_2iQ = (point_affine*)&_2iQ_Alice;
Tab = TabAlice;
/* scalar <= k/3 mod rA */
multiply((digit_t*)m, _3inv_oA, tmp, NWORDS_ORDER);
copy_words(tmp, scalar, NWORDS_ORDER);
scalar[NWORDS_ORDER-1] &= (digit_t)(-1) >> 12; // Hardcoded value
}
else if (AliceOrBob == BOB)
{
nbits = CurveIsogeny->oBbits;
fpcopy751(SA_X,S);
fpcopy751(QB_SA_X,Q_S);
fpcopy751(_2iQ_mu_Bob,_2iQ_mu[0]); fpzero751(_2iQ_mu[1]);
_2iQ = (point_affine*)&_2iQ_Bob;
Tab = TabBob;
/* scalar <= k/4 mod rB */
uint64_t Montgomery_Rprime[NWORDS64_ORDER] = {0x1A55482318541298, 0x070A6370DFA12A03, 0xCB1658E0E3823A40, 0xB3B7384EB5DEF3F9, 0xCBCA952F7006EA33, 0x00569EF8EC94864C}; // Value (2^384)^2 mod 3^239
uint64_t Montgomery_rprime[NWORDS64_ORDER] = {0x48062A91D3AB563D, 0x6CE572751303C2F5, 0x5D1319F3F160EC9D, 0xE35554E8C2D5623A, 0xCA29300232BC79A5, 0x8AAD843D646D78C5}; // Value -(3^239)^-1 mod 2^384
to_Montgomery_mod_order((digit_t*)m, (digit_t*)scalar, CurveIsogeny->Border, (digit_t*)&Montgomery_rprime, (digit_t*)&Montgomery_Rprime);
Montgomery_multiply_mod_order(scalar, _4inv_oB, scalar, CurveIsogeny->Border, (digit_t*)&Montgomery_rprime);
from_Montgomery_mod_order(scalar, scalar, CurveIsogeny->Border, (digit_t*)&Montgomery_rprime); // Converting back from Montgomery representation
}
else
{
return CRYPTO_ERROR_INVALID_PARAMETER;
}
// Initializing with the points
// R1 <= S/1; R2 <= (Q-S)/1
fpcopy751( S, R1.X[0]); fpzero751(R1.X[1]); fpcopy751(CurveIsogeny->Montgomery_one, R1.Z[0]); fpzero751(R1.Z[1]);
fpcopy751(Q_S, R2.X[0]); fpzero751(R2.X[1]); fpcopy751(CurveIsogeny->Montgomery_one, R2.Z[0]); fpzero751(R2.Z[1]);
prev = 0;
for (i = 0; i < nbits; i++) {
bit = (unsigned int)(scalar[0]&0x1);
mp_shiftr1(scalar, NWORDS_ORDER);
mask = 0 - (digit_t)(bit^prev);
swap_points(&R1, &R2, mask);
prev = bit;
fpcopy751(&Tab[NWORDS64_FIELD*i],dif[0]); fpzero751(dif[1]);
xADD(&R2, &R1, dif);
}
mask = 0 - (digit_t)(prev);
swap_points(&R1, &R2, mask);
f2elm_t A,C;
/* Clearing low-order point S */
if (AliceOrBob == ALICE) {
/* Since A=0 C=1 then A24=1, C24=2*/
fpcopy751(CurveIsogeny->Montgomery_one, C[0]);
fp2add751(C,C,A);
fp2add751(A,A,C);
xTPL(&R1,&R1,A,C);
xTPL(&R2,&R2,A,C);
} else if (AliceOrBob == BOB) {
xDBL(&R1,&R1,A,C);xDBL(&R1,&R1,A,C);
xDBL(&R2,&R2,A,C);xDBL(&R2,&R2,A,C);
} else {
return CRYPTO_ERROR_INVALID_PARAMETER;
}
/* Recovering y-coordinate of [k]Q */
fpcopy751(R1.X[0],R3.X[0]); fpcopy751(R1.Z[0],R3.Z[0]);
xADD(&R3, &R2, _2iQ_mu);
// fpcopy751(R1.X,pj_R1.X[0]); fpzero751(pj_R1.X[1]);
// fpcopy751(R1.Z,pj_R1.Z[0]); fpzero751(pj_R1.Z[1]);
//
// fpcopy751(R2.X,pj_R2.X[0]); fpzero751(pj_R2.X[1]);
// fpcopy751(R2.Z,pj_R2.Z[0]); fpzero751(pj_R2.Z[1]);
//
// fpcopy751(R3.X,pj_R3.X[0]); fpzero751(pj_R3.X[1]);
// fpcopy751(R3.Z,pj_R3.Z[0]); fpzero751(pj_R3.Z[1]);
y_recovering(_2iQ,&R1,&R3,&R2,&kQ);
f2elm_t A0;
fp2zero751(A0);
if (AliceOrBob == ALICE) {
point_affine _2PA_2QAi = { /* _2PA_2QAi = 2*PA+3*2^(eAbits-2)*QA; */
.x = {
{
0x9C2A3654B5CAA8FC,0x71DA703BC6A72437,0x7C2E74D8DDF07EB9,0x14217ED4E09CB339,0x8979DB08CD0BF8,0xF7EFF46D132F195,
0x34CCAA9BDA7F3D69,0x298AD9EAEA699D0C,0xAC00D3102D2A4EB3,0xFC8BC59EA987B1DB,0xA51628CA48A04F89,0x47F4AD49111
},
{
0xEBE72AA7350D53F3,0x2AD5E4A6D724E6DB,0xCCE815F093693BE3,0x39D90FD543297A89,0xC872A970ED7B8EB8,0x775A3DEC00752075,
0x27781E17B1BBAA5D,0x510799DF5F2459F9,0x7DE81DCC8F9B3707,0x9281E4D1F57F8702,0x558B8BD7875B80C2,0x2D8E2039BB0F
}
},
.y = {
{
0x25721FD413D36128,0xB75301F94F201B0F,0xC6EF91C078E0D072,0x68A5D29518238704,0xF5D834D34B74D319,0x678077DFBCE10E1A,
0xBEB070CAF807BF23,0x75C6E430503A161F,0x9B4ADD18950606D,0x91E33834B80D997A,0xB44B7306707EEEC0,0xFB19CC973A9
},
{
0x13D0AF76B43A118F,0xC672DFCFB94383BB,0xF2A3AA58CF0F901D,0x59AF31AA0ED00AF7,0x7EAA87DC3D3D6A6B,0x6D0CEDD2D710E948,
0xCF54D57071F61B7,0x4C75E7D3E23045BA,0x7B980701F77A722B,0xC9DE6336249F9EAF,0x1FD1804308731AC3,0xADE1FCC7D75
}
}
};
point_affine _2PA = { /* _2PA = 2PA */
.x = {
{
0x6B58A68059FEE240,0x6A0CDDE101D53945,0x9BB390C970A96718,0x85F5DA2AE04CF8F3,0x29309A3504F4A87B,0x1469F3B3BC429496,
0x2F4F81DF3819923B,0xFE8B21AEE84BDC2C,0x829370C4AC4B87D5,0x3321889EE50D93EF,0x98E0C58286164C12,0x3F6D9F200574
},
{
0
}
},
.y = {
{
0x9FC84D7F1EBBFA5E,0xA631CAFE224EBA60,0x728CFCBC6895235,0x4380806397C0395A,0x938F3ACAB92026AD,0xAE4A897D357119B0,
0xC28737A9D7335F15,0xFD32897FA8CCF5AE,0xEB4C22E97E465C6B,0xAA293637BB4FFC0F,0xFCB838FCB1CFC885,0x1FD4B77C0283
},
{
0
}
}
};
point_affine _PA_2QAi = { /* _PA_2QAi = -PA+3*2^(eAbits-1)*QA */
.x = {
{
0xBCF56AA1F988F05B,0xA1904A7DF12D502B,0x8863E83BF9FB4EE,0xDD7F60C3C87CF762,0x936A936EA8ED9BA7,0x30280B6D28F7A522,
0x6561E996C54FB269,0x896D12A3F7308B87,0x47433574585DE774,0xD1014375B4E01E89,0xD245D4550DE52C62,0x62B8D74B81BB
},
{
0
}
},
.y = {
{
0x98E2719E38FF6403,0x7205773D6A942623,0xC0EC772D3544BD8D,0x1C36DC8B14612DA1,0xA9BD0817778D6956,0x99EF9DD47C4BBA32,
0x84258BDBBB228935,0xC602FA25EEC263B5,0xCA7937EC11B30CE5,0x3DB83145BE87175E,0xCE5632B5CAB5EA2D,0x51986B424567
},
{
0
}
}
};
point_affine _PA = { /* _PA = -PA */
.x = {
{
0xD56FE52627914862,0x1FAD60DC96B5BAEA,0x1E137D0BF07AB91,0x404D3E9252161964,0x3C5385E4CD09A337,0x4476426769E4AF73,
0x9790C6DB989DFE33,0xE06E1C04D2AA8B5E,0x38C08185EDEA73B9,0xAA41F678A4396CA6,0x92B9259B2229E9A0,0x2F9326818BE0
},
{
0
}
},
.y = {
{
0x332BD16FBE3D7739,0x7E5E20FF2319E3DB,0xEA856234AEFBD81B,0xE016DF7D6D071283,0x8AE42796F73CD34F,0x6364B408A4774575,
0xA71C97F17CE99497,0xDA03CDD9AA0CBE71,0xE52B4FDA195BD56F,0xDAC41F811FCE0A46,0x9333720F0EE84A61,0x1399F006E578
},
{
0
}
}
};
f2elm_t one;
fpcopy751(CurveIsogeny->Montgomery_one,one[0]);fpzero751(one[1]);
bit = (unsigned int)(scalar[0]&0x1);
mp_shiftr1(scalar, NWORDS_ORDER);
mask = 0 - (digit_t)(bit);
swap_points_affine(&_2PA,&_2PA_2QAi,mask);
ADD(&kQ,_2PA.x,_2PA.y,one,A0,&full_R);
bit = (unsigned int)(scalar[0]&0x1);
mp_shiftr1(scalar, NWORDS_ORDER);
mask = 0 - (digit_t)(bit);
swap_points_affine(&_PA,&_PA_2QAi,mask);
ADD(&full_R,_PA.x,_PA.y,one,A0,&full_R);
fp2copy751(full_R.X,R->X);
fp2copy751(full_R.Z,R->Z);
}
else if (AliceOrBob == BOB) {
fpcopy751(P->x,full_P.X[0]); fpzero751(full_P.X[1]);
fpcopy751(P->y,full_P.Y[0]); fpzero751(full_P.Y[1]);
fpcopy751(CurveIsogeny->Montgomery_one,full_P.Z[0]); fpzero751(full_P.Z[1]);
ADD(&kQ,full_P.X,full_P.Y,full_P.Z,A0,&full_R);
fp2copy751(full_R.X,R->X);
fp2copy751(full_R.Z,R->Z);
}
else {
return CRYPTO_ERROR_INVALID_PARAMETER;
}
return CRYPTO_SUCCESS;
}
CRYPTO_STATUS original_secret_pt(const point_basefield_t P, const digit_t* m, const unsigned int AliceOrBob, point_proj_t R, PCurveIsogenyStruct CurveIsogeny)
{ // Computes key generation entirely in the base field by exploiting a 1-dimensional Montgomery ladder in the trace zero subgroup and
// recovering the y-coordinate for the addition. All operations in the base field GF(p).
// Input: The scalar m, point P = (x,y) on E in the base field subgroup and Q = (x1,y1*i) on E in the trace-zero subgroup.
// x,y,x1,y1 are all in the base field.
// Output: R = (RX0+RX1*i)/RZ0 (the x-coordinate of P+[m]Q).
unsigned int nbits;
point_basefield_t Q;
point_basefield_proj_t S, T;
digit_t *X0 = (digit_t*)S->X, *Z0 = (digit_t*)S->Z, *X1 = (digit_t*)T->X, *Z1 = (digit_t*)T->Z;
digit_t *x = (digit_t*)P->x, *y = (digit_t*)P->y, *x1 = (digit_t*)Q->x, *y1 = (digit_t*)Q->y;
digit_t scalar[NWORDS_ORDER];
felm_t t0, t1, t2, A24 = {0};
digit_t *RX0 = (digit_t*)R->X[0], *RX1 = (digit_t*)R->X[1], *RZ0 = (digit_t*)R->Z[0], *RZ1 = (digit_t*)R->Z[1];
fpcopy751(P->x, Q->x); // Q = (-XP,YP)
fpcopy751(P->y, Q->y);
fpneg751(Q->x);
if (AliceOrBob == ALICE) {
nbits = CurveIsogeny->oAbits;
} else if (AliceOrBob == BOB) {
nbits = CurveIsogeny->oBbits;
} else {
return CRYPTO_ERROR_INVALID_PARAMETER;
}
// Setting curve constant to one (in standard representation), used in xDBLADD_basefield() in the ladder computation
A24[0] = 1;
copy_words(m, scalar, NWORDS_ORDER);
ladder(Q->x, scalar, S, T, A24, nbits, CurveIsogeny->owordbits, CurveIsogeny);
//RX0 = (2*y*y1*Z0^2*Z1 + Z1*(X0*x1+Z0)*(X0+x1*Z0) - X1*(X0-x1*Z0)^2)*(2*y*y1*Z0^2*Z1 - Z1*(X0*x1+Z0)*(X0+x1*Z0) + X1*(X0-x1*Z0)^2) - 4*y1^2*Z0*Z1^2*(X0+x*Z0)*(X0-x*Z0)^2;
//RX1 = 4*y*y1*Z0^2*Z1*(Z1*(X0*x1+Z0)*(X0+x1*Z0) - X1*(X0-x1*Z0)^2);
//RZ0 = 4*y1^2*Z0^2*Z1^2*(X0-x*Z0)^2;
fpmul751_mont(x1, Z0, RX1);
fpmul751_mont(X0, x1, RX0);
fpsub751(X0, RX1, t0);
fpadd751(X0, RX1, RX1);
fpsqr751_mont(t0, t0);
fpadd751(RX0, Z0, RX0);
fpmul751_mont(t0, X1, t0);
fpmul751_mont(RX0, RX1, RX0);
fpmul751_mont(y1, Z1, t2);
fpmul751_mont(y, Z0, t1);
fpadd751(t2, t2, t2);
fpmul751_mont(t2, Z0, RX1);
fpmul751_mont(RX0, Z1, RX0);
fpsub751(RX0, t0, RX0);
fpmul751_mont(t1, RX1, t1);
fpsqr751_mont(RX1, t0);
fpmul751_mont(t2, RX1, t2);
fpmul751_mont(t1, RX0, RX1);
fpadd751(t1, RX0, RZ0);
fpadd751(RX1, RX1, RX1);
fpsub751(t1, RX0, t1);
fpmul751_mont(x, Z0, RX0);
fpmul751_mont(t1, RZ0, t1);
fpsub751(X0, RX0, RZ0);
fpadd751(X0, RX0, RX0);
fpsqr751_mont(RZ0, RZ0);
fpmul751_mont(t2, RX0, t2);
fpmul751_mont(t2, RZ0, t2);
fpmul751_mont(RZ0, t0, RZ0);
fpsub751(t1, t2, RX0);
fpzero751(RZ1);
return CRYPTO_SUCCESS;
}
CRYPTO_STATUS right2left_secret_pt(const point_basefield_t P, const digit_t* m, const unsigned int AliceOrBob, point_proj_t R, PCurveIsogenyStruct CurveIsogeny)
{ // Computes key generation entirely in the base field by exploiting a 1-dimensional Montgomery ladder in the trace zero subgroup and
// recovering the y-coordinate for the addition. All operations in the base field GF(p).
// Input: The scalar m, point P = (x,y) on E in the base field subgroup and Q = (x1,y1*i) on E in the trace-zero subgroup.
// x,y,x1,y1 are all in the base field.
// Output: R = (RX0+RX1*i)/RZ0 (the x-coordinate of P+[m]Q).
felm_t S,Q_S,_2iQ_mu;
point_affine *_2iQ;
point_basefield_proj R1,R2,R3;
point_proj pj_R1,pj_R2,pj_R3;
point_full_proj kQ,full_P,full_R;
digit_t scalar[NWORDS_ORDER];
digit_t mask,prev;
const uint64_t * Tab;
unsigned int bit = 0, nbits;
unsigned int i;
const digit_t _3inv_oA[NWORDS_ORDER] = { /* 1/3 mod oA */
0xAAAAAAAAAAAAAAAB,0xAAAAAAAAAAAAAAAA,0xAAAAAAAAAAAAAAAA,
0xAAAAAAAAAAAAAAAA,0xAAAAAAAAAAAAAAAA,0x000AAAAAAAAAAAAA
};
const digit_t _4inv_oB[NWORDS_ORDER] = { /* (1/4 mod oB)*2^384 mod (3^239) */
0xEB5506643C20F9BD,0xD8C154C49CFD1FCA,0xCA45971B4D3D250B,
0x7A2D8BDBAEE77B53,0x8EA63AA6E40654F8,0x10EB80AE5003815
};
const felm_t SA_X = { /* (x,y) = SA; SA_X = to_mont(x) */
0xFFFFFFFFFFFDB652,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0x6B9FFFFFFFFFFFFF,
0x8EC4E4A1129C0C42,0x711E035AD4A8A7A6,0x3FB0E0B52A8F9A48,0x38BE00CA8AAAEDF1,0xFD1AFE3322A8147D,0x428AB0851139
};
const felm_t SB_X = { /* (x,y) = SB; SB_X = to_mont(x) */
0x84485B5F84F87723,0x23C95C099F84DE4F,0xB8AF1E7118B30D0F,0xB9B1CE62F48547F9,0x3DA55BB3FB08A700,0xB5817ADE5B91F597,
0xE79E1FCF0AF940C9,0xBB60344181114E9D,0x0F7BAD5863C28A11,0x3FE7A11898419A7C,0x1070C40098C51E33,0x122697AF2B19
};
const felm_t QA_SB_X = { /* (x,y) = QA-SB; QA-SB_X = to_mont(x) */
0xBDB500AF28BF19C6,0xA9DBA0DC9C500777,0x555587C5242091F8,0xB2F79AE2BAD40CDF,0x9A53DFEFCCA87778,0xCB63F90D3B29E592,
0xFDA7FE26D6BE7CE8,0xB4C80AA87FF27B01,0xBA2873AF800A8E11,0x734ACDAA1430C012,0x6F4563B6927C6BA,0x4FCACEFE6BA2
};
const felm_t QB_SA_X = { /* (x,y) = QB-SA; QB-SA_X = to_mont(x) */
0xBD0C0516F26F441B,0x400260A336ADFA44,0x5C8493B76AD372C7,0xC5C0C661C9D57C0D,0xD4545A50CEC58A98,0xCDE3DFC6D1739EF5,
0x3B00EF1493497D3F,0x3B9823CA7356AEF4,0x4E8B29ED0EE9DE2D,0xA0355CA6B1B38187,0x690010CA832C9F01,0x1F72CAA86A34
};
/* Alice
* xx,yy:= ((3*2^370)*QA)[1];
* to_mont(xx);
* to_mont(yy);
*/
const point_affine _2iQ_Alice = {
.x = {
{
0xFFFFFFFFFFFDB652,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0xFFFFFFFFFFFFFFFF,0x6B9FFFFFFFFFFFFF,
0x8EC4E4A1129C0C42,0x711E035AD4A8A7A6,0x3FB0E0B52A8F9A48,0x38BE00CA8AAAEDF1,0xFD1AFE3322A8147D,0x428AB0851139
},
{0}
},
.y = {
{0},
{
0x957EBAD489F67C6F,0x948C7DD037C464D5,0x525626E3E5A93782,0x168CF49C8D303980,0x3D3CF159A048DAA,0x478404EE5A178BC9,
0x4E61C8483CD8F624,0x49DF712544173CDE,0x566370793BAF611D,0x7E290975577CF148,0x97E421BC9BE88A62,0x5370762DBFA4
}
}
};
/* Bob
* xx,yy:= ((2^381)*QB)[1];
* to_mont(xx);
* to_mont(yy);
*/
const point_affine _2iQ_Bob = {
.x = {
{
0xE7D83DC64F12E2B9,0x6EE99F795F1A9233,0x414D2A3584E41776,0x40045C1254A2FAF5,0x762325D4B9940E0B,0x85A9D3293308D273,
0x8BBDDE6675B0738,0x2AFB36F660844EFC,0xF126084E14464CB2,0xBEA6574730D48163,0xA01A811F3E8320,0x214AB2679FC4
},
{0}
},
.y = {
{0},
{
0x649C4220DC572954,0x7AF81067892DEAD6,0x9A4A55D7D9B9FC47,0x1088D8BA9EEBB1D8,0x7CEFD297E1571129,0xA062161EA25B0712,
0x163F00B337478492,0x7E7ECB08F7AAFB3A,0xB45BD73348434A07,0xC22E8036A1EE60C2,0xE433D634E26AB3B9,0x3E1E09D93302
}
}
};
/* Alice
* xx:= (3*(2^370)*QA)[1];
* mu:=(xx+1)/(xx-1)
* to_mont(mu);
*/
const felm_t _2iQ_mu_Alice = {0};
/* Bob
* xx:= ((2^381)*QB)[1];
* mu:=(xx+1)/(xx-1)
* to_mont(mu);
*/
const felm_t _2iQ_mu_Bob = {
0x7A2EF704BA86438A,0x65745351CB267E9A,0x365EBFE7AB67F2A6,0x716A6B3EB9686110,0xCD12A4C8FC52419,0x741300535188FA50,
0x47AFE3AE69F2FEED,0x51D646896C5D4D44,0x15A655580B962731,0x3D76FEE0903A845A,0x9BC37064137AF0EC,0x1001531B0EED
};
if (AliceOrBob == ALICE)
{
digit_t tmp[2*NWORDS_ORDER];
nbits = CurveIsogeny->oAbits-2;
fpcopy751(SB_X,S);
fpcopy751(QA_SB_X,Q_S);
fpcopy751(_2iQ_mu_Alice,_2iQ_mu);
_2iQ = (point_affine*)&_2iQ_Alice;
Tab = TabAlice;
/* scalar <= k/3 mod rA */
multiply((digit_t*)m, _3inv_oA, tmp, NWORDS_ORDER);
copy_words(tmp, scalar, NWORDS_ORDER);
scalar[NWORDS_ORDER-1] &= (digit_t)(-1) >> 12; // Hardcoded value
}
else if (AliceOrBob == BOB)
{
nbits = CurveIsogeny->oBbits;
fpcopy751(SA_X,S);
fpcopy751(QB_SA_X,Q_S);
fpcopy751(_2iQ_mu_Bob,_2iQ_mu);
_2iQ = (point_affine*)&_2iQ_Bob;
Tab = TabBob;
/* scalar <= k/4 mod rB */
uint64_t Montgomery_Rprime[NWORDS64_ORDER] = {0x1A55482318541298, 0x070A6370DFA12A03, 0xCB1658E0E3823A40, 0xB3B7384EB5DEF3F9, 0xCBCA952F7006EA33, 0x00569EF8EC94864C}; // Value (2^384)^2 mod 3^239
uint64_t Montgomery_rprime[NWORDS64_ORDER] = {0x48062A91D3AB563D, 0x6CE572751303C2F5, 0x5D1319F3F160EC9D, 0xE35554E8C2D5623A, 0xCA29300232BC79A5, 0x8AAD843D646D78C5}; // Value -(3^239)^-1 mod 2^384
to_Montgomery_mod_order((digit_t*)m, (digit_t*)scalar, CurveIsogeny->Border, (digit_t*)&Montgomery_rprime, (digit_t*)&Montgomery_Rprime);
Montgomery_multiply_mod_order(scalar, _4inv_oB, scalar, CurveIsogeny->Border, (digit_t*)&Montgomery_rprime);
from_Montgomery_mod_order(scalar, scalar, CurveIsogeny->Border, (digit_t*)&Montgomery_rprime); // Converting back from Montgomery representation
}
else
{
return CRYPTO_ERROR_INVALID_PARAMETER;
}
// Initializing with the points
// R1 <= S/1; R2 <= (Q-S)/1
fpcopy751( S, R1.X); fpcopy751(CurveIsogeny->Montgomery_one, R1.Z);
fpcopy751(Q_S, R2.X); fpcopy751(CurveIsogeny->Montgomery_one, R2.Z);
prev = 0;
for (i = 0; i < nbits; i++) {
bit = (unsigned int)(scalar[0]&0x1);
mp_shiftr1(scalar, NWORDS_ORDER);
mask = 0 - (digit_t)(bit^prev);
swap_points_basefield(&R1, &R2, mask);
prev = bit;
DADD_basefield(&R2, &R1, &Tab[NWORDS64_FIELD*i]);
}
mask = 0 - (digit_t)(prev);
swap_points_basefield(&R1, &R2, mask);
/* Clearing low-order point S */
if (AliceOrBob == ALICE) {
/**
* The new tripling formulas require
* A24=A-2C, C24=2C
* Therefore,
* since A=0 C=1 then
* A24=-2,, C24=2
**/
felm_t A,C;
fpcopy751(CurveIsogeny->Montgomery_one, C);
fpadd751(C,C,C);
fpcopy751(C,A);
fpneg751(A);
xTPL_basefield(&R1,&R1,A,C);
xTPL_basefield(&R2,&R2,A,C);
} else if (AliceOrBob == BOB) {
xDBL_basefield(&R1,&R1);xDBL_basefield(&R1,&R1);
xDBL_basefield(&R2,&R2);xDBL_basefield(&R2,&R2);
} else {
return CRYPTO_ERROR_INVALID_PARAMETER;
}
/* Recovering y-coordinate of [k]Q */
fpcopy751(R1.X,R3.X); fpcopy751(R1.Z,R3.Z);
DADD_basefield(&R3, &R2, _2iQ_mu);
fpcopy751(R1.X,pj_R1.X[0]); fpzero751(pj_R1.X[1]);
fpcopy751(R1.Z,pj_R1.Z[0]); fpzero751(pj_R1.Z[1]);
fpcopy751(R2.X,pj_R2.X[0]); fpzero751(pj_R2.X[1]);
fpcopy751(R2.Z,pj_R2.Z[0]); fpzero751(pj_R2.Z[1]);
fpcopy751(R3.X,pj_R3.X[0]); fpzero751(pj_R3.X[1]);
fpcopy751(R3.Z,pj_R3.Z[0]); fpzero751(pj_R3.Z[1]);
y_recovering(_2iQ,&pj_R1,&pj_R3,&pj_R2,&kQ);
f2elm_t A0;
fp2zero751(A0);
if (AliceOrBob == ALICE) {
point_affine _2PA_2QAi = { /* _2PA_2QAi = 2*PA+3*2^(eAbits-2)*QA; */
.x = {
{
0x9C2A3654B5CAA8FC,0x71DA703BC6A72437,0x7C2E74D8DDF07EB9,0x14217ED4E09CB339,0x8979DB08CD0BF8,0xF7EFF46D132F195,
0x34CCAA9BDA7F3D69,0x298AD9EAEA699D0C,0xAC00D3102D2A4EB3,0xFC8BC59EA987B1DB,0xA51628CA48A04F89,0x47F4AD49111
},
{
0xEBE72AA7350D53F3,0x2AD5E4A6D724E6DB,0xCCE815F093693BE3,0x39D90FD543297A89,0xC872A970ED7B8EB8,0x775A3DEC00752075,
0x27781E17B1BBAA5D,0x510799DF5F2459F9,0x7DE81DCC8F9B3707,0x9281E4D1F57F8702,0x558B8BD7875B80C2,0x2D8E2039BB0F
}
},
.y = {
{
0x25721FD413D36128,0xB75301F94F201B0F,0xC6EF91C078E0D072,0x68A5D29518238704,0xF5D834D34B74D319,0x678077DFBCE10E1A,
0xBEB070CAF807BF23,0x75C6E430503A161F,0x9B4ADD18950606D,0x91E33834B80D997A,0xB44B7306707EEEC0,0xFB19CC973A9
},
{
0x13D0AF76B43A118F,0xC672DFCFB94383BB,0xF2A3AA58CF0F901D,0x59AF31AA0ED00AF7,0x7EAA87DC3D3D6A6B,0x6D0CEDD2D710E948,
0xCF54D57071F61B7,0x4C75E7D3E23045BA,0x7B980701F77A722B,0xC9DE6336249F9EAF,0x1FD1804308731AC3,0xADE1FCC7D75
}
}
};
point_affine _2PA = { /* _2PA = 2PA */
.x = {
{
0x6B58A68059FEE240,0x6A0CDDE101D53945,0x9BB390C970A96718,0x85F5DA2AE04CF8F3,0x29309A3504F4A87B,0x1469F3B3BC429496,
0x2F4F81DF3819923B,0xFE8B21AEE84BDC2C,0x829370C4AC4B87D5,0x3321889EE50D93EF,0x98E0C58286164C12,0x3F6D9F200574
},
{
0
}
},
.y = {
{
0x9FC84D7F1EBBFA5E,0xA631CAFE224EBA60,0x728CFCBC6895235,0x4380806397C0395A,0x938F3ACAB92026AD,0xAE4A897D357119B0,
0xC28737A9D7335F15,0xFD32897FA8CCF5AE,0xEB4C22E97E465C6B,0xAA293637BB4FFC0F,0xFCB838FCB1CFC885,0x1FD4B77C0283
},
{
0
}
}
};
point_affine _PA_2QAi = { /* _PA_2QAi = -PA+3*2^(eAbits-1)*QA */
.x = {
{
0xBCF56AA1F988F05B,0xA1904A7DF12D502B,0x8863E83BF9FB4EE,0xDD7F60C3C87CF762,0x936A936EA8ED9BA7,0x30280B6D28F7A522,
0x6561E996C54FB269,0x896D12A3F7308B87,0x47433574585DE774,0xD1014375B4E01E89,0xD245D4550DE52C62,0x62B8D74B81BB
},
{
0
}
},
.y = {
{
0x98E2719E38FF6403,0x7205773D6A942623,0xC0EC772D3544BD8D,0x1C36DC8B14612DA1,0xA9BD0817778D6956,0x99EF9DD47C4BBA32,
0x84258BDBBB228935,0xC602FA25EEC263B5,0xCA7937EC11B30CE5,0x3DB83145BE87175E,0xCE5632B5CAB5EA2D,0x51986B424567
},
{
0
}
}
};
point_affine _PA = { /* _PA = -PA */
.x = {
{
0xD56FE52627914862,0x1FAD60DC96B5BAEA,0x1E137D0BF07AB91,0x404D3E9252161964,0x3C5385E4CD09A337,0x4476426769E4AF73,
0x9790C6DB989DFE33,0xE06E1C04D2AA8B5E,0x38C08185EDEA73B9,0xAA41F678A4396CA6,0x92B9259B2229E9A0,0x2F9326818BE0
},
{
0
}
},
.y = {
{
0x332BD16FBE3D7739,0x7E5E20FF2319E3DB,0xEA856234AEFBD81B,0xE016DF7D6D071283,0x8AE42796F73CD34F,0x6364B408A4774575,
0xA71C97F17CE99497,0xDA03CDD9AA0CBE71,0xE52B4FDA195BD56F,0xDAC41F811FCE0A46,0x9333720F0EE84A61,0x1399F006E578
},
{
0
}
}
};
f2elm_t one;
fpcopy751(CurveIsogeny->Montgomery_one,one[0]);fpzero751(one[1]);
bit = (unsigned int)(scalar[0]&0x1);
mp_shiftr1(scalar, NWORDS_ORDER);
mask = 0 - (digit_t)(bit);
swap_points_affine(&_2PA,&_2PA_2QAi,mask);
ADD(&kQ,_2PA.x,_2PA.y,one,A0,&full_R);
bit = (unsigned int)(scalar[0]&0x1);
mp_shiftr1(scalar, NWORDS_ORDER);
mask = 0 - (digit_t)(bit);
swap_points_affine(&_PA,&_PA_2QAi,mask);
ADD(&full_R,_PA.x,_PA.y,one,A0,&full_R);
fp2copy751(full_R.X,R->X);
fp2copy751(full_R.Z,R->Z);
}
else if (AliceOrBob == BOB) {
fpcopy751(P->x,full_P.X[0]); fpzero751(full_P.X[1]);
fpcopy751(P->y,full_P.Y[0]); fpzero751(full_P.Y[1]);
fpcopy751(CurveIsogeny->Montgomery_one,full_P.Z[0]); fpzero751(full_P.Z[1]);
ADD(&kQ,full_P.X,full_P.Y,full_P.Z,A0,&full_R);
fp2copy751(full_R.X,R->X);
fp2copy751(full_R.Z,R->Z);
}
else {
return CRYPTO_ERROR_INVALID_PARAMETER;
}