void do_line(element_ptr e, element_ptr edenom, element_ptr A, element_ptr B)
{
element_ptr Ax = curve_x_coord(A);
element_ptr Ay = curve_y_coord(A);
element_ptr Bx = curve_x_coord(B);
element_ptr By = curve_y_coord(B);
element_sub(b, Bx, Ax);
element_sub(a, Ay, By);
element_mul(c, Ax, By);
element_mul(e0, Ay, Bx);
element_sub(c, c, e0);
element_mul(e0, a, numx);
element_mul(e1, b, numy);
element_add(e0, e0, e1);
element_add(e0, e0, c);
element_mul(e, e, e0);
element_mul(e0, a, denomx);
element_mul(e1, b, denomy);
element_add(e0, e0, e1);
element_add(e0, e0, c);
element_mul(edenom, edenom, e0);
}
static void e_miller_proj(element_t res, element_t P,
element_ptr QR, element_ptr R,
e_pairing_data_ptr p) {
//collate divisions
int n;
element_t v, vd;
element_t v1, vd1;
element_t Z, Z1;
element_t a, b, c;
const element_ptr cca = curve_a_coeff(P);
element_t e0, e1;
const element_ptr e2 = a, e3 = b;
element_t z, z2;
int i;
element_ptr Zx, Zy;
const element_ptr Px = curve_x_coord(P);
const element_ptr numx = curve_x_coord(QR);
const element_ptr numy = curve_y_coord(QR);
const element_ptr denomx = curve_x_coord(R);
const element_ptr denomy = curve_y_coord(R);
//convert Z from weighted projective (Jacobian) to affine
//i.e. (X, Y, Z) --> (X/Z^2, Y/Z^3)
//also sets z to 1
void to_affine(void)
{
element_invert(z, z);
element_square(e0, z);
element_mul(Zx, Zx, e0);
element_mul(e0, e0, z);
element_mul(Zy, Zy, e0);
element_set1(z);
element_set1(z2);
}
static void do_line(element_ptr z, element_ptr V, element_ptr P, element_ptr Q)
{
element_ptr Vx = curve_x_coord(V);
element_ptr Vy = curve_y_coord(V);
element_ptr Px = curve_x_coord(P);
element_ptr Py = curve_y_coord(P);
element_ptr Qx = curve_x_coord(Q);
element_ptr Qy = curve_y_coord(Q);
element_t a, b, c, e0;
element_init_same_as(a, Vx);
element_init_same_as(b, Vx);
element_init_same_as(c, Vx);
element_init_same_as(e0, Vx);
//a = -(B.y - A.y) / (B.x - A.x);
//b = 1;
//c = -(A.y + a * A.x);
element_sub(a, Py, Vy);
element_sub(b, Vx, Px);
element_div(a, a, b);
element_set1(b);
element_mul(c, a, Vx);
element_add(c, c, Vy);
element_neg(c, c);
/*
//but we could multiply by B.x - A.x to avoid division, so
//a = -(By - Ay)
//b = Bx - Ax
//c = -(Ay b + a Ax);
element_sub(a, Vy, Py);
element_sub(b, Px, Vx);
element_mul(c, Vx, Py);
element_mul(e0, Vy, Px);
element_sub(c, c, e0);
//
//actually no, since fasterweil won't work if we do this
*/
element_printf("line at %B: %B %B %B\n", V, a, b, c);
element_mul(a, a, Qx);
element_mul(b, b, Qy);
element_add(c, c, a);
element_add(z, c, b);
element_printf(" = %B\n", z);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(e0);
}
static void do_tangent(element_ptr z, element_ptr V, element_ptr Q)
{
element_ptr Vx = curve_x_coord(V);
element_ptr Vy = curve_y_coord(V);
element_ptr Qx = curve_x_coord(Q);
element_ptr Qy = curve_y_coord(Q);
element_t a, b, c;
element_init_same_as(a, Vx);
element_init_same_as(b, Vx);
element_init_same_as(c, Vx);
//a = -slope_tangent(V.x, V.y);
//b = 1;
//c = -(V.y + aV.x);
/*
//we could multiply by -2*V.y to avoid division so:
//a = -(3 Vx^2 + cc->a)
//b = 2 * Vy
//c = -(2 Vy^2 + a Vx);
//
//actually no, since fasterweil won't work if we do this
*/
element_square(a, Vx);
//element_mul_si(a, a, 3);
element_add(b, a, a);
element_add(a, b, a);
element_set1(b);
element_add(a, a, b);
element_neg(a, a);
element_double(b, Vy);
element_div(a, a, b);
element_set1(b);
element_mul(c, a, Vx);
element_add(c, c, Vy);
element_neg(c, c);
element_printf("tan at %B: %B %B %B\n", V, a, b, c);
element_mul(a, a, Qx);
element_mul(b, b, Qy);
element_add(c, c, a);
element_add(z, c, b);
element_printf("tan eval = %B\n", z);
element_clear(a);
element_clear(b);
element_clear(c);
}
void cc_pairings_affine(element_ptr out, element_t in1[], element_t in2[],
int n_prod, pairing_t pairing) {
element_ptr Qbase;
element_t* Qx = pbc_malloc(sizeof(element_t)*n_prod);
element_t* Qy = pbc_malloc(sizeof(element_t)*n_prod);
pptr p = pairing->data;
int i;
for(i=0; i<n_prod; i++){
element_init(Qx[i], p->Fqd);
element_init(Qy[i], p->Fqd);
Qbase = in2[i];
// Twist: (x, y) --> (v^-1 x, v^-(3/2) y)
// where v is the quadratic nonresidue used to construct the twist.
element_mul(Qx[i], curve_x_coord(Qbase), p->nqrinv);
// v^-3/2 = v^-2 * v^1/2
element_mul(Qy[i], curve_y_coord(Qbase), p->nqrinv2);
}
cc_millers_no_denom_affine(out, pairing->r, in1, Qx, Qy, n_prod);
cc_tatepower(out, out, pairing);
for(i=0; i<n_prod; i++){
element_clear(Qx[i]);
element_clear(Qy[i]);
}
pbc_free(Qx);
pbc_free(Qy);
}
static void do_vert(element_ptr z, element_ptr V, element_ptr Q)
{
element_ptr Vx = curve_x_coord(V);
element_ptr Qx = curve_x_coord(Q);
element_ptr Qy = curve_y_coord(Q);
element_t a, b, c;
element_init_same_as(a, Vx);
element_init_same_as(b, Vx);
element_init_same_as(c, Vx);
//a = 1
//b = 0;
//c = -Vx
element_set1(a);
element_set0(b);
element_neg(c, Vx);
element_printf("vert at %B: %B %B %B\n", Vx, a, b, c);
element_mul(a, a, Qx);
element_mul(b, b, Qy);
element_add(c, c, a);
element_add(z, c, b);
element_printf("vert eval = %B\n", z);
element_clear(a);
element_clear(b);
element_clear(c);
}
void miller(element_t res, mpz_t q, element_t P, element_ptr Qx, element_ptr Qy) {
int m;
element_t v;
element_t Z;
element_t a, b, c;
element_t t0;
element_t e0;
const element_ptr cca = curve_a_coeff(P);
const element_ptr Px = curve_x_coord(P);
const element_ptr Py = curve_y_coord(P);
element_ptr Zx, Zy;
void do_tangent(void) {
// a = -(3 Zx^2 + cc->a)
// b = 2 * Zy
// c = -(2 Zy^2 + a Zx);
element_square(a, Zx); mult1++;
element_mul_si(a, a, 3); add1++; add1++; add1++;
element_add(a, a, cca); add1++;
element_neg(a, a);
element_add(b, Zy, Zy); add1++;
element_mul(t0, b, Zy); mult1++;
element_mul(c, a, Zx); mult1++;
element_add(c, c, t0); add1++;
element_neg(c, c);
d_miller_evalfn(e0, a, b, c, Qx, Qy);
element_mul(v, v, e0); multk++;
}
static int cc_is_almost_coddh(element_ptr a, element_ptr b,
element_ptr c, element_ptr d,
pairing_t pairing) {
int res = 0;
element_t t0, t1, t2;
element_t cx, cy;
element_t dx, dy;
pptr p = pairing->data;
element_init(cx, p->Fqd);
element_init(cy, p->Fqd);
element_init(dx, p->Fqd);
element_init(dy, p->Fqd);
element_init(t0, p->Fqk);
element_init(t1, p->Fqk);
element_init(t2, p->Fqk);
// Twist: (x, y) --> (v^-1 x, v^-(3/2) y)
// where v is the quadratic nonresidue used to construct the twist.
element_mul(cx, curve_x_coord(c), p->nqrinv);
element_mul(dx, curve_x_coord(d), p->nqrinv);
// v^-3/2 = v^-2 * v^1/2
element_mul(cy, curve_y_coord(c), p->nqrinv2);
element_mul(dy, curve_y_coord(d), p->nqrinv2);
cc_miller_no_denom_fn(t0, pairing->r, a, dx, dy);
cc_miller_no_denom_fn(t1, pairing->r, b, cx, cy);
cc_tatepower(t0, t0, pairing);
cc_tatepower(t1, t1, pairing);
element_mul(t2, t0, t1);
if (element_is1(t2)) res = 1; // We were given g, g^x, h, h^-x.
else {
// Cheaply check the other case.
element_invert(t1, t1);
element_mul(t2, t0, t1);
if (element_is1(t2)) res = 1; // We were given g, g^x, h, h^x.
}
element_clear(cx);
element_clear(cy);
element_clear(dx);
element_clear(dy);
element_clear(t0);
element_clear(t1);
element_clear(t2);
return res;
}
static void sn_pairing(element_ptr out, element_ptr in1, element_ptr in2,
pairing_t pairing) {
sn_pairing_data_ptr p = pairing->data;
element_ptr Q = in2;
element_t e0;
element_t R, QR;
element_init(R, p->Eq);
element_init(QR, p->Eq);
element_random(R);
element_init(e0, out->field);
element_add(QR, Q, R);
sn_miller(out, pairing->r, in1, curve_x_coord(QR), curve_y_coord(QR));
sn_miller(e0, pairing->r, in1, curve_x_coord(R), curve_y_coord(R));
element_invert(e0, e0);
element_mul(out, out, e0);
//element_pow_mpz(out, out, p->tateexp);
element_clear(R);
element_clear(QR);
}
static void d_pairing_pp_apply(element_ptr out, element_ptr in2,
pairing_pp_t p) {
mpz_ptr q = p->pairing->r;
pptr info = p->pairing->data;
mp_bitcnt_t m = (mp_bitcnt_t)mpz_sizeinbase(q, 2);
m = (m > 2 ? m - 2 : 0);
pp_coeff_t *coeff = (pp_coeff_t *) p->data;
pp_coeff_ptr pp = coeff[0];
element_ptr Qbase = in2;
element_t e0;
element_t Qx, Qy;
element_t v;
element_init_same_as(e0, out);
element_init_same_as(v, out);
element_init(Qx, info->Fqd);
element_init(Qy, info->Fqd);
// Twist: (x, y) --> (v^-1 x, v^-(3/2) y)
// where v is the quadratic nonresidue used to construct the twist
element_mul(Qx, curve_x_coord(Qbase), info->nqrinv);
// v^-3/2 = v^-2 * v^1/2
element_mul(Qy, curve_y_coord(Qbase), info->nqrinv2);
element_set1(out);
for(;;) {
d_miller_evalfn(e0, pp->a, pp->b, pp->c, Qx, Qy);
element_mul(out, out, e0);
pp++;
if (!m) break;
if (mpz_tstbit(q, m)) {
d_miller_evalfn(e0, pp->a, pp->b, pp->c, Qx, Qy);
element_mul(out, out, e0);
pp++;
}
m--;
element_square(out, out);
}
cc_tatepower(out, out, p->pairing);
element_clear(e0);
element_clear(Qx);
element_clear(Qy);
element_clear(v);
}
static void cc_pairing(element_ptr out, element_ptr in1, element_ptr in2,
pairing_t pairing) {
element_ptr Qbase = in2;
element_t Qx, Qy;
pptr p = pairing->data;
element_init(Qx, p->Fqd);
element_init(Qy, p->Fqd);
// Twist: (x, y) --> (v^-1 x, v^-(3/2) y)
// where v is the quadratic nonresidue used to construct the twist.
element_mul(Qx, curve_x_coord(Qbase), p->nqrinv);
// v^-3/2 = v^-2 * v^1/2
element_mul(Qy, curve_y_coord(Qbase), p->nqrinv2);
cc_miller_no_denom_fn(out, pairing->r, in1, Qx, Qy);
cc_tatepower(out, out, pairing);
element_clear(Qx);
element_clear(Qy);
}
static void f_pairing(element_ptr out, element_ptr in1, element_ptr in2,
pairing_t pairing) {
element_ptr Qbase = in2;
element_t x, y;
f_pairing_data_ptr p = pairing->data;
element_init(x, p->Fq2);
element_init(y, p->Fq2);
//map from twist: (x, y) --> (v^-2 x, v^-3 y)
//where v is the sixth root used to construct the twist
//i.e. v^6 = -alpha
//thus v^-2 = -alpha^-1 v^4
//and v^-3 = -alpha^-1 v^3
element_mul(x, curve_x_coord(Qbase), p->negalphainv);
element_mul(y, curve_y_coord(Qbase), p->negalphainv);
cc_miller_no_denom(out, pairing->r, in1, x, y, p->negalpha);
element_clear(x);
element_clear(y);
f_tateexp(out);
}
// Same as above, but with affine coordinates.
static void cc_miller_no_denom_affine(element_t res, mpz_t q, element_t P,
element_ptr Qx, element_ptr Qy) {
mp_bitcnt_t m;
element_t v;
element_t Z;
element_t a, b, c;
element_t t0;
element_t e0;
const element_ptr cca = curve_a_coeff(P);
const element_ptr Px = curve_x_coord(P);
const element_ptr Py = curve_y_coord(P);
element_ptr Zx, Zy;
/* TODO: when exactly is this not needed?
void do_vertical() {
mapbase(e0, Z->x);
element_sub(e0, Qx, e0);
element_mul(v, v, e0);
}
*/
#define do_tangent() { \
/* a = -(3 Zx^2 + cc->a) */ \
/* b = 2 * Zy */ \
/* c = -(2 Zy^2 + a Zx); */ \
\
element_square(a, Zx); \
element_mul_si(a, a, 3); \
element_add(a, a, cca); \
element_neg(a, a); \
\
element_add(b, Zy, Zy); \
\
element_mul(t0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, t0); \
element_neg(c, c); \
\
d_miller_evalfn(e0, a, b, c, Qx, Qy); \
element_mul(v, v, e0); \
}
#define do_line() { \
/* a = -(B.y - A.y) / (B.x - A.x); */ \
/* b = 1; */ \
/* c = -(A.y + a * A.x); */ \
/* but we multiply by B.x - A.x to avoid division. */ \
\
element_sub(b, Px, Zx); \
element_sub(a, Zy, Py); \
element_mul(t0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, t0); \
element_neg(c, c); \
\
d_miller_evalfn(e0, a, b, c, Qx, Qy); \
element_mul(v, v, e0); \
}
element_init(a, Px->field);
element_init(b, a->field);
element_init(c, a->field);
element_init(t0, a->field);
element_init(e0, res->field);
element_init(v, res->field);
element_init(Z, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_set1(v);
m = (mp_bitcnt_t)mpz_sizeinbase(q, 2);
m = (m > 2 ? m - 2 : 0);
for(;;) {
do_tangent();
if (!m) break;
element_double(Z, Z);
if (mpz_tstbit(q, m)) {
do_line();
element_add(Z, Z, P);
}
m--;
element_square(v, v);
}
element_set(res, v);
element_clear(v);
element_clear(Z);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(t0);
element_clear(e0);
#undef do_tangent
#undef do_line
}
static void miller(element_t res, element_t P, element_ptr QR, element_ptr R, int n) {
// Collate divisions.
mp_bitcnt_t m;
element_t v, vd;
element_t Z;
element_t a, b, c;
const element_ptr cca = curve_a_coeff(P);
const element_ptr Px = curve_x_coord(P);
const element_ptr Py = curve_y_coord(P);
element_t e0, e1;
mpz_t q;
element_ptr Zx, Zy;
const element_ptr numx = curve_x_coord(QR);
const element_ptr numy = curve_y_coord(QR);
const element_ptr denomx = curve_x_coord(R);
const element_ptr denomy = curve_y_coord(R);
#define do_vertical(e, edenom) { \
element_sub(e0, numx, Zx); \
element_mul((e), (e), e0); \
\
element_sub(e0, denomx, Zx); \
element_mul((edenom), (edenom), e0); \
}
#define do_tangent(e, edenom) { \
/*a = -slope_tangent(A.x, A.y); \
b = 1; \
c = -(A.y + a * A.x); \
but we multiply by 2*A.y to avoid division*/ \
\
/*a = -Ax * (Ax + Ax + Ax + twicea_2) - a_4; \
Common curves: a2 = 0 (and cc->a is a_4), so \
a = -(3 Ax^2 + cc->a) \
b = 2 * Ay \
c = -(2 Ay^2 + a Ax); */ \
\
if (element_is0(Zy)) { \
do_vertical((e), (edenom)); \
} else { \
element_square(a, Zx); \
element_mul_si(a, a, 3); \
element_add(a, a, cca); \
element_neg(a, a); \
\
element_add(b, Zy, Zy); \
\
element_mul(e0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, e0); \
element_neg(c, c); \
\
element_mul(e0, a, numx); \
element_mul(e1, b, numy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul((e), (e), e0); \
\
element_mul(e0, a, denomx); \
element_mul(e1, b, denomy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul((edenom), (edenom), e0); \
} \
}
#define do_line(e, edenom) { \
if (!element_cmp(Zx, Px)) { \
if (!element_cmp(Zy, Py)) { \
do_tangent(e, edenom); \
} else { \
do_vertical(e, edenom); \
} \
} else { \
element_sub(b, Px, Zx); \
element_sub(a, Zy, Py); \
element_mul(c, Zx, Py); \
element_mul(e0, Zy, Px); \
element_sub(c, c, e0); \
\
element_mul(e0, a, numx); \
element_mul(e1, b, numy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(e, e, e0); \
\
element_mul(e0, a, denomx); \
element_mul(e1, b, denomy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(edenom, edenom, e0); \
} \
}
element_init(a, res->field);
element_init(b, res->field);
element_init(c, res->field);
element_init(e0, res->field);
element_init(e1, res->field);
element_init(v, res->field);
element_init(vd, res->field);
element_init(Z, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_set1(v);
element_set1(vd);
mpz_init(q);
mpz_set_ui(q, n);
m = (mp_bitcnt_t)mpz_sizeinbase(q, 2);
m = (m > 2 ? m - 2 : 0);
for (;;) {
element_square(v, v);
element_square(vd, vd);
do_tangent(v, vd);
element_double(Z, Z);
do_vertical(vd, v);
if (mpz_tstbit(q, m)) {
do_line(v, vd);
element_add(Z, Z, P);
if (m) {
do_vertical(vd, v);
}
}
if (!m) break;
m--;
}
mpz_clear(q);
element_invert(vd, vd);
element_mul(res, v, vd);
element_clear(v);
element_clear(vd);
element_clear(Z);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(e0);
element_clear(e1);
#undef do_vertical
#undef do_tangent
#undef do_line
}
static void e_miller_proj(element_t res, element_t P,
element_ptr QR, element_ptr R,
e_pairing_data_ptr p) {
//collate divisions
int n;
element_t v, vd;
element_t v1, vd1;
element_t Z, Z1;
element_t a, b, c;
const element_ptr cca = curve_a_coeff(P);
element_t e0, e1;
const element_ptr e2 = a, e3 = b;
element_t z, z2;
int i;
element_ptr Zx, Zy;
const element_ptr Px = curve_x_coord(P);
const element_ptr numx = curve_x_coord(QR);
const element_ptr numy = curve_y_coord(QR);
const element_ptr denomx = curve_x_coord(R);
const element_ptr denomy = curve_y_coord(R);
//convert Z from weighted projective (Jacobian) to affine
//i.e. (X, Y, Z) --> (X/Z^2, Y/Z^3)
//also sets z to 1
#define to_affine() { \
element_invert(z, z); \
element_square(e0, z); \
element_mul(Zx, Zx, e0); \
element_mul(e0, e0, z); \
element_mul(Zy, Zy, e0); \
element_set1(z); \
element_set1(z2); \
}
#define proj_double() { \
const element_ptr x = Zx; \
const element_ptr y = Zy; \
/* e0 = 3x^2 + (cc->a) z^4 */ \
element_square(e0, x); \
/* element_mul_si(e0, e0, 3); */ \
element_double(e1, e0); \
element_add(e0, e0, e1); \
element_square(e1, z2); \
element_mul(e1, e1, cca); \
element_add(e0, e0, e1); \
\
/* z_out = 2 y z */ \
element_mul(z, y, z); \
/* element_mul_si(z, z, 2); */ \
element_double(z, z); \
element_square(z2, z); \
\
/* e1 = 4 x y^2 */ \
element_square(e2, y); \
element_mul(e1, x, e2); \
/* element_mul_si(e1, e1, 4); */ \
element_double(e1, e1); \
element_double(e1, e1); \
\
/* x_out = e0^2 - 2 e1 */ \
/* element_mul_si(e3, e1, 2); */ \
element_double(e3, e1); \
element_square(x, e0); \
element_sub(x, x, e3); \
\
/* e2 = 8y^4 */ \
element_square(e2, e2); \
/* element_mul_si(e2, e2, 8); */ \
element_double(e2, e2); \
element_double(e2, e2); \
element_double(e2, e2); \
\
/* y_out = e0(e1 - x_out) - e2 */ \
element_sub(e1, e1, x); \
element_mul(e0, e0, e1); \
element_sub(y, e0, e2); \
}
#define do_tangent(e, edenom) { \
/* a = -(3x^2 + cca z^4) */ \
/* b = 2 y z^3 */ \
/* c = -(2 y^2 + x a) */ \
/* a = z^2 a */ \
element_square(a, z2); \
element_mul(a, a, cca); \
element_square(b, Zx); \
/* element_mul_si(b, b, 3); */ \
element_double(e0, b); \
element_add(b, b, e0); \
element_add(a, a, b); \
element_neg(a, a); \
\
/* element_mul_si(e0, Zy, 2); */ \
element_double(e0, Zy); \
element_mul(b, e0, z2); \
element_mul(b, b, z); \
\
element_mul(c, Zx, a); \
element_mul(a, a, z2); \
element_mul(e0, e0, Zy); \
element_add(c, c, e0); \
element_neg(c, c); \
\
element_mul(e0, a, numx); \
element_mul(e1, b, numy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(e, e, e0); \
\
element_mul(e0, a, denomx); \
element_mul(e1, b, denomy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(edenom, edenom, e0); \
}
#define do_vertical(e, edenom, Ax) { \
element_mul(e0, numx, z2); \
element_sub(e0, e0, Ax); \
element_mul(e, e, e0); \
\
element_mul(e0, denomx, z2); \
element_sub(e0, e0, Ax); \
element_mul(edenom, edenom, e0); \
}
#define do_line(e, edenom, A, B) { \
element_ptr Ax = curve_x_coord(A); \
element_ptr Ay = curve_y_coord(A); \
element_ptr Bx = curve_x_coord(B); \
element_ptr By = curve_y_coord(B); \
\
element_sub(b, Bx, Ax); \
element_sub(a, Ay, By); \
element_mul(c, Ax, By); \
element_mul(e0, Ay, Bx); \
element_sub(c, c, e0); \
\
element_mul(e0, a, numx); \
element_mul(e1, b, numy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(e, e, e0); \
\
element_mul(e0, a, denomx); \
element_mul(e1, b, denomy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(edenom, edenom, e0); \
}
element_init(a, res->field);
element_init(b, res->field);
element_init(c, res->field);
element_init(e0, res->field);
element_init(e1, res->field);
element_init(z, res->field);
element_init(z2, res->field);
element_set1(z);
element_set1(z2);
element_init(v, res->field);
element_init(vd, res->field);
element_init(v1, res->field);
element_init(vd1, res->field);
element_init(Z, P->field);
element_init(Z1, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_set1(v);
element_set1(vd);
element_set1(v1);
element_set1(vd1);
n = p->exp1;
for (i=0; i<n; i++) {
element_square(v, v);
element_square(vd, vd);
do_tangent(v, vd);
proj_double();
do_vertical(vd, v, Zx);
}
to_affine();
if (p->sign1 < 0) {
element_set(v1, vd);
element_set(vd1, v);
do_vertical(vd1, v1, Zx);
element_neg(Z1, Z);
} else {
element_set(v1, v);
element_set(vd1, vd);
element_set(Z1, Z);
}
n = p->exp2;
for (; i<n; i++) {
element_square(v, v);
element_square(vd, vd);
do_tangent(v, vd);
proj_double();
do_vertical(vd, v, Zx);
}
to_affine();
element_mul(v, v, v1);
element_mul(vd, vd, vd1);
do_line(v, vd, Z, Z1);
element_add(Z, Z, Z1);
do_vertical(vd, v, Zx);
if (p->sign0 > 0) {
do_vertical(v, vd, Px);
}
element_invert(vd, vd);
element_mul(res, v, vd);
element_clear(v);
element_clear(vd);
element_clear(v1);
element_clear(vd1);
element_clear(z);
element_clear(z2);
element_clear(Z);
element_clear(Z1);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(e0);
element_clear(e1);
#undef to_affine
#undef proj_double
#undef do_tangent
#undef do_vertical
#undef do_line
}
//TODO: the following code is useless as the Tate pairing is degenerate on singular curves
static void sn_miller(element_t res, mpz_t q, element_t P,
element_ptr Qx, element_ptr Qy) {
//collate divisions
int m;
element_t v, vd;
element_t Z;
element_t a, b, c;
element_t e0, e1;
element_ptr Zx;
element_ptr Zy;
const element_ptr Px = curve_x_coord(P);
const element_ptr Py = curve_y_coord(P);
#define do_vertical(e) \
element_sub(e0, Qx, Zx); \
element_mul(e, e, e0);
//a = -slope_tangent(Z.x, Z.y);
//b = 1;
//c = -(Z.y + a * Z.x);
//but we multiply by 2*Z.y to avoid division
//a = -Zx * (Zx + Zx + Zx + 2)
//b = 2 * Zy
//c = -(2 Zy^2 + a Zx);
#define do_tangent(e) \
element_double(e0, Zx); \
element_add(a, Zx, e0); \
element_set_si(e0, 2); \
element_add(a, a, e0); \
element_mul(a, a, Zx); \
element_neg(a, a); \
element_add(b, Zy, Zy); \
element_mul(e0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, e0); \
element_neg(c, c); \
element_mul(e0, a, Qx); \
element_mul(e1, b, Qy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(e, e, e0);
//a = -(B.y - A.y) / (B.x - A.x);
//b = 1;
//c = -(A.y + a * A.x);
//but we'll multiply by B.x - A.x to avoid division
#define do_line(e) \
element_sub(b, Px, Zx); \
element_sub(a, Zy, Py); \
element_mul(e0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, e0); \
element_neg(c, c); \
element_mul(e0, a, Qx); \
element_mul(e1, b, Qy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(e, e, e0);
element_init(a, Px->field);
element_init(b, Px->field);
element_init(c, Px->field);
element_init(e0, res->field);
element_init(e1, res->field);
element_init(v, res->field);
element_init(vd, res->field);
element_init(Z, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_set1(v);
element_set1(vd);
m = mpz_sizeinbase(q, 2) - 2;
while(m >= 0) {
element_mul(v, v, v);
element_mul(vd, vd, vd);
do_tangent(v);
element_double(Z, Z);
do_vertical(vd);
if (mpz_tstbit(q, m)) {
do_line(v);
element_add(Z, Z, P);
do_vertical(vd);
}
m--;
}
#undef do_tangent
#undef do_vertical
#undef do_line
element_invert(vd, vd);
element_mul(res, v, vd);
element_clear(v);
element_clear(vd);
element_clear(Z);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(e0);
element_clear(e1);
}
static void e_miller_affine(element_t res, element_t P,
element_ptr QR, element_ptr R,
e_pairing_data_ptr p) {
//collate divisions
int n;
element_t v, vd;
element_t v1, vd1;
element_t Z, Z1;
element_t a, b, c;
element_t e0, e1;
const element_ptr Px = curve_x_coord(P);
const element_ptr cca = curve_a_coeff(P);
element_ptr Zx, Zy;
int i;
const element_ptr numx = curve_x_coord(QR);
const element_ptr numy = curve_y_coord(QR);
const element_ptr denomx = curve_x_coord(R);
const element_ptr denomy = curve_y_coord(R);
#define do_vertical(e, edenom, Ax) { \
element_sub(e0, numx, Ax); \
element_mul(e, e, e0); \
\
element_sub(e0, denomx, Ax); \
element_mul(edenom, edenom, e0); \
}
#define do_tangent(e, edenom) { \
/* a = -slope_tangent(A.x, A.y); */ \
/* b = 1; */ \
/* c = -(A.y + a * A.x); */ \
/* but we multiply by 2*A.y to avoid division */ \
\
/* a = -Ax * (Ax + Ax + Ax + twicea_2) - a_4; */ \
/* Common curves: a2 = 0 (and cc->a is a_4), so */ \
/* a = -(3 Ax^2 + cc->a) */ \
/* b = 2 * Ay */ \
/* c = -(2 Ay^2 + a Ax); */ \
\
element_square(a, Zx); \
element_mul_si(a, a, 3); \
element_add(a, a, cca); \
element_neg(a, a); \
\
element_add(b, Zy, Zy); \
\
element_mul(e0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, e0); \
element_neg(c, c); \
\
element_mul(e0, a, numx); \
element_mul(e1, b, numy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(e, e, e0); \
\
element_mul(e0, a, denomx); \
element_mul(e1, b, denomy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(edenom, edenom, e0); \
}
#define do_line(e, edenom, A, B) { \
element_ptr Ax = curve_x_coord(A); \
element_ptr Ay = curve_y_coord(A); \
element_ptr Bx = curve_x_coord(B); \
element_ptr By = curve_y_coord(B); \
\
element_sub(b, Bx, Ax); \
element_sub(a, Ay, By); \
element_mul(c, Ax, By); \
element_mul(e0, Ay, Bx); \
element_sub(c, c, e0); \
\
element_mul(e0, a, numx); \
element_mul(e1, b, numy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(e, e, e0); \
\
element_mul(e0, a, denomx); \
element_mul(e1, b, denomy); \
element_add(e0, e0, e1); \
element_add(e0, e0, c); \
element_mul(edenom, edenom, e0); \
}
element_init(a, res->field);
element_init(b, res->field);
element_init(c, res->field);
element_init(e0, res->field);
element_init(e1, res->field);
element_init(v, res->field);
element_init(vd, res->field);
element_init(v1, res->field);
element_init(vd1, res->field);
element_init(Z, P->field);
element_init(Z1, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_set1(v);
element_set1(vd);
element_set1(v1);
element_set1(vd1);
n = p->exp1;
for (i=0; i<n; i++) {
element_square(v, v);
element_square(vd, vd);
do_tangent(v, vd);
element_double(Z, Z);
do_vertical(vd, v, Zx);
}
if (p->sign1 < 0) {
element_set(v1, vd);
element_set(vd1, v);
do_vertical(vd1, v1, Zx);
element_neg(Z1, Z);
} else {
element_set(v1, v);
element_set(vd1, vd);
element_set(Z1, Z);
}
n = p->exp2;
for (; i<n; i++) {
element_square(v, v);
element_square(vd, vd);
do_tangent(v, vd);
element_double(Z, Z);
do_vertical(vd, v, Zx);
}
element_mul(v, v, v1);
element_mul(vd, vd, vd1);
do_line(v, vd, Z, Z1);
element_add(Z, Z, Z1);
do_vertical(vd, v, Zx);
if (p->sign0 > 0) {
do_vertical(v, vd, Px);
}
element_invert(vd, vd);
element_mul(res, v, vd);
element_clear(v);
element_clear(vd);
element_clear(v1);
element_clear(vd1);
element_clear(Z);
element_clear(Z1);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(e0);
element_clear(e1);
#undef do_vertical
#undef do_tangent
#undef do_line
}
// in1, in2 are from E(F_q), out from F_q^2.
// Pairing via elliptic nets (see Stange).
static void e_pairing_ellnet(element_ptr out, element_ptr in1, element_ptr in2,
pairing_t pairing) {
const element_ptr a = curve_a_coeff(in1);
const element_ptr b = curve_b_coeff(in1);
element_ptr x = curve_x_coord(in1);
element_ptr y = curve_y_coord(in1);
element_ptr x2 = curve_x_coord(in2);
element_ptr y2 = curve_y_coord(in2);
//notation: cmi means c_{k-i}, ci means c_{k+i}
element_t cm3, cm2, cm1, c0, c1, c2, c3, c4;
element_t dm1, d0, d1;
element_t A, B, C;
element_init_same_as(cm3, x);
element_init_same_as(cm2, x);
element_init_same_as(cm1, x);
element_init_same_as(c0, x);
element_init_same_as(c1, x);
element_init_same_as(c2, x);
element_init_same_as(c3, x);
element_init_same_as(c4, x);
element_init_same_as(C, x);
element_init_same_as(dm1, out);
element_init_same_as(d0, out);
element_init_same_as(d1, out);
element_init_same_as(A, x);
element_init_same_as(B, out);
// c1 = 2y
// cm3 = -2y
element_double(c1, y);
element_neg(cm3, c1);
//use c0, cm1, cm2, C, c4 as temp variables for now
//compute c3, c2
element_square(cm2, x);
element_square(C, cm2);
element_mul(cm1, b, x);
element_double(cm1, cm1);
element_square(c4, a);
element_mul(c2, cm1, cm2);
element_double(c2, c2);
element_mul(c0, a, C);
element_add(c2, c2, c0);
element_mul(c0, c4, cm2);
element_sub(c2, c2, c0);
element_double(c0, c2);
element_double(c0, c0);
element_add(c2, c2, c0);
element_mul(c0, cm1, a);
element_square(c3, b);
element_double(c3, c3);
element_double(c3, c3);
element_add(c0, c0, c3);
element_double(c0, c0);
element_mul(c3, a, c4);
element_add(c0, c0, c3);
element_sub(c2, c2, c0);
element_mul(c0, cm2, C);
element_add(c3, c0, c2);
element_mul(c3, c3, c1);
element_double(c3, c3);
element_mul(c0, a, cm2);
element_add(c0, c0, cm1);
element_double(c0, c0);
element_add(c0, c0, C);
element_double(c2, c0);
element_add(c0, c0, c2);
element_sub(c2, c0, c4);
// c0 = 1
// cm2 = -1
element_set1(c0);
element_neg(cm2, c0);
// c4 = c_5 = c_2^3 c_4 - c_3^3 = c1^3 c3 - c2^3
element_square(C, c1);
element_mul(c4, C, c1);
element_mul(c4, c4, c3);
element_square(C, c2);
element_mul(C, C, c2);
element_sub(c4, c4, C);
//compute A, B, d1 (which is d_2 since k = 1)
element_sub(A, x, x2);
element_double(C, x);
element_add(C, C, x2);
element_square(cm1, A);
element_mul(cm1, C, cm1);
element_add(d1, y, y2);
element_square(d1, d1);
element_sub(B, cm1, d1);
element_invert(B, B);
element_invert(A, A);
element_sub(d1, y, y2);
element_mul(d1, d1, A);
element_square(d1, d1);
element_sub(d1, C, d1);
// cm1 = 0
// C = (2y)^-1
element_set0(cm1);
element_invert(C, c1);
element_set1(dm1);
element_set1(d0);
element_t sm2, sm1;
element_t s0, s1, s2, s3;
element_t tm2, tm1;
element_t t0, t1, t2, t3;
element_t e0, e1;
element_t u, v;
element_init_same_as(sm2, x);
element_init_same_as(sm1, x);
element_init_same_as(s0, x);
element_init_same_as(s1, x);
element_init_same_as(s2, x);
element_init_same_as(s3, x);
element_init_same_as(tm2, x);
element_init_same_as(tm1, x);
element_init_same_as(t0, x);
element_init_same_as(t1, x);
element_init_same_as(t2, x);
element_init_same_as(t3, x);
element_init_same_as(e0, x);
element_init_same_as(e1, x);
element_init_same_as(u, d0);
element_init_same_as(v, d0);
int m = mpz_sizeinbase(pairing->r, 2) - 2;
for (;;) {
element_square(sm2, cm2);
element_square(sm1, cm1);
element_square(s0, c0);
element_square(s1, c1);
element_square(s2, c2);
element_square(s3, c3);
element_mul(tm2, cm3, cm1);
element_mul(tm1, cm2, c0);
element_mul(t0, cm1, c1);
element_mul(t1, c0, c2);
element_mul(t2, c1, c3);
element_mul(t3, c2, c4);
element_square(u, d0);
element_mul(v, dm1, d1);
if (mpz_tstbit(pairing->r, m)) {
//double-and-add
element_mul(e0, t0, sm2);
element_mul(e1, tm2, s0);
element_sub(cm3, e0, e1);
element_mul(cm3, cm3, C);
element_mul(e0, t0, sm1);
element_mul(e1, tm1, s0);
element_sub(cm2, e0, e1);
element_mul(e0, t1, sm1);
element_mul(e1, tm1, s1);
element_sub(cm1, e0, e1);
element_mul(cm1, cm1, C);
element_mul(e0, t1, s0);
element_mul(e1, t0, s1);
element_sub(c0, e0, e1);
element_mul(e0, t2, s0);
element_mul(e1, t0, s2);
element_sub(c1, e0, e1);
element_mul(c1, c1, C);
element_mul(e0, t2, s1);
element_mul(e1, t1, s2);
element_sub(c2, e0, e1);
element_mul(e0, t3, s1);
element_mul(e1, t1, s3);
element_sub(c3, e0, e1);
element_mul(c3, c3, C);
element_mul(e0, t3, s2);
element_mul(e1, t2, s3);
element_sub(c4, e0, e1);
element_mul(out, u, t0);
element_mul(dm1, v, s0);
element_sub(dm1, dm1, out);
element_mul(out, u, t1);
element_mul(d0, v, s1);
element_sub(d0, d0, out);
element_mul(d0, d0, A);
element_mul(out, u, t2);
element_mul(d1, v, s2);
element_sub(d1, d1, out);
element_mul(d1, d1, B);
} else {
//double
element_mul(e0, tm1, sm2);
element_mul(e1, tm2, sm1);
element_sub(cm3, e0, e1);
element_mul(e0, t0, sm2);
element_mul(e1, tm2, s0);
element_sub(cm2, e0, e1);
element_mul(cm2, cm2, C);
element_mul(e0, t0, sm1);
element_mul(e1, tm1, s0);
element_sub(cm1, e0, e1);
element_mul(e0, t1, sm1);
element_mul(e1, tm1, s1);
element_sub(c0, e0, e1);
element_mul(c0, c0, C);
element_mul(e0, t1, s0);
element_mul(e1, t0, s1);
element_sub(c1, e0, e1);
element_mul(e0, t2, s0);
element_mul(e1, t0, s2);
element_sub(c2, e0, e1);
element_mul(c2, c2, C);
element_mul(e0, t2, s1);
element_mul(e1, t1, s2);
element_sub(c3, e0, e1);
element_mul(e0, t3, s1);
element_mul(e1, t1, s3);
element_sub(c4, e0, e1);
element_mul(c4, c4, C);
element_mul(out, u, tm1);
element_mul(dm1, v, sm1);
element_sub(dm1, dm1, out);
element_mul(out, u, t0);
element_mul(d0, v, s0);
element_sub(d0, d0, out);
element_mul(out, u, t1);
element_mul(d1, v, s1);
element_sub(d1, d1, out);
element_mul(d1, d1, A);
}
if (!m) break;
m--;
}
element_invert(c1, c1);
element_mul(d1, d1, c1);
element_pow_mpz(out, d1, pairing->phikonr);
element_clear(dm1);
element_clear(d0);
element_clear(d1);
element_clear(cm3);
element_clear(cm2);
element_clear(cm1);
element_clear(c0);
element_clear(c1);
element_clear(c2);
element_clear(c3);
element_clear(c4);
element_clear(sm2);
element_clear(sm1);
element_clear(s0);
element_clear(s1);
element_clear(s2);
element_clear(s3);
element_clear(tm2);
element_clear(tm1);
element_clear(t0);
element_clear(t1);
element_clear(t2);
element_clear(t3);
element_clear(e0);
element_clear(e1);
element_clear(A);
element_clear(B);
element_clear(C);
element_clear(u);
element_clear(v);
}
static void d_pairing_pp_init(pairing_pp_t p, element_ptr in1, pairing_t pairing) {
element_ptr P = in1;
const element_ptr Px = curve_x_coord(P);
const element_ptr Py = curve_y_coord(P);
element_t Z;
mp_bitcnt_t m;
pptr info = pairing->data;
element_t t0;
element_t a, b, c;
field_ptr Fq = info->Fq;
pp_coeff_t *coeff;
mpz_ptr q = pairing->r;
pp_coeff_ptr pp;
const element_ptr cca = curve_a_coeff(P);
element_ptr Zx;
element_ptr Zy;
#define store_abc() { \
element_init(pp->a, Fq); \
element_init(pp->b, Fq); \
element_init(pp->c, Fq); \
element_set(pp->a, a); \
element_set(pp->b, b); \
element_set(pp->c, c); \
pp++; \
}
#define do_tangent() { \
/* a = -slope_tangent(Z.x, Z.y); */ \
/* b = 1; */ \
/* c = -(Z.y + a * Z.x); */ \
/* but we multiply by 2*Z.y to avoid division. */ \
\
/* a = -Zx * (3 Zx + twicea_2) - a_4; */ \
/* Common curves: a2 = 0 (and cc->a is a_4), so */ \
/* a = -(3 Zx^2 + cc->a) */ \
/* b = 2 * Zy */ \
/* c = -(2 Zy^2 + a Zx); */ \
\
element_square(a, Zx); \
element_double(t0, a); \
element_add(a, a, t0); \
element_add(a, a, cca); \
element_neg(a, a); \
\
element_add(b, Zy, Zy); \
\
element_mul(t0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, t0); \
element_neg(c, c); \
\
store_abc(); \
}
#define do_line() { \
/* a = -(B.y - A.y) / (B.x - A.x); */ \
/* b = 1; */ \
/* c = -(A.y + a * A.x); */ \
/* but we'll multiply by B.x - A.x to avoid division */ \
\
element_sub(b, Px, Zx); \
element_sub(a, Zy, Py); \
element_mul(t0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, t0); \
element_neg(c, c); \
\
store_abc(); \
}
element_init(Z, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_init(t0, Fq);
element_init(a, Fq);
element_init(b, Fq);
element_init(c, Fq);
m = (mp_bitcnt_t)mpz_sizeinbase(q, 2);
m = (m > 2 ? m - 2 : 0);
p->data = pbc_malloc(sizeof(pp_coeff_t) * 2 * m);
coeff = (pp_coeff_t *) p->data;
pp = coeff[0];
for(;;) {
do_tangent();
if (!m) break;
element_double(Z, Z);
if (mpz_tstbit(q, m)) {
do_line();
element_add(Z, Z, P);
}
m--;
}
element_clear(t0);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(Z);
#undef store_abc
#undef do_tangent
#undef do_line
}
element_init(e1, res->field);
element_init(z, res->field);
element_init(z2, res->field);
element_set1(z);
element_set1(z2);
element_init(v, res->field);
element_init(vd, res->field);
element_init(v1, res->field);
element_init(vd1, res->field);
element_init(Z, P->field);
element_init(Z1, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_set1(v);
element_set1(vd);
element_set1(v1);
element_set1(vd1);
n = p->exp1;
for (i=0; i<n; i++) {
element_square(v, v);
element_square(vd, vd);
do_tangent(v, vd);
proj_double();
do_vertical(vd, v, Zx);
}
to_affine();
void shipseystange(element_t z, element_t P, element_t Q)
{
mpz_t q1r;
mpz_init(q1r);
mpz_set_ui(q1r, 696);
element_ptr x = curve_x_coord(P);
element_ptr y = curve_y_coord(P);
element_ptr x2 = curve_x_coord(Q);
element_ptr y2 = curve_y_coord(Q);
element_t v0m1, v0m2, v0m3;
element_t v00, v01, v02, v03, v04;
element_t v1m1, v10, v11;
element_t t0, t1, t2;
element_t W20inv;
element_t Wm11inv;
element_t W2m1inv;
element_t sm2, sm1, s0, s1, s2, s3;
element_t pm2, pm1, p0, p1, p2, p3;
element_init_same_as(sm2, z);
element_init_same_as(sm1, z);
element_init_same_as(s0, z);
element_init_same_as(s1, z);
element_init_same_as(s2, z);
element_init_same_as(s3, z);
element_init_same_as(pm2, z);
element_init_same_as(pm1, z);
element_init_same_as(p0, z);
element_init_same_as(p1, z);
element_init_same_as(p2, z);
element_init_same_as(p3, z);
element_init_same_as(v0m3, z);
element_init_same_as(v0m2, z);
element_init_same_as(v0m1, z);
element_init_same_as(v00, z);
element_init_same_as(v01, z);
element_init_same_as(v02, z);
element_init_same_as(v03, z);
element_init_same_as(v04, z);
element_init_same_as(v1m1, z);
element_init_same_as(v10, z);
element_init_same_as(v11, z);
element_init_same_as(W20inv, z);
element_init_same_as(Wm11inv, z);
element_init_same_as(W2m1inv, z);
element_init_same_as(t0, z);
element_init_same_as(t1, z);
element_init_same_as(t2, z);
element_set0(v0m1);
element_set1(v00);
element_neg(v0m2, v00);
element_double(v01, y);
element_neg(v0m3, v01);
element_invert(W20inv, v01);
element_sub(Wm11inv, x, x2);
element_square(t1, Wm11inv);
element_invert(Wm11inv, Wm11inv);
element_double(t0, x);
element_add(t0, t0, x2);
element_mul(t1, t0, t1);
element_add(t0, y, y2);
element_square(t0, t0);
element_sub(t0, t0, t1);
element_invert(W2m1inv, t0);
/* Let P=(x,y) since A=1, B=0 we have:
* W(3,0) = 3x^4 + 6x^2 - 1
* W(4,0) = 4y(x^6 + 5x^4 - 5x^2 - 1)
*/
//t0 = x^2
element_square(t0, x);
//t1 = x^4
element_square(t1, t0);
//t2 = x^4 + 2 x^2
element_double(t2, t0);
element_add(t2, t2, t1);
//v02 = W(3,0)
element_double(v02, t2);
element_add(v02, v02, t2);
element_add(v02, v02, v0m2);
//t2 = x^4 - x^2
element_sub(t2, t1, t0);
//v03 = 5(x^4 - x^2)
element_double(v03, t2);
element_double(v03, v03);
element_add(v03, v03, t2);
//t2 = x^6
element_mul(t2, t0, t1);
//v03 = W(4,0)
element_add(v03, v03, t2);
element_add(v03, v03, v0m2);
element_double(v03, v03);
element_double(v03, v03);
element_mul(v03, v03, y);
//v04 = W(5,0) = W(2,0)^3 W(4,0) - W(3,0)^3
element_square(t0, v01);
element_mul(t0, t0, v01);
element_mul(v04, t0, v03);
element_square(t0, v02);
element_mul(t0, t0, v02);
element_sub(v04, v04, t0);
element_set1(v1m1);
element_set1(v10);
element_printf("x y: %B %B\n", x, y);
element_printf("x2 y2: %B %B\n", x2, y2);
element_sub(t0, x2, x);
element_sub(t1, y2, y);
element_div(t0, t1, t0);
element_square(t0, t0);
element_double(v11, x);
element_add(v11, v11, x2);
element_sub(v11, v11, t0);
element_printf("VEC1: %B %B %B\n", v1m1, v10, v11);
element_printf("VEC0: %B %B %B %B %B %B %B %B\n",
v0m3, v0m2, v0m1, v00, v01, v02, v03, v04);
//Double
element_square(sm2, v0m2);
element_square(sm1, v0m1);
element_square(s0, v00);
element_square(s1, v01);
element_square(s2, v02);
element_square(s3, v03);
element_mul(pm2, v0m3, v0m1);
element_mul(pm1, v0m2, v00);
element_mul(p0, v0m1, v01);
element_mul(p1, v00, v02);
element_mul(p2, v01, v03);
element_mul(p3, v02, v04);
element_mul(t0, pm1, sm2);
element_mul(t1, pm2, sm1);
element_sub(v0m3, t0, t1);
element_mul(t1, pm2, s0);
element_mul(t0, p0, sm2);
element_sub(v0m2, t0, t1);
element_mul(v0m2, v0m2, W20inv);
element_mul(t0, p0, sm1);
element_mul(t1, pm1, s0);
element_sub(v0m1, t0, t1);
element_mul(t1, pm1, s1);
element_mul(t0, p1, sm1);
element_sub(v00, t0, t1);
element_mul(v00, v00, W20inv);
element_mul(t0, p1, s0);
element_mul(t1, p0, s1);
element_sub(v01, t0, t1);
element_mul(t1, p0, s2);
element_mul(t0, p2, s0);
element_sub(v02, t0, t1);
element_mul(v02, v02, W20inv);
element_mul(t0, p2, s1);
element_mul(t1, p1, s2);
element_sub(v03, t0, t1);
element_mul(t1, p1, s3);
element_mul(t0, p3, s1);
element_sub(v04, t0, t1);
element_mul(v04, v04, W20inv);
element_square(t0, v10);
element_mul(t1, v1m1, v11);
element_mul(t2, pm1, t0);
element_mul(v1m1, t1, sm1);
element_sub(v1m1, v1m1, t2);
element_mul(t2, p0, t0);
element_mul(v10, t1, s0);
element_sub(v10, v10, t2);
element_mul(t2, p1, t0);
element_mul(v11, t1, s1);
element_sub(v11, v11, t2);
element_mul(v11, v11, Wm11inv);
element_printf("VEC1: %B %B %B\n", v1m1, v10, v11);
element_printf("VEC0: %B %B %B %B %B %B %B %B\n",
v0m3, v0m2, v0m1, v00, v01, v02, v03, v04);
//DoubleAdd
element_square(sm2, v0m2);
element_square(sm1, v0m1);
element_square(s0, v00);
element_square(s1, v01);
element_square(s2, v02);
element_square(s3, v03);
element_mul(pm2, v0m3, v0m1);
element_mul(pm1, v0m2, v00);
element_mul(p0, v0m1, v01);
element_mul(p1, v00, v02);
element_mul(p2, v01, v03);
element_mul(p3, v02, v04);
element_mul(t1, pm2, s0);
element_mul(t0, p0, sm2);
element_sub(v0m3, t0, t1);
element_mul(v0m3, v0m3, W20inv);
element_mul(t0, p0, sm1);
element_mul(t1, pm1, s0);
element_sub(v0m2, t0, t1);
element_mul(t1, pm1, s1);
element_mul(t0, p1, sm1);
element_sub(v0m1, t0, t1);
element_mul(v0m1, v0m1, W20inv);
element_mul(t0, p1, s0);
element_mul(t1, p0, s1);
element_sub(v00, t0, t1);
element_mul(t1, p0, s2);
element_mul(t0, p2, s0);
element_sub(v01, t0, t1);
element_mul(v01, v01, W20inv);
element_mul(t0, p2, s1);
element_mul(t1, p1, s2);
element_sub(v02, t0, t1);
element_mul(t1, p1, s3);
element_mul(t0, p3, s1);
element_sub(v03, t0, t1);
element_mul(v03, v03, W20inv);
element_mul(t0, p3, s2);
element_mul(t1, p2, s3);
element_sub(v04, t0, t1);
element_square(t0, v10);
element_mul(t1, v1m1, v11);
element_mul(t2, p0, t0);
element_mul(v1m1, t1, s0);
element_sub(v1m1, v1m1, t2);
element_mul(t2, p1, t0);
element_mul(v10, t1, s1);
element_sub(v10, v10, t2);
element_mul(v10, v10, Wm11inv);
element_mul(t2, t1, s2);
element_mul(v11, p2, t0);
element_sub(v11, v11, t2);
element_mul(v11, v11, W2m1inv);
element_printf("VEC1: %B %B %B\n", v1m1, v10, v11);
element_printf("VEC0: %B %B %B %B %B %B %B %B\n",
v0m3, v0m2, v0m1, v00, v01, v02, v03, v04);
element_div(z, v11, v01);
element_printf("prepow: %B\n", z);
element_pow_mpz(z, z, q1r);
mpz_clear(q1r);
}
// Miller's algorithm, assuming we can ignore the denominator. We can do this
// with careful group selection when the embedding degree is even. See thesis.
// This version uses projective coordinates, which don't seem much faster.
static void cc_miller_no_denom_proj(element_t res, mpz_t q, element_t P,
element_ptr Qx, element_ptr Qy) {
mp_bitcnt_t m;
element_t v;
element_t Z;
element_t a, b, c;
element_t t0, t1;
element_ptr t2 = a, t3 = b, t4 = c;
element_t e0;
element_t z, z2;
element_ptr Zx, Zy;
const element_ptr curve_a = curve_a_coeff(P);
const element_ptr Px = curve_x_coord(P);
const element_ptr Py = curve_y_coord(P);
#define proj_double() { \
/* t0 = 3x^2 + (curve_a) z^4 */ \
element_square(t0, Zx); \
/* element_mul_si(t0, t0, 3); */ \
element_double(t1, t0); \
element_add(t0, t0, t1); \
element_square(t1, z2); \
element_mul(t1, t1, curve_a); \
element_add(t0, t0, t1); \
\
/* z_out = 2 y z */ \
element_mul(z, Zy, z); \
/* element_mul_si(z, z, 2); */ \
element_double(z, z); \
element_square(z2, z); \
\
/* t1 = 4 x y^2 */ \
element_square(t2, Zy); \
element_mul(t1, Zx, t2); \
/* element_mul_si(t1, t1, 4); */ \
element_double(t1, t1); \
element_double(t1, t1); \
\
/* x_out = t0^2 - 2 t1 */ \
/* element_mul_si(t3, t1, 2); */ \
element_double(t3, t1); \
element_square(Zx, t0); \
element_sub(Zx, Zx, t3); \
\
/* t2 = 8y^4 */ \
element_square(t2, t2); \
/* element_mul_si(t2, t2, 8); */ \
element_double(t2, t2); \
element_double(t2, t2); \
element_double(t2, t2); \
\
/* y_out = t0(t1 - x_out) - t2 */ \
element_sub(t1, t1, Zx); \
element_mul(t0, t0, t1); \
element_sub(Zy, t0, t2); \
}
#define proj_mixin() { \
/* t2 = Px z^2 */ \
element_mul(t2, z2, Px); \
\
/* t3 = Zx - t2 */ \
element_sub(t3, Zx, t2); \
\
/* t0 = Py z^3 */ \
element_mul(t0, z2, Py); \
element_mul(t0, t0, z); \
\
/* t1 = Zy - t0 */ \
element_sub(t1, Zy, t0); \
\
/* e7 = Zx + t2, use t2 to double for e7 */ \
element_add(t2, Zx, t2); \
\
/* e8 = Zy + t0, use t0 to double for e8 */ \
element_add(t0, Zy, t0); \
\
/* z = z t3 */ \
element_mul(z, z, t3); \
element_square(z2, z); \
\
/* Zx = t1^2 - e7 t3^2 */ \
/* t3 now holds t3^3, */ \
/* t4 holds e7 t3^2. */ \
element_square(t4, t3); \
element_mul(t3, t4, t3); \
element_square(Zx, t1); \
element_mul(t4, t2, t4); \
element_sub(Zx, Zx, t4); \
\
/* t4 = e7 t3^2 - 2 Zx */ \
element_sub(t4, t4, Zx); \
element_sub(t4, t4, Zx); \
\
/* Zy = (t4 t1 - e8 t3^3)/2 */ \
element_mul(t4, t4, t1); \
element_mul(t0, t0, t3); \
element_sub(t4, t4, t0); \
element_halve(Zy, t4); \
}
#define do_tangent() { \
/* a = -(3x^2 + cca z^4) */ \
/* b = 2 y z^3 */ \
/* c = -(2 y^2 + x a) */ \
/* a = z^2 a */ \
element_square(a, z2); \
element_mul(a, a, curve_a); \
element_square(b, Zx); \
/* element_mul_si(b, b, 3); */ \
element_double(t0, b); \
element_add(b, b, t0); \
element_add(a, a, b); \
element_neg(a, a); \
\
element_mul(b, z, z2); \
element_mul(b, b, Zy); \
element_mul_si(b, b, 2); \
\
element_mul(c, Zx, a); \
element_mul(a, a, z2); \
element_square(t0, Zy); \
element_mul_si(t0, t0, 2); \
element_add(c, c, t0); \
element_neg(c, c); \
\
d_miller_evalfn(e0, a, b, c, Qx, Qy); \
element_mul(v, v, e0); \
}
#define do_line() { \
/* a = -(Py z^3 - Zy) */ \
/* b = Px z^3 - Zx z */ \
/* c = Zx z Py - Zy Px; */ \
\
element_mul(t0, Zx, z); \
element_mul(t1, z2, z); \
\
element_mul(a, Py, t1); \
element_sub(a, Zy, a); \
\
element_mul(b, Px, t1); \
element_sub(b, b, t0); \
\
element_mul(t0, t0, Py); \
element_mul(c, Zy, Px); \
element_sub(c, t0, c); \
\
d_miller_evalfn(e0, a, b, c, Qx, Qy); \
element_mul(v, v, e0); \
}
element_init(a, Px->field);
element_init(b, a->field);
element_init(c, a->field);
element_init(t0, a->field);
element_init(t1, a->field);
element_init(e0, res->field);
element_init(z, a->field);
element_init(z2, a->field);
element_set1(z);
element_set1(z2);
element_init(v, res->field);
element_init(Z, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_x_coord(Z);
element_set1(v);
m = (mp_bitcnt_t)mpz_sizeinbase(q, 2);
m = (m > 2 ? m - 2 : 0);
for(;;) {
do_tangent();
if (!m) break;
proj_double();
if (mpz_tstbit(q, m)) {
do_line();
proj_mixin();
}
m--;
element_square(v, v);
}
element_set(res, v);
element_clear(v);
element_clear(Z);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(t0);
element_clear(t1);
element_clear(e0);
element_clear(z);
element_clear(z2);
#undef proj_double
#undef proj_mixin
#undef do_tangent
#undef do_line
}
static void cc_miller_no_denom(element_t res, mpz_t q, element_t P,
element_ptr Qx, element_ptr Qy, element_t negalpha) {
mp_bitcnt_t m;
element_t v;
element_t Z;
element_t a, b, c;
element_t t0;
element_t e0, e1;
element_ptr Zx, Zy;
const element_ptr Px = curve_x_coord(P);
const element_ptr Py = curve_y_coord(P);
#define do_term(i, j, k, flag) { \
element_ptr e2; \
e2 = element_item(e0, i); \
element_mul(e1, element_item(v, j), Qx); \
if (flag == 1) element_mul(e1, e1, negalpha); \
element_mul(element_x(e1), element_x(e1), a); \
element_mul(element_y(e1), element_y(e1), a); \
element_mul(e2, element_item(v, k), Qy); \
element_mul(element_x(e2), element_x(e2), b); \
element_mul(element_y(e2), element_y(e2), b); \
element_add(e2, e2, e1); \
if (flag == 2) element_mul(e2, e2, negalpha); \
element_mul(element_x(e1), element_x(element_item(v, i)), c); \
element_mul(element_y(e1), element_y(element_item(v, i)), c); \
element_add(e2, e2, e1); \
}
// a, b, c lie in Fq
// Qx, Qy lie in Fq^2
// Qx is coefficient of x^4
// Qy is coefficient of x^3
//
// computes v *= (a Qx x^4 + b Qy x^3 + c)
//
// recall x^6 = -alpha thus
// x^4 (u0 + u1 x^1 + ... + u5 x^5) =
// u0 x^4 + u1 x^5
// - alpha u2 - alpha u3 x - alpha u4 x^2 - alpha u5 x^3
// and
// x^4 (u0 + u1 x^1 + ... + u5 x^5) =
// u0 x^3 + u1 x^4 + u2 x^5
// - alpha u3 - alpha u4 x - alpha u5 x^2
#define f_miller_evalfn() { \
do_term(0, 2, 3, 2); \
do_term(1, 3, 4, 2); \
do_term(2, 4, 5, 2); \
do_term(3, 5, 0, 1); \
do_term(4, 0, 1, 0); \
do_term(5, 1, 2, 0); \
element_set(v, e0); \
}
/*
element_ptr e1;
e1 = element_item(e0, 4);
element_mul(element_x(e1), element_x(Qx), a);
element_mul(element_y(e1), element_y(Qx), a);
e1 = element_item(e0, 3);
element_mul(element_x(e1), element_x(Qy), b);
element_mul(element_y(e1), element_y(Qy), b);
element_set(element_x(element_item(e0, 0)), c);
element_mul(v, v, e0);
*/
//a = -3 Zx^2 since cc->a is 0 for D = 3
//b = 2 * Zy
//c = -(2 Zy^2 + a Zx);
#define do_tangent() { \
element_square(a, Zx); \
element_mul_si(a, a, 3); \
element_neg(a, a); \
\
element_add(b, Zy, Zy); \
\
element_mul(t0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, t0); \
element_neg(c, c); \
\
f_miller_evalfn(); \
}
//a = -(B.y - A.y) / (B.x - A.x);
//b = 1;
//c = -(A.y + a * A.x);
//but we'll multiply by B.x - A.x to avoid division
#define do_line() { \
element_sub(b, Px, Zx); \
element_sub(a, Zy, Py); \
element_mul(t0, b, Zy); \
element_mul(c, a, Zx); \
element_add(c, c, t0); \
element_neg(c, c); \
\
f_miller_evalfn(); \
}
element_init(a, Px->field);
element_init(b, a->field);
element_init(c, a->field);
element_init(t0, a->field);
element_init(e0, res->field);
element_init(e1, Qx->field);
element_init(v, res->field);
element_init(Z, P->field);
element_set(Z, P);
Zx = curve_x_coord(Z);
Zy = curve_y_coord(Z);
element_set1(v);
m = (mp_bitcnt_t)mpz_sizeinbase(q, 2);
m = (m > 2 ? m - 2 : 0);
//TODO: sliding NAF
for(;;) {
do_tangent();
if (!m) break;
element_double(Z, Z);
if (mpz_tstbit(q, m)) {
do_line();
element_add(Z, Z, P);
}
m--;
element_square(v, v);
}
element_set(res, v);
element_clear(v);
element_clear(Z);
element_clear(a);
element_clear(b);
element_clear(c);
element_clear(t0);
element_clear(e0);
element_clear(e1);
#undef do_term
#undef f_miller_evalfn
#undef do_tangent
#undef do_line
}
