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);
}
}
/**
* 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 fp6_inv(fp6_t c, fp6_t a) {
fp2_t v0;
fp2_t v1;
fp2_t v2;
fp2_t t0;
fp2_null(v0);
fp2_null(v1);
fp2_null(v2);
fp2_null(t0);
TRY {
fp2_new(v0);
fp2_new(v1);
fp2_new(v2);
fp2_new(t0);
/* v0 = a_0^2 - E * a_1 * a_2. */
fp2_sqr(t0, a[0]);
fp2_mul(v0, a[1], a[2]);
fp2_mul_nor(v2, v0);
fp2_sub(v0, t0, v2);
/* v1 = E * a_2^2 - a_0 * a_1. */
fp2_sqr(t0, a[2]);
fp2_mul_nor(v2, t0);
fp2_mul(v1, a[0], a[1]);
fp2_sub(v1, v2, v1);
/* v2 = a_1^2 - a_0 * a_2. */
fp2_sqr(t0, a[1]);
fp2_mul(v2, a[0], a[2]);
fp2_sub(v2, t0, v2);
fp2_mul(t0, a[1], v2);
fp2_mul_nor(c[1], t0);
fp2_mul(c[0], a[0], v0);
fp2_mul(t0, a[2], v1);
fp2_mul_nor(c[2], t0);
fp2_add(t0, c[0], c[1]);
fp2_add(t0, t0, c[2]);
fp2_inv(t0, t0);
fp2_mul(c[0], v0, t0);
fp2_mul(c[1], v1, t0);
fp2_mul(c[2], v2, t0);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(v0);
fp2_free(v1);
fp2_free(v2);
fp2_free(t0);
}
}
void fp2_norm_low(fp2_t c, fp2_t a) {
fp2_t t;
bn_t b;
fp2_null(t);
bn_null(b);
TRY {
fp2_new(t);
bn_new(b);
#if FP_PRIME == 158
fp_dbl(t[0], a[0]);
fp_dbl(t[0], t[0]);
fp_sub(t[0], t[0], a[1]);
fp_dbl(t[1], a[1]);
fp_dbl(t[1], t[1]);
fp_add(c[1], a[0], t[1]);
fp_copy(c[0], t[0]);
#elif defined(FP_QNRES)
/* If p = 3 mod 8, (1 + i) is a QNR/CNR. */
fp_neg(t[0], a[1]);
fp_add(c[1], a[0], a[1]);
fp_add(c[0], t[0], a[0]);
#else
switch (fp_prime_get_mod8()) {
case 3:
/* If p = 3 mod 8, (1 + u) is a QNR/CNR. */
fp_neg(t[0], a[1]);
fp_add(c[1], a[0], a[1]);
fp_add(c[0], t[0], a[0]);
break;
case 5:
/* If p = 5 mod 8, (u) is a QNR/CNR. */
fp2_mul_art(c, a);
break;
case 7:
/* If p = 7 mod 8, we choose (2^(lg_4(b-1)) + u) as QNR/CNR. */
fp2_mul_art(t, a);
fp2_dbl(c, a);
fp_prime_back(b, ep_curve_get_b());
for (int i = 1; i < bn_bits(b) / 2; i++) {
fp2_dbl(c, c);
}
fp2_add(c, c, t);
break;
default:
THROW(ERR_NO_VALID);
break;
}
#endif
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t);
bn_free(b);
}
}
/**
* Doubles a point represented in affine coordinates on an ordinary prime
* elliptic curve.
*
* @param[out] r - the result.
* @param[in] p - the point to double.
*/
static void ep2_dbl_projc_imp(ep2_t r, ep2_t p) {
fp2_t t0, t1, t2, t3;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
fp2_null(t3);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
fp2_new(t3);
fp2_sqr(t0, p->x);
fp2_add(t2, t0, t0);
fp2_add(t0, t2, t0);
fp2_sqr(t3, p->y);
fp2_mul(t1, t3, p->x);
fp2_add(t1, t1, t1);
fp2_add(t1, t1, t1);
fp2_sqr(r->x, t0);
fp2_add(t2, t1, t1);
fp2_sub(r->x, r->x, t2);
fp2_mul(r->z, p->z, p->y);
fp2_add(r->z, r->z, r->z);
fp2_add(t3, t3, t3);
fp2_sqr(t3, t3);
fp2_add(t3, t3, t3);
fp2_sub(t1, t1, r->x);
fp2_mul(r->y, t0, t1);
fp2_sub(r->y, r->y, t3);
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
fp2_free(t3);
}
}
void fp2_mul_nor_basic(fp2_t c, fp2_t a) {
fp2_t t;
bn_t b;
fp2_null(t);
bn_null(b);
TRY {
fp2_new(t);
bn_new(b);
#ifdef FP_QNRES
/* If p = 3 mod 8, (1 + i) is a QNR/CNR. */
fp_neg(t[0], a[1]);
fp_add(c[1], a[0], a[1]);
fp_add(c[0], t[0], a[0]);
#else
switch (fp_prime_get_mod8()) {
case 3:
/* If p = 3 mod 8, (1 + u) is a QNR/CNR. */
fp_neg(t[0], a[1]);
fp_add(c[1], a[0], a[1]);
fp_add(c[0], t[0], a[0]);
break;
case 1:
case 5:
/* If p = 5 mod 8, (u) is a QNR/CNR. */
fp2_mul_art(c, a);
break;
case 7:
/* If p = 7 mod 8, we choose (4 + u) is a QNR/CNR. */
fp2_mul_art(t, a);
fp2_dbl(c, a);
fp2_dbl(c, c);
fp2_add(c, c, t);
break;
default:
THROW(ERR_NO_VALID);
}
#endif
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t);
bn_free(b);
}
}
/**
* 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);
}
}
/**
* Doubles a point represented in affine coordinates on an ordinary prime
* elliptic curve.
*
* @param[out] r - the result.
* @param[in] p - the point to double.
*/
static void ep2_dbl_projc_imp(ep2_t r, ep2_t p) {
fp2_t t0, t1, t2, t3, t4, t5;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
fp2_null(t3);
fp2_null(t4);
fp2_null(t5);
TRY {
if (ep_curve_opt_a() == OPT_ZERO) {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
fp2_new(t3);
fp2_new(t4);
fp2_new(t5);
fp2_sqr(t0, p->x);
fp2_add(t2, t0, t0);
fp2_add(t0, t2, t0);
fp2_sqr(t3, p->y);
fp2_mul(t1, t3, p->x);
fp2_add(t1, t1, t1);
fp2_add(t1, t1, t1);
fp2_sqr(r->x, t0);
fp2_add(t2, t1, t1);
fp2_sub(r->x, r->x, t2);
fp2_mul(r->z, p->z, p->y);
fp2_add(r->z, r->z, r->z);
fp2_add(t3, t3, t3);
fp2_sqr(t3, t3);
fp2_add(t3, t3, t3);
fp2_sub(t1, t1, r->x);
fp2_mul(r->y, t0, t1);
fp2_sub(r->y, r->y, t3);
} else {
/* dbl-2007-bl formulas: 1M + 8S + 1*a + 10add + 1*8 + 2*2 + 1*3 */
/* http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl */
/* t0 = x1^2, t1 = y1^2, t2 = y1^4. */
fp2_sqr(t0, p->x);
fp2_sqr(t1, p->y);
fp2_sqr(t2, t1);
if (!p->norm) {
/* t3 = z1^2. */
fp2_sqr(t3, p->z);
if (ep_curve_get_a() == OPT_ZERO) {
/* z3 = 2 * y1 * z1. */
fp2_mul(r->z, p->y, p->z);
fp2_dbl(r->z, r->z);
} else {
/* z3 = (y1 + z1)^2 - y1^2 - z1^2. */
fp2_add(r->z, p->y, p->z);
fp2_sqr(r->z, r->z);
fp2_sub(r->z, r->z, t1);
fp2_sub(r->z, r->z, t3);
}
} else {
/* z3 = 2 * y1. */
fp2_dbl(r->z, p->y);
}
/* t4 = S = 2*((x1 + y1^2)^2 - x1^2 - y1^4). */
fp2_add(t4, p->x, t1);
fp2_sqr(t4, t4);
fp2_sub(t4, t4, t0);
fp2_sub(t4, t4, t2);
fp2_dbl(t4, t4);
/* t5 = M = 3 * x1^2 + a * z1^4. */
fp2_dbl(t5, t0);
fp2_add(t5, t5, t0);
ep2_curve_get_a(t0);
if (!p->norm) {
fp2_sqr(t3, t3);
fp2_mul(t1, t0, t3);
fp2_add(t5, t5, t1);
} else {
fp2_add(t5, t5, t0);
}
/* x3 = T = M^2 - 2 * S. */
fp2_sqr(r->x, t5);
fp2_dbl(t1, t4);
fp2_sub(r->x, r->x, t1);
/* y3 = M * (S - T) - 8 * y1^4. */
fp2_dbl(t2, t2);
fp2_dbl(t2, t2);
fp2_dbl(t2, t2);
fp2_sub(t4, t4, r->x);
fp2_mul(t5, t5, t4);
fp2_sub(r->y, t5, t2);
}
r->norm = 0;
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);
}
}
/**
* Adds two points represented in projective coordinates on an ordinary prime
* elliptic curve.
*
* @param r - the result.
* @param p - the first point to add.
* @param q - the second point to add.
*/
static void ep2_add_projc_imp(ep2_t r, ep2_t p, ep2_t q) {
#if defined(EP_MIXED) && defined(STRIP)
ep2_add_projc_mix(r, p, q);
#else /* General addition. */
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 (q->norm) {
ep2_add_projc_mix(r, p, q);
} else {
/* t0 = z1^2. */
fp2_sqr(t0, p->z);
/* t1 = z2^2. */
fp2_sqr(t1, q->z);
/* t2 = U1 = x1 * z2^2. */
fp2_mul(t2, p->x, t1);
/* t3 = U2 = x2 * z1^2. */
fp2_mul(t3, q->x, t0);
/* t6 = z1^2 + z2^2. */
fp2_add(t6, t0, t1);
/* t0 = S2 = y2 * z1^3. */
fp2_mul(t0, t0, p->z);
fp2_mul(t0, t0, q->y);
/* t1 = S1 = y1 * z2^3. */
fp2_mul(t1, t1, q->z);
fp2_mul(t1, t1, p->y);
/* t3 = H = U2 - U1. */
fp2_sub(t3, t3, t2);
/* t0 = R = 2 * (S2 - S1). */
fp2_sub(t0, t0, t1);
fp2_dbl(t0, t0);
/* If E is zero. */
if (fp2_is_zero(t3)) {
if (fp2_is_zero(t0)) {
/* 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 {
/* t4 = I = (2*H)^2. */
fp2_dbl(t4, t3);
fp2_sqr(t4, t4);
/* t5 = J = H * I. */
fp2_mul(t5, t3, t4);
/* t4 = V = U1 * I. */
fp2_mul(t4, t2, t4);
/* x3 = R^2 - J - 2 * V. */
fp2_sqr(r->x, t0);
fp2_sub(r->x, r->x, t5);
fp2_dbl(t2, t4);
fp2_sub(r->x, r->x, t2);
/* y3 = R * (V - x3) - 2 * S1 * J. */
fp2_sub(t4, t4, r->x);
fp2_mul(t4, t4, t0);
fp2_mul(t1, t1, t5);
fp2_dbl(t1, t1);
fp2_sub(r->y, t4, t1);
/* z3 = ((z1 + z2)^2 - z1^2 - z2^2) * H. */
fp2_add(r->z, p->z, q->z);
fp2_sqr(r->z, r->z);
fp2_sub(r->z, r->z, t6);
fp2_mul(r->z, 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);
}
#endif
}
void fp6_add(fp6_t c, fp6_t a, fp6_t b) {
fp2_add(c[0], a[0], b[0]);
fp2_add(c[1], a[1], b[1]);
fp2_add(c[2], a[2], b[2]);
}
void pp_dbl_lit_k12(fp12_t l, ep_t r, ep_t p, ep2_t q) {
fp_t t0, t1, t2, t3, t4, t5, t6;
int one = 1, zero = 0;
fp_null(t0);
fp_null(t1);
fp_null(t2);
fp_null(t3);
fp_null(t4);
fp_null(t5);
fp_null(t6);
TRY {
fp_new(t0);
fp_new(t1);
fp_new(t2);
fp_new(t3);
fp_new(t4);
fp_new(t5);
fp_new(t6);
fp_sqr(t0, p->x);
fp_sqr(t1, p->y);
fp_sqr(t2, p->z);
fp_mul(t4, ep_curve_get_b(), t2);
fp_dbl(t3, t4);
fp_add(t3, t3, t4);
fp_add(t4, p->x, p->y);
fp_sqr(t4, t4);
fp_sub(t4, t4, t0);
fp_sub(t4, t4, t1);
fp_add(t5, p->y, p->z);
fp_sqr(t5, t5);
fp_sub(t5, t5, t1);
fp_sub(t5, t5, t2);
fp_dbl(t6, t3);
fp_add(t6, t6, t3);
fp_sub(r->x, t1, t6);
fp_mul(r->x, r->x, t4);
fp_add(r->y, t1, t6);
fp_sqr(r->y, r->y);
fp_sqr(t4, t3);
fp_dbl(t6, t4);
fp_add(t6, t6, t4);
fp_dbl(t6, t6);
fp_dbl(t6, t6);
fp_sub(r->y, r->y, t6);
fp_mul(r->z, t1, t5);
fp_dbl(r->z, r->z);
fp_dbl(r->z, r->z);
r->norm = 0;
if (ep2_curve_is_twist() == EP_MTYPE) {
one ^= 1;
zero ^= 1;
}
fp2_dbl(l[zero][one], q->x);
fp2_add(l[zero][one], l[zero][one], q->x);
fp_mul(l[zero][one][0], l[zero][one][0], t0);
fp_mul(l[zero][one][1], l[zero][one][1], t0);
fp_sub(l[zero][zero][0], t3, t1);
fp_zero(l[zero][zero][1]);
fp_mul(l[one][one][0], q->y[0], t5);
fp_mul(l[one][one][1], q->y[1], t5);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t0);
fp_free(t1);
fp_free(t2);
fp_free(t3);
fp_free(t4);
fp_free(t5);
fp_free(t6);
}
}
void pp_dbl_k12_projc_lazyr(fp12_t l, ep2_t r, ep2_t q, ep_t p) {
fp2_t t0, t1, t2, t3, t4, t5, t6;
dv2_t u0, u1;
int one = 1, zero = 0;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
fp2_null(t3);
fp2_null(t4);
fp2_null(t5);
fp2_null(t6);
dv2_null(u0);
dv2_null(u1);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
fp2_new(t3);
fp2_new(t4);
fp2_new(t5);
fp2_new(t6);
dv2_new(u0);
dv2_new(u1);
if (ep2_curve_is_twist() == EP_MTYPE) {
one ^= 1;
zero ^= 1;
}
if (ep_curve_opt_b() == RLC_TWO) {
/* C = z1^2. */
fp2_sqr(t0, q->z);
/* B = y1^2. */
fp2_sqr(t1, q->y);
/* t5 = B + C. */
fp2_add(t5, t0, t1);
/* t3 = E = 3b'C = 3C * (1 - i). */
fp2_dbl(t3, t0);
fp2_add(t0, t0, t3);
fp_add(t2[0], t0[0], t0[1]);
fp_sub(t2[1], t0[1], t0[0]);
/* t0 = x1^2. */
fp2_sqr(t0, q->x);
/* t4 = A = (x1 * y1)/2. */
fp2_mul(t4, q->x, q->y);
fp_hlv(t4[0], t4[0]);
fp_hlv(t4[1], t4[1]);
/* t3 = F = 3E. */
fp2_dbl(t3, t2);
fp2_add(t3, t3, t2);
/* x3 = A * (B - F). */
fp2_sub(r->x, t1, t3);
fp2_mul(r->x, r->x, t4);
/* G = (B + F)/2. */
fp2_add(t3, t1, t3);
fp_hlv(t3[0], t3[0]);
fp_hlv(t3[1], t3[1]);
/* y3 = G^2 - 3E^2. */
fp2_sqrn_low(u0, t2);
fp2_addd_low(u1, u0, u0);
fp2_addd_low(u1, u1, u0);
fp2_sqrn_low(u0, t3);
fp2_subc_low(u0, u0, u1);
/* H = (Y + Z)^2 - B - C. */
fp2_add(t3, q->y, q->z);
fp2_sqr(t3, t3);
fp2_sub(t3, t3, t5);
fp2_rdcn_low(r->y, u0);
/* z3 = B * H. */
fp2_mul(r->z, t1, t3);
/* l11 = E - B. */
fp2_sub(l[1][1], t2, t1);
/* l10 = (3 * xp) * t0. */
fp_mul(l[one][zero][0], p->x, t0[0]);
fp_mul(l[one][zero][1], p->x, t0[1]);
/* l01 = F * (-yp). */
fp_mul(l[zero][zero][0], t3[0], p->y);
fp_mul(l[zero][zero][1], t3[1], p->y);
} else {
/* A = x1^2. */
fp2_sqr(t0, q->x);
/* B = y1^2. */
fp2_sqr(t1, q->y);
/* C = z1^2. */
fp2_sqr(t2, q->z);
/* D = 3bC, for general b. */
fp2_dbl(t3, t2);
fp2_add(t3, t3, t2);
ep2_curve_get_b(t4);
fp2_mul(t3, t3, t4);
/* E = (x1 + y1)^2 - A - B. */
fp2_add(t4, q->x, q->y);
fp2_sqr(t4, t4);
fp2_sub(t4, t4, t0);
fp2_sub(t4, t4, t1);
/* F = (y1 + z1)^2 - B - C. */
fp2_add(t5, q->y, q->z);
fp2_sqr(t5, t5);
fp2_sub(t5, t5, t1);
fp2_sub(t5, t5, t2);
/* G = 3D. */
fp2_dbl(t6, t3);
fp2_add(t6, t6, t3);
/* x3 = E * (B - G). */
fp2_sub(r->x, t1, t6);
fp2_mul(r->x, r->x, t4);
/* y3 = (B + G)^2 -12D^2. */
fp2_add(t6, t6, t1);
fp2_sqr(t6, t6);
fp2_sqr(t2, t3);
fp2_dbl(r->y, t2);
fp2_dbl(t2, r->y);
fp2_dbl(r->y, t2);
fp2_add(r->y, r->y, t2);
fp2_sub(r->y, t6, r->y);
/* z3 = 4B * F. */
fp2_dbl(r->z, t1);
fp2_dbl(r->z, r->z);
fp2_mul(r->z, r->z, t5);
/* l00 = D - B. */
fp2_sub(l[one][one], t3, t1);
/* l10 = (3 * xp) * A. */
fp_mul(l[one][zero][0], p->x, t0[0]);
fp_mul(l[one][zero][1], p->x, t0[1]);
/* l01 = F * (-yp). */
fp_mul(l[zero][zero][0], t5[0], p->y);
fp_mul(l[zero][zero][1], t5[1], p->y);
}
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);
dv2_free(u0);
dv2_free(u1);
}
}
void fp12_sqr_pck_basic(fp12_t c, fp12_t a) {
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);
fp2_sqr(t0, a[0][1]);
fp2_sqr(t1, a[1][2]);
fp2_add(t5, a[0][1], a[1][2]);
fp2_sqr(t2, t5);
fp2_add(t3, t0, t1);
fp2_sub(t5, t2, t3);
fp2_add(t6, a[1][0], a[0][2]);
fp2_sqr(t3, t6);
fp2_sqr(t2, a[1][0]);
fp2_mul_nor(t6, t5);
fp2_add(t5, t6, a[1][0]);
fp2_dbl(t5, t5);
fp2_add(c[1][0], t5, t6);
fp2_mul_nor(t4, t1);
fp2_add(t5, t0, t4);
fp2_sub(t6, t5, a[0][2]);
fp2_sqr(t1, a[0][2]);
fp2_dbl(t6, t6);
fp2_add(c[0][2], t6, t5);
fp2_mul_nor(t4, t1);
fp2_add(t5, t2, t4);
fp2_sub(t6, t5, a[0][1]);
fp2_dbl(t6, t6);
fp2_add(c[0][1], t6, t5);
fp2_add(t0, t2, t1);
fp2_sub(t5, t3, t0);
fp2_add(t6, t5, a[1][2]);
fp2_dbl(t6, t6);
fp2_add(c[1][2], t5, t6);
} 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);
}
}
void fp12_sqr_cyc_basic(fp12_t c, fp12_t a) {
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);
/* Define z = sqrt(E) */
/* Now a is seen as (t0,t1) + (t2,t3) * w + (t4,t5) * w^2 */
/* (t0, t1) = (a00 + a11*z)^2. */
fp2_sqr(t2, a[0][0]);
fp2_sqr(t3, a[1][1]);
fp2_add(t1, a[0][0], a[1][1]);
fp2_mul_nor(t0, t3);
fp2_add(t0, t0, t2);
fp2_sqr(t1, t1);
fp2_sub(t1, t1, t2);
fp2_sub(t1, t1, t3);
fp2_sub(c[0][0], t0, a[0][0]);
fp2_add(c[0][0], c[0][0], c[0][0]);
fp2_add(c[0][0], t0, c[0][0]);
fp2_add(c[1][1], t1, a[1][1]);
fp2_add(c[1][1], c[1][1], c[1][1]);
fp2_add(c[1][1], t1, c[1][1]);
fp2_sqr(t0, a[0][1]);
fp2_sqr(t1, a[1][2]);
fp2_add(t5, a[0][1], a[1][2]);
fp2_sqr(t2, t5);
fp2_add(t3, t0, t1);
fp2_sub(t5, t2, t3);
fp2_add(t6, a[1][0], a[0][2]);
fp2_sqr(t3, t6);
fp2_sqr(t2, a[1][0]);
fp2_mul_nor(t6, t5);
fp2_add(t5, t6, a[1][0]);
fp2_dbl(t5, t5);
fp2_add(c[1][0], t5, t6);
fp2_mul_nor(t4, t1);
fp2_add(t5, t0, t4);
fp2_sub(t6, t5, a[0][2]);
fp2_sqr(t1, a[0][2]);
fp2_dbl(t6, t6);
fp2_add(c[0][2], t6, t5);
fp2_mul_nor(t4, t1);
fp2_add(t5, t2, t4);
fp2_sub(t6, t5, a[0][1]);
fp2_dbl(t6, t6);
fp2_add(c[0][1], t6, t5);
fp2_add(t0, t2, t1);
fp2_sub(t5, t3, t0);
fp2_add(t6, t5, a[1][2]);
fp2_dbl(t6, t6);
fp2_add(c[1][2], t5, t6);
} 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);
}
}