/**
* Doubles a point represented in affine coordinates on an ordinary prime
* elliptic curve.
*
* @param[out] r - the result.
* @param[out] s - the resulting slope.
* @param[in] p - the point to double.
*/
static void ep2_dbl_basic_imp(ep2_t r, fp2_t s, ep2_t p) {
fp2_t t0, t1, t2;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
/* t0 = 1/(2 * y1). */
fp2_dbl(t0, p->y);
fp2_inv(t0, t0);
/* t1 = 3 * x1^2 + a. */
fp2_sqr(t1, p->x);
fp2_copy(t2, t1);
fp2_dbl(t1, t1);
fp2_add(t1, t1, t2);
ep2_curve_get_a(t2);
fp2_add(t1, t1, t2);
/* t1 = (3 * x1^2 + a)/(2 * y1). */
fp2_mul(t1, t1, t0);
if (s != NULL) {
fp2_copy(s, t1);
}
/* t2 = t1^2. */
fp2_sqr(t2, t1);
/* x3 = t1^2 - 2 * x1. */
fp2_dbl(t0, p->x);
fp2_sub(t0, t2, t0);
/* y3 = t1 * (x1 - x3) - y1. */
fp2_sub(t2, p->x, t0);
fp2_mul(t1, t1, t2);
fp2_sub(r->y, t1, p->y);
fp2_copy(r->x, t0);
fp2_copy(r->z, p->z);
r->norm = 1;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
}
}
void ep2_neg_projc(ep2_t r, ep2_t p) {
if (ep2_is_infty(p)) {
ep2_set_infty(r);
return;
}
if (r != p) {
fp2_copy(r->x, p->x);
fp2_copy(r->z, p->z);
}
fp2_neg(r->y, p->y);
r->norm = p->norm;
}
void ep2_curve_set(fp2_t a, fp2_t b, ep2_t g, bn_t r, bn_t h) {
ctx_t *ctx = core_get();
ctx->ep2_is_twist = 0;
fp2_copy(ctx->ep2_a, a);
fp2_copy(ctx->ep2_b, b);
ep2_norm(&(ctx->ep2_g), g);
bn_copy(&(ctx->ep2_r), r);
bn_copy(&(ctx->ep2_h), h);
#if defined(EP_PRECO)
ep2_mul_pre((ep2_t *)ep2_curve_get_tab(), &(ctx->ep2_g));
#endif
}
void ep2_rhs(fp2_t rhs, ep2_t p) {
fp2_t t0;
fp2_t t1;
fp2_null(t0);
fp2_null(t1);
TRY {
fp2_new(t0);
fp2_new(t1);
/* t0 = x1^2. */
fp2_sqr(t0, p->x);
/* t1 = x1^3. */
fp2_mul(t1, t0, p->x);
ep2_curve_get_a(t0);
fp2_mul(t0, p->x, t0);
fp2_add(t1, t1, t0);
ep2_curve_get_b(t0);
fp2_add(t1, t1, t0);
fp2_copy(rhs, t1);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t0);
fp2_free(t1);
}
}
void fp2_pck(fp2_t c, fp2_t a) {
int b = fp_get_bit(a[1], 0);
if (fp2_test_uni(c)) {
fp_copy(c[0], a[0]);
fp_zero(c[1]);
fp_set_bit(c[1], 0, b);
} else {
fp2_copy(c, a);
}
}
void fp2_inv_sim(fp2_t *c, fp2_t *a, int n) {
int i;
fp2_t u, t[n];
for (i = 0; i < n; i++) {
fp2_null(t[i]);
}
fp2_null(u);
TRY {
for (i = 0; i < n; i++) {
fp2_new(t[i]);
}
fp2_new(u);
fp2_copy(c[0], a[0]);
fp2_copy(t[0], a[0]);
for (i = 1; i < n; i++) {
fp2_copy(t[i], a[i]);
fp2_mul(c[i], c[i - 1], t[i]);
}
fp2_inv(u, c[n - 1]);
for (i = n - 1; i > 0; i--) {
fp2_mul(c[i], c[i - 1], u);
fp2_mul(u, u, t[i]);
}
fp2_copy(c[0], u);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
for (i = 0; i < n; i++) {
fp2_free(t[i]);
}
fp2_free(u);
}
}
/**
* Doubles a point represented in affine coordinates on an ordinary prime
* elliptic curve.
*
* @param[out] r - the result.
* @param[out] s - the resulting slope.
* @param[in] p - the point to double.
*/
static void ep2_dbl_basic_imp(ep2_t r, fp2_t s, ep2_t p) {
fp2_t t0, t1, t2;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
/* t0 = 1/(2 * y1). */
fp2_dbl(t0, p->y);
fp2_inv(t0, t0);
/* t1 = 3 * x1^2 + a. */
fp2_sqr(t1, p->x);
fp2_copy(t2, t1);
fp2_dbl(t1, t1);
fp2_add(t1, t1, t2);
if (ep2_curve_is_twist()) {
switch (ep_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fp_set_dig(t2[0], 1);
fp2_mul_art(t2, t2);
fp2_mul_art(t2, t2);
fp2_add(t1, t1, t2);
break;
case OPT_DIGIT:
fp_set_dig(t2[0], ep_curve_get_a()[0]);
fp2_mul_art(t2, t2);
fp2_mul_art(t2, t2);
fp2_add(t1, t1, t2);
break;
default:
fp_copy(t2[0], ep_curve_get_a());
fp_zero(t2[1]);
fp2_mul_art(t2, t2);
fp2_mul_art(t2, t2);
fp2_add(t1, t1, t2);
break;
}
}
/* t1 = (3 * x1^2 + a)/(2 * y1). */
fp2_mul(t1, t1, t0);
if (s != NULL) {
fp2_copy(s, t1);
}
/* t2 = t1^2. */
fp2_sqr(t2, t1);
/* x3 = t1^2 - 2 * x1. */
fp2_dbl(t0, p->x);
fp2_sub(t0, t2, t0);
/* y3 = t1 * (x1 - x3) - y1. */
fp2_sub(t2, p->x, t0);
fp2_mul(t1, t1, t2);
fp2_sub(r->y, t1, p->y);
fp2_copy(r->x, t0);
fp2_copy(r->z, p->z);
r->norm = 1;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
}
}
void fp12_mul_dxs_lazyr(fp12_t c, fp12_t a, fp12_t b) {
fp6_t t0;
dv6_t u0, u1, u2;
fp6_null(t0);
dv6_null(u0);
dv6_null(u1);
dv6_null(u2);
TRY {
fp6_new(t0);
dv6_new(u0);
dv6_new(u1);
dv6_new(u2);
if (ep2_curve_is_twist() == EP_DTYPE) {
#if EP_ADD == BASIC
/* t0 = a_0 * b_0. */
fp_muln_low(u0[0][0], a[0][0][0], b[0][0][0]);
fp_muln_low(u0[0][1], a[0][0][1], b[0][0][0]);
fp_muln_low(u0[1][0], a[0][1][0], b[0][0][0]);
fp_muln_low(u0[1][1], a[0][1][1], b[0][0][0]);
fp_muln_low(u0[2][0], a[0][2][0], b[0][0][0]);
fp_muln_low(u0[2][1], a[0][2][1], b[0][0][0]);
/* t2 = b_0 + b_1. */
fp_add(t0[0][0], b[0][0][0], b[1][0][0]);
fp_copy(t0[0][1], b[1][0][1]);
fp2_copy(t0[1], b[1][1]);
#elif EP_ADD == PROJC
/* t0 = a_0 * b_0. */
#ifdef RLC_FP_ROOM
fp2_mulc_low(u0[0], a[0][0], b[0][0]);
fp2_mulc_low(u0[1], a[0][1], b[0][0]);
fp2_mulc_low(u0[2], a[0][2], b[0][0]);
#else
fp2_muln_low(u0[0], a[0][0], b[0][0]);
fp2_muln_low(u0[1], a[0][1], b[0][0]);
fp2_muln_low(u0[2], a[0][2], b[0][0]);
#endif
/* t2 = b_0 + b_1. */
fp2_add(t0[0], b[0][0], b[1][0]);
fp2_copy(t0[1], b[1][1]);
#endif
/* t1 = a_1 * b_1. */
fp6_mul_dxs_unr_lazyr(u1, a[1], b[1]);
} else {
/* t0 = a_0 * b_0. */
fp6_mul_dxs_unr_lazyr(u0, a[0], b[0]);
#if EP_ADD == BASIC
/* t0 = a_0 * b_0. */
fp_muln_low(u1[1][0], a[1][2][0], b[1][1][0]);
fp_muln_low(u1[1][1], a[1][2][1], b[1][1][0]);
fp2_nord_low(u1[0], u1[1]);
fp_muln_low(u1[1][0], a[1][0][0], b[1][1][0]);
fp_muln_low(u1[1][1], a[1][0][1], b[1][1][0]);
fp_muln_low(u1[2][0], a[1][1][0], b[1][1][0]);
fp_muln_low(u1[2][1], a[1][1][1], b[1][1][0]);
/* t2 = b_0 + b_1. */
fp2_copy(t0[0], b[0][0]);
fp_add(t0[1][0], b[0][1][0], b[1][1][0]);
fp_copy(t0[1][1], b[0][1][1]);
#elif EP_ADD == PROJC
/* t1 = a_1 * b_1. */
fp2_muln_low(u1[1], a[1][2], b[1][1]);
fp2_nord_low(u1[0], u1[1]);
fp2_muln_low(u1[1], a[1][0], b[1][1]);
fp2_muln_low(u1[2], a[1][1], b[1][1]);
/* t2 = b_0 + b_1. */
fp2_copy(t0[0], b[0][0]);
fp2_add(t0[1], b[0][1], b[1][1]);
#endif
}
/* c_1 = a_0 + a_1. */
fp6_add(c[1], a[0], a[1]);
/* c_1 = (a_0 + a_1) * (b_0 + b_1) */
fp6_mul_dxs_unr_lazyr(u2, c[1], t0);
for (int i = 0; i < 3; i++) {
fp2_subc_low(u2[i], u2[i], u0[i]);
fp2_subc_low(u2[i], u2[i], u1[i]);
}
fp2_rdcn_low(c[1][0], u2[0]);
fp2_rdcn_low(c[1][1], u2[1]);
fp2_rdcn_low(c[1][2], u2[2]);
fp2_nord_low(u2[0], u1[2]);
fp2_addc_low(u0[0], u0[0], u2[0]);
fp2_addc_low(u0[1], u0[1], u1[0]);
fp2_addc_low(u0[2], u0[2], u1[1]);
/* c_0 = a_0b_0 + v * a_1b_1. */
fp2_rdcn_low(c[0][0], u0[0]);
fp2_rdcn_low(c[0][1], u0[1]);
fp2_rdcn_low(c[0][2], u0[2]);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp6_free(t0);
dv6_free(u0);
dv6_free(u1);
dv6_free(u2);
}
}
void fp12_mul_dxs_basic(fp12_t c, fp12_t a, fp12_t b) {
fp6_t t0, t1, t2;
fp6_null(t0);
fp6_null(t1);
fp6_null(t2);
TRY {
fp6_new(t0);
fp6_new(t1);
fp6_new(t2);
if (ep2_curve_is_twist() == EP_DTYPE) {
#if EP_ADD == BASIC
/* t0 = a_0 * b_0 */
fp_mul(t0[0][0], a[0][0][0], b[0][0][0]);
fp_mul(t0[0][1], a[0][0][1], b[0][0][0]);
fp_mul(t0[1][0], a[0][1][0], b[0][0][0]);
fp_mul(t0[1][1], a[0][1][1], b[0][0][0]);
fp_mul(t0[2][0], a[0][2][0], b[0][0][0]);
fp_mul(t0[2][1], a[0][2][1], b[0][0][0]);
/* t2 = b_0 + b_1. */
fp_add(t2[0][0], b[0][0][0], b[1][0][0]);
fp_copy(t2[0][1], b[1][0][1]);
fp2_copy(t2[1], b[1][1]);
#elif EP_ADD == PROJC
/* t0 = a_0 * b_0 */
fp2_mul(t0[0], a[0][0], b[0][0]);
fp2_mul(t0[1], a[0][1], b[0][0]);
fp2_mul(t0[2], a[0][2], b[0][0]);
/* t2 = b_0 + b_1. */
fp2_add(t2[0], b[0][0], b[1][0]);
fp2_copy(t2[1], b[1][1]);
#endif
/* t1 = a_1 * b_1. */
fp6_mul_dxs(t1, a[1], b[1]);
} else {
/* t0 = a_0 * b_0. */
fp6_mul_dxs(t0, a[0], b[0]);
#if EP_ADD == BASIC
/* t1 = a_1 * b_1. */
fp_mul(t2[0][0], a[1][2][0], b[1][1][0]);
fp_mul(t2[0][1], a[1][2][1], b[1][1][0]);
fp2_mul_nor(t1[0], t2[0]);
fp_mul(t1[1][0], a[1][0][0], b[1][1][0]);
fp_mul(t1[1][1], a[1][0][1], b[1][1][0]);
fp_mul(t1[2][0], a[1][1][0], b[1][1][0]);
fp_mul(t1[2][1], a[1][1][1], b[1][1][0]);
/* t2 = b_0 + b_1. */
fp2_copy(t2[0], b[0][0]);
fp_add(t2[1][0], b[0][1][0], b[1][1][0]);
fp_copy(t2[1][1], b[0][1][1]);
#elif EP_ADD == PROJC
/* t1 = a_1 * b_1. */
fp2_mul(t2[0], a[1][2], b[1][1]);
fp2_mul_nor(t1[0], t2[0]);
fp2_mul(t1[1], a[1][0], b[1][1]);
fp2_mul(t1[2], a[1][1], b[1][1]);
/* t2 = b_0 + b_1. */
fp2_copy(t2[0], b[0][0]);
fp2_add(t2[1], b[0][1], b[1][1]);
#endif
}
/* c_1 = a_0 + a_1. */
fp6_add(c[1], a[0], a[1]);
/* c_1 = (a_0 + a_1) * (b_0 + b_1) - a_0 * b_0 - a_1 * b_1. */
fp6_mul_dxs(c[1], c[1], t2);
fp6_sub(c[1], c[1], t0);
fp6_sub(c[1], c[1], t1);
/* c_0 = a_0 * b_0 + v * a_1 * b_1. */
fp6_mul_art(t1, t1);
fp6_add(c[0], t0, t1);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp6_free(t0);
fp6_free(t1);
fp6_free(t2);
}
}
/**
* Adds two points represented in affine coordinates on an ordinary prime
* elliptic curve.
*
* @param r - the result.
* @param s - the resulting slope.
* @param p - the first point to add.
* @param q - the second point to add.
*/
static void ep2_add_basic_imp(ep2_t r, fp2_t s, ep2_t p, ep2_t q) {
fp2_t t0, t1, t2;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
/* t0 = x2 - x1. */
fp2_sub(t0, q->x, p->x);
/* t1 = y2 - y1. */
fp2_sub(t1, q->y, p->y);
/* If t0 is zero. */
if (fp2_is_zero(t0)) {
if (fp2_is_zero(t1)) {
/* If t1 is zero, q = p, should have doubled. */
ep2_dbl_basic(r, p);
} else {
/* If t1 is not zero and t0 is zero, q = -p and r = infty. */
ep2_set_infty(r);
}
} else {
/* t2 = 1/(x2 - x1). */
fp2_inv(t2, t0);
/* t2 = lambda = (y2 - y1)/(x2 - x1). */
fp2_mul(t2, t1, t2);
/* x3 = lambda^2 - x2 - x1. */
fp2_sqr(t1, t2);
fp2_sub(t0, t1, p->x);
fp2_sub(t0, t0, q->x);
/* y3 = lambda * (x1 - x3) - y1. */
fp2_sub(t1, p->x, t0);
fp2_mul(t1, t2, t1);
fp2_sub(r->y, t1, p->y);
fp2_copy(r->x, t0);
fp2_copy(r->z, p->z);
if (s != NULL) {
fp2_copy(s, t2);
}
r->norm = 1;
}
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
}
}
/**
* Adds a point represented in affine coordinates to a point represented in
* projective coordinates.
*
* @param r - the result.
* @param s - the slope.
* @param p - the affine point.
* @param q - the projective point.
*/
static void ep2_add_projc_mix(ep2_t r, ep2_t p, ep2_t q) {
fp2_t t0, t1, t2, t3, t4, t5, t6;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
fp2_null(t3);
fp2_null(t4);
fp2_null(t5);
fp2_null(t6);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
fp2_new(t3);
fp2_new(t4);
fp2_new(t5);
fp2_new(t6);
if (!p->norm) {
/* t0 = z1^2. */
fp2_sqr(t0, p->z);
/* t3 = U2 = x2 * z1^2. */
fp2_mul(t3, q->x, t0);
/* t1 = S2 = y2 * z1^3. */
fp2_mul(t1, t0, p->z);
fp2_mul(t1, t1, q->y);
/* t3 = H = U2 - x1. */
fp2_sub(t3, t3, p->x);
/* t1 = R = 2 * (S2 - y1). */
fp2_sub(t1, t1, p->y);
} else {
/* H = x2 - x1. */
fp2_sub(t3, q->x, p->x);
/* t1 = R = 2 * (y2 - y1). */
fp2_sub(t1, q->y, p->y);
}
/* t2 = HH = H^2. */
fp2_sqr(t2, t3);
/* If E is zero. */
if (fp2_is_zero(t3)) {
if (fp2_is_zero(t1)) {
/* If I is zero, p = q, should have doubled. */
ep2_dbl_projc(r, p);
} else {
/* If I is not zero, q = -p, r = infinity. */
ep2_set_infty(r);
}
} else {
/* t5 = J = H * HH. */
fp2_mul(t5, t3, t2);
/* t4 = V = x1 * HH. */
fp2_mul(t4, p->x, t2);
/* x3 = R^2 - J - 2 * V. */
fp2_sqr(r->x, t1);
fp2_sub(r->x, r->x, t5);
fp2_dbl(t6, t4);
fp2_sub(r->x, r->x, t6);
/* y3 = R * (V - x3) - Y1 * J. */
fp2_sub(t4, t4, r->x);
fp2_mul(t4, t4, t1);
fp2_mul(t1, p->y, t5);
fp2_sub(r->y, t4, t1);
if (!p->norm) {
/* z3 = z1 * H. */
fp2_mul(r->z, p->z, t3);
} else {
/* z3 = H. */
fp2_copy(r->z, t3);
}
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
fp2_free(t3);
fp2_free(t4);
fp2_free(t5);
fp2_free(t6);
}
}
/**
* Computes the constantes required for evaluating Frobenius maps.
*/
static void fp2_calc() {
bn_t e;
fp2_t t0;
fp2_t t1;
ctx_t *ctx = core_get();
bn_null(e);
fp2_null(t0);
fp2_null(t1);
TRY {
bn_new(e);
fp2_new(t0);
fp2_new(t1);
fp2_zero(t0);
fp_set_dig(t0[0], 1);
fp2_mul_nor(t0, t0);
e->used = FP_DIGS;
dv_copy(e->dp, fp_prime_get(), FP_DIGS);
bn_sub_dig(e, e, 1);
bn_div_dig(e, e, 6);
fp2_exp(t0, t0, e);
#if ALLOC == AUTO
fp2_copy(ctx->fp2_p[0], t0);
fp2_sqr(ctx->fp2_p[1], ctx->fp2_p[0]);
fp2_mul(ctx->fp2_p[2], ctx->fp2_p[1], ctx->fp2_p[0]);
fp2_sqr(ctx->fp2_p[3], ctx->fp2_p[1]);
fp2_mul(ctx->fp2_p[4], ctx->fp2_p[3], ctx->fp2_p[0]);
#else
fp_copy(ctx->fp2_p[0][0], t0[0]);
fp_copy(ctx->fp2_p[0][1], t0[1]);
fp2_sqr(t1, t0);
fp_copy(ctx->fp2_p[1][0], t1[0]);
fp_copy(ctx->fp2_p[1][1], t1[1]);
fp2_mul(t1, t1, t0);
fp_copy(ctx->fp2_p[2][0], t1[0]);
fp_copy(ctx->fp2_p[2][1], t1[1]);
fp2_sqr(t1, t0);
fp2_sqr(t1, t1);
fp_copy(ctx->fp2_p[3][0], t1[0]);
fp_copy(ctx->fp2_p[3][1], t1[1]);
fp2_mul(t1, t1, t0);
fp_copy(ctx->fp2_p[4][0], t1[0]);
fp_copy(ctx->fp2_p[4][1], t1[1]);
#endif
fp2_frb(t1, t0, 1);
fp2_mul(t0, t1, t0);
fp_copy(ctx->fp2_p2[0], t0[0]);
fp_sqr(ctx->fp2_p2[1], ctx->fp2_p2[0]);
fp_mul(ctx->fp2_p2[2], ctx->fp2_p2[1], ctx->fp2_p2[0]);
fp_sqr(ctx->fp2_p2[3], ctx->fp2_p2[1]);
for (int i = 0; i < 5; i++) {
fp_mul(ctx->fp2_p3[i][0], ctx->fp2_p2[i % 3], ctx->fp2_p[i][0]);
fp_mul(ctx->fp2_p3[i][1], ctx->fp2_p2[i % 3], ctx->fp2_p[i][1]);
}
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
bn_free(e);
fp2_free(t0);
fp2_free(t1);
}
}
void fp6_copy(fp6_t c, fp6_t a) {
fp2_copy(c[0], a[0]);
fp2_copy(c[1], a[1]);
fp2_copy(c[2], a[2]);
}
void ep2_copy(ep2_t r, ep2_t p) {
fp2_copy(r->x, p->x);
fp2_copy(r->y, p->y);
fp2_copy(r->z, p->z);
r->norm = p->norm;
}
