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);
}
}
void fp2_sqr_basic(fp2_t c, fp2_t a) {
fp_t t0, t1, t2;
fp_null(t0);
fp_null(t1);
fp_null(t2);
TRY {
fp_new(t0);
fp_new(t1);
fp_new(t2);
/* t0 = (a_0 + a_1). */
fp_add(t0, a[0], a[1]);
/* t1 = (a_0 - a_1). */
fp_sub(t1, a[0], a[1]);
/* t1 = a_0 + u^2 * a_1. */
for (int i = -1; i > fp_prime_get_qnr(); i--) {
fp_sub(t1, t1, a[1]);
}
for (int i = 0; i <= fp_prime_get_qnr(); i++) {
fp_add(t1, t1, a[1]);
}
if (fp_prime_get_qnr() == -1) {
/* t2 = 2 * a_0. */
fp_dbl(t2, a[0]);
/* c_1 = 2 * a_0 * a_1. */
fp_mul(c[1], t2, a[1]);
/* c_0 = a_0^2 + a_1^2 * u^2. */
fp_mul(c[0], t0, t1);
} else {
/* c_1 = a_0 * a_1. */
fp_mul(c[1], a[0], a[1]);
/* c_0 = a_0^2 + a_1^2 * u^2. */
fp_mul(c[0], t0, t1);
for (int i = -1; i > fp_prime_get_qnr(); i--) {
fp_add(c[0], c[0], c[1]);
}
for (int i = 0; i <= fp_prime_get_qnr(); i++) {
fp_sub(c[0], c[0], c[1]);
}
/* c_1 = 2 * a_0 * a_1. */
fp_dbl(c[1], c[1]);
}
/* c = c_0 + c_1 * u. */
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t0);
fp_free(t1);
fp_free(t2);
}
}
int fp2_srt(fp2_t c, fp2_t a) {
int r = 0;
fp_t t1;
fp_t t2;
fp_t t3;
fp_null(t1);
fp_null(t2);
fp_null(t3);
TRY {
fp_new(t1);
fp_new(t2);
fp_new(t3);
/* t1 = a[0]^2 - u^2 * a[1]^2 */
fp_sqr(t1, a[0]);
fp_sqr(t2, a[1]);
for (int i = -1; i > fp_prime_get_qnr(); i--) {
fp_add(t1, t1, t2);
}
for (int i = 0; i <= fp_prime_get_qnr(); i++) {
fp_sub(t1, t1, t2);
}
fp_add(t1, t1, t2);
if (fp_srt(t2, t1)) {
/* t1 = (a_0 + sqrt(t1)) / 2 */
fp_add(t1, a[0], t2);
fp_set_dig(t3, 2);
fp_inv(t3, t3);
fp_mul(t1, t1, t3);
if (!fp_srt(t3, t1)) {
/* t1 = (a_0 - sqrt(t1)) / 2 */
fp_sub(t1, a[0], t2);
fp_set_dig(t3, 2);
fp_inv(t3, t3);
fp_mul(t1, t1, t3);
fp_srt(t3, t1);
}
/* c_0 = sqrt(t1) */
fp_copy(c[0], t3);
/* c_1 = a_1 / (2 * sqrt(t1)) */
fp_dbl(t3, t3);
fp_inv(t3, t3);
fp_mul(c[1], a[1], t3);
r = 1;
}
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t1);
fp_free(t2);
fp_free(t3);
}
return r;
}
void fp2_inv(fp2_t c, fp2_t a) {
fp_t t0, t1;
fp_null(t0);
fp_null(t1);
TRY {
fp_new(t0);
fp_new(t1);
/* t0 = a_0^2, t1 = a_1^2. */
fp_sqr(t0, a[0]);
fp_sqr(t1, a[1]);
/* t1 = 1/(a_0^2 + a_1^2). */
#ifndef FP_QNRES
if (fp_prime_get_qnr() != -1) {
if (fp_prime_get_qnr() == -2) {
fp_dbl(t1, t1);
fp_add(t0, t0, t1);
} else {
if (fp_prime_get_qnr() < 0) {
fp_mul_dig(t1, t1, -fp_prime_get_qnr());
fp_add(t0, t0, t1);
} else {
fp_mul_dig(t1, t1, fp_prime_get_qnr());
fp_sub(t0, t0, t1);
}
}
} else {
fp_add(t0, t0, t1);
}
#else
fp_add(t0, t0, t1);
#endif
fp_inv(t1, t0);
/* c_0 = a_0/(a_0^2 + a_1^2). */
fp_mul(c[0], a[0], t1);
/* c_1 = - a_1/(a_0^2 + a_1^2). */
fp_mul(c[1], a[1], t1);
fp_neg(c[1], c[1]);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t0);
fp_free(t1);
}
}
void fp3_mul_art(fp3_t c, fp3_t a) {
fp_t t;
fp_null(t);
TRY {
fp_new(t);
/* (a_0 + a_1 * u + a_1 * u^2) * u = a_0 * u + a_1 * u^2 + a_1 * u^3. */
fp_copy(t, a[0]);
fp_dbl(c[0], a[2]);
fp_neg(c[0], c[0]);
fp_copy(c[2], a[1]);
fp_copy(c[1], t);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t);
}
}
void fp2_dblm_low(fp2_t c, fp2_t a) {
/* 2 * (a0 + a1 * u) = 2 * a0 + 2 * a1 * u. */
fp_dbl(c[0], a[0]);
fp_dbl(c[1], a[1]);
}
void fp3_sqr_basic(fp3_t c, fp3_t a) {
dv_t t0, t1, t2, t3, t4, t5;
dv_null(t0);
dv_null(t1);
dv_null(t2);
dv_null(t3);
dv_null(t4);
dv_null(t5);
TRY {
dv_new(t0);
dv_new(t1);
dv_new(t2);
dv_new(t3);
dv_new(t4);
dv_new(t5);
/* t0 = a_0^2. */
fp_sqrn_low(t0, a[0]);
/* t1 = 2 * a_1 * a_2. */
fp_dbl(t2, a[1]);
fp_muln_low(t1, t2, a[2]);
/* t2 = a_2^2. */
fp_sqrn_low(t2, a[2]);
/* t3 = (a_0 + a_2 + a_1)^2, t4 = (a_0 + a_2 - a_1)^2. */
fp_add(t3, a[0], a[2]);
fp_add(t4, t3, a[1]);
fp_sub(t5, t3, a[1]);
fp_sqrn_low(t3, t4);
fp_sqrn_low(t4, t5);
/* t4 = (t4 + t3)/2. */
fp_addd_low(t4, t4, t3);
fp_hlvd_low(t4, t4);
/* t3 = t3 - t4 - t1. */
fp_addc_low(t5, t1, t4);
fp_subc_low(t3, t3, t5);
/* c_2 = t4 - t0 - t2. */
fp_addc_low(t5, t0, t2);
fp_subc_low(t4, t4, t5);
fp_rdc(c[2], t4);
/* c_0 = t0 + t1 * B. */
fp_subc_low(t0, t0, t1);
for (int i = -1; i > fp_prime_get_cnr(); i--) {
fp_subc_low(t0, t0, t1);
}
fp_rdc(c[0], t0);
/* c_1 = t3 + t2 * B. */
fp_subc_low(t3, t3, t2);
for (int i = -1; i > fp_prime_get_cnr(); i--) {
fp_subc_low(t3, t3, t2);
}
fp_rdc(c[1], t3);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
dv_free(t0);
dv_free(t1);
dv_free(t2);
dv_free(t3);
dv_free(t4);
dv_free(t5);
}
}
void fp3_dbl_basic(fp2_t c, fp2_t a) {
/* 2 * (a_0 + a_1 * u) = 2 * a_0 + 2 * a_1 * u. */
fp_dbl(c[0], a[0]);
fp_dbl(c[1], a[1]);
fp_dbl(c[2], a[2]);
}
/**
* Doubles a point represented in affine coordinates on an ordinary prime
* elliptic curve.
*
* @param[out] r - the result.
* @param[out] s - the slope.
* @param[in] p - the point to double.
*/
static void ep_dbl_basic_imp(ep_t r, fp_t s, const ep_t p) {
fp_t t0, t1, t2;
fp_null(t0);
fp_null(t1);
fp_null(t2);
TRY {
fp_new(t0);
fp_new(t1);
fp_new(t2);
/* t0 = 1/2 * y1. */
fp_dbl(t0, p->y);
fp_inv(t0, t0);
/* t1 = 3 * x1^2 + a. */
fp_sqr(t1, p->x);
fp_copy(t2, t1);
fp_dbl(t1, t1);
fp_add(t1, t1, t2);
switch (ep_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fp_add_dig(t1, t1, (dig_t)1);
break;
#if FP_RDC != MONTY
case OPT_DIGIT:
fp_add_dig(t1, t1, ep_curve_get_a()[0]);
break;
#endif
default:
fp_add(t1, t1, ep_curve_get_a());
break;
}
/* t1 = (3 * x1^2 + a)/(2 * y1). */
fp_mul(t1, t1, t0);
if (s != NULL) {
fp_copy(s, t1);
}
/* t2 = t1^2. */
fp_sqr(t2, t1);
/* x3 = t1^2 - 2 * x1. */
fp_dbl(t0, p->x);
fp_sub(t0, t2, t0);
/* y3 = t1 * (x1 - x3) - y1. */
fp_sub(t2, p->x, t0);
fp_mul(t1, t1, t2);
fp_sub(r->y, t1, p->y);
fp_copy(r->x, t0);
fp_copy(r->z, p->z);
r->norm = 1;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t0);
fp_free(t1);
fp_free(t2);
}
}
/**
* Doubles a point represented in projective coordinates on an ordinary prime
* elliptic curve.
*
* @param r - the result.
* @param p - the point to double.
*/
static void ep_dbl_projc_imp(ep_t r, const ep_t p) {
fp_t t0, t1, t2, t3, t4, t5;
fp_null(t1);
fp_null(t2);
fp_null(t3);
fp_null(t4);
fp_null(t5);
TRY {
fp_new(t0);
fp_new(t1);
fp_new(t2);
fp_new(t3);
fp_new(t4);
fp_new(t5);
if (!p->norm && ep_curve_opt_a() == OPT_MINUS3) {
/* dbl-2001-b formulas: 3M + 5S + 8add + 1*4 + 2*8 + 1*3 */
/* http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b */
/* t0 = delta = z1^2. */
fp_sqr(t0, p->z);
/* t1 = gamma = y1^2. */
fp_sqr(t1, p->y);
/* t2 = beta = x1 * y1^2. */
fp_mul(t2, p->x, t1);
/* t3 = alpha = 3 * (x1 - z1^2) * (x1 + z1^2). */
fp_sub(t3, p->x, t0);
fp_add(t4, p->x, t0);
fp_mul(t4, t3, t4);
fp_dbl(t3, t4);
fp_add(t3, t3, t4);
/* x3 = alpha^2 - 8 * beta. */
fp_dbl(t2, t2);
fp_dbl(t2, t2);
fp_dbl(t5, t2);
fp_sqr(r->x, t3);
fp_sub(r->x, r->x, t5);
/* z3 = (y1 + z1)^2 - gamma - delta. */
fp_add(r->z, p->y, p->z);
fp_sqr(r->z, r->z);
fp_sub(r->z, r->z, t1);
fp_sub(r->z, r->z, t0);
/* y3 = alpha * (4 * beta - x3) - 8 * gamma^2. */
fp_dbl(t1, t1);
fp_sqr(t1, t1);
fp_dbl(t1, t1);
fp_sub(r->y, t2, r->x);
fp_mul(r->y, r->y, t3);
fp_sub(r->y, r->y, t1);
} else if (ep_curve_opt_a() == OPT_ZERO) {
/* dbl-2009-l formulas: 2M + 5S + 6add + 1*8 + 3*2 + 1*3. */
/* http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l */
/* A = X1^2 */
fp_sqr(t0, p->x);
/* B = Y1^2 */
fp_sqr(t1, p->y);
/* C = B^2 */
fp_sqr(t2, t1);
/* D = 2*((X1+B)^2-A-C) */
fp_add(t1, t1, p->x);
fp_sqr(t1, t1);
fp_sub(t1, t1, t0);
fp_sub(t1, t1, t2);
fp_dbl(t1, t1);
/* E = 3*A */
fp_dbl(t3, t0);
fp_add(t0, t3, t0);
/* F = E^2 */
fp_sqr(t3, t0);
/* Z3 = 2*Y1*Z1 */
fp_mul(r->z, p->y, p->z);
fp_dbl(r->z, r->z);
/* X3 = F-2*D */
fp_sub(r->x, t3, t1);
fp_sub(r->x, r->x, t1);
/* Y3 = E*(D-X3)-8*C */
fp_sub(r->y, t1, r->x);
fp_mul(r->y, r->y, t0);
fp_dbl(t2, t2);
fp_dbl(t2, t2);
fp_dbl(t2, t2);
fp_sub(r->y, r->y, t2);
} 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. */
fp_sqr(t0, p->x);
fp_sqr(t1, p->y);
fp_sqr(t2, t1);
if (!p->norm) {
/* t3 = z1^2. */
fp_sqr(t3, p->z);
if (ep_curve_get_a() == OPT_ZERO) {
/* z3 = 2 * y1 * z1. */
fp_mul(r->z, p->y, p->z);
fp_dbl(r->z, r->z);
} else {
/* z3 = (y1 + z1)^2 - y1^2 - z1^2. */
fp_add(r->z, p->y, p->z);
fp_sqr(r->z, r->z);
fp_sub(r->z, r->z, t1);
fp_sub(r->z, r->z, t3);
}
} else {
/* z3 = 2 * y1. */
fp_dbl(r->z, p->y);
}
/* t4 = S = 2*((x1 + y1^2)^2 - x1^2 - y1^4). */
fp_add(t4, p->x, t1);
fp_sqr(t4, t4);
fp_sub(t4, t4, t0);
fp_sub(t4, t4, t2);
fp_dbl(t4, t4);
/* t5 = M = 3 * x1^2 + a * z1^4. */
fp_dbl(t5, t0);
fp_add(t5, t5, t0);
if (!p->norm) {
fp_sqr(t3, t3);
switch (ep_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fp_add(t5, t5, ep_curve_get_a());
break;
case OPT_DIGIT:
fp_mul_dig(t1, t3, ep_curve_get_a()[0]);
fp_add(t5, t5, t1);
break;
default:
fp_mul(t1, ep_curve_get_a(), t3);
fp_add(t5, t5, t1);
break;
}
} else {
switch (ep_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fp_add_dig(t5, t5, (dig_t)1);
break;
case OPT_DIGIT:
fp_add_dig(t5, t5, ep_curve_get_a()[0]);
break;
default:
fp_add(t5, t5, ep_curve_get_a());
break;
}
}
/* x3 = T = M^2 - 2 * S. */
fp_sqr(r->x, t5);
fp_dbl(t1, t4);
fp_sub(r->x, r->x, t1);
/* y3 = M * (S - T) - 8 * y1^4. */
fp_dbl(t2, t2);
fp_dbl(t2, t2);
fp_dbl(t2, t2);
fp_sub(t4, t4, r->x);
fp_mul(t5, t5, t4);
fp_sub(r->y, t5, t2);
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t0);
fp_free(t1);
fp_free(t2);
fp_free(t3);
fp_free(t4);
fp_free(t5);
}
}
void fp2_sqrn_low(dv2_t c, fp2_t a) {
align dig_t t0[2 * FP_DIGS], t1[2 * FP_DIGS], t2[2 * FP_DIGS];
/* t0 = (a0 + a1). */
#ifdef FP_SPACE
/* if we have room for carries, we can avoid reductions here. */
fp_addn_low(t0, a[0], a[1]);
#else
fp_addm_low(t0, a[0], a[1]);
#endif
/* t1 = (a0 - a1). */
fp_subm_low(t1, a[0], a[1]);
#ifdef FP_QNRES
#ifdef FP_SPACE
fp_dbln_low(t2, a[0]);
#else
fp_dblm_low(t2, a[0]);
#endif
/* c1 = 2 * a0 * a1. */
fp_muln_low(c[1], t2, a[1]);
/* c_0 = a_0^2 + a_1^2 * u^2. */
fp_muln_low(c[0], t0, t1);
#else /* !FP_QNRES */
/* t1 = u^2 * (a1 * b1). */
for (int i = -1; i > fp_prime_get_qnr(); i--) {
fp_subm_low(t1, t1, a[1]);
}
if (fp_prime_get_qnr() == -1) {
/* t2 = 2 * a0. */
fp_dbl(t2, a[0]);
/* c1 = 2 * a0 * a1. */
fp_muln_low(c[1], t2, a[1]);
/* c0 = a0^2 + b_0^2 * u^2. */
fp_muln_low(c[0], t0, t1);
} else {
/* c1 = a0 * a1. */
fp_muln_low(c[1], a[0], a[1]);
/* c0 = a0^2 + b_0^2 * u^2. */
fp_muln_low(c[0], t0, t1);
#ifdef FP_SPACE
for (int i = -1; i > fp_prime_get_qnr(); i--) {
fp_addd_low(c[0], c[0], c[1]);
}
/* c1 = 2 * a0 * a1. */
fp_addd_low(c[1], c[1], c[1]);
#else
for (int i = -1; i > fp_prime_get_qnr(); i--) {
fp_addc_low(c[0], c[0], c[1]);
}
/* c1 = 2 * a0 * a1. */
fp_addc_low(c[1], c[1], c[1]);
#endif
}
#endif
/* c = c0 + c1 * u. */
}
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_k2_projc_lazyr(fp2_t l, ep_t r, ep_t p, ep_t q) {
fp_t t0, t1, t2, t3, t4, t5;
dv_t u0, u1;
fp_null(t0);
fp_null(t1);
fp_null(t2);
fp_null(t3);
fp_null(t4);
fp_null(t5);
dv_null(u0);
dv_null(u1);
TRY {
fp_new(t0);
fp_new(t1);
fp_new(t2);
fp_new(t3);
fp_new(t4);
fp_new(t5);
dv_new(u0);
dv_new(u1);
/* For these curves, we always can choose a = -3. */
/* dbl-2001-b formulas: 3M + 5S + 8add + 1*4 + 2*8 + 1*3 */
/* http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b */
/* t0 = delta = z1^2. */
fp_sqr(t0, p->z);
/* t1 = gamma = y1^2. */
fp_sqr(t1, p->y);
/* t2 = beta = x1 * y1^2. */
fp_mul(t2, p->x, t1);
/* t3 = alpha = 3 * (x1 - z1^2) * (x1 + z1^2). */
fp_sub(t3, p->x, t0);
fp_add(t4, p->x, t0);
fp_mul(t4, t3, t4);
fp_dbl(t3, t4);
fp_add(t3, t3, t4);
/* t2 = 4 * beta. */
fp_dbl(t2, t2);
fp_dbl(t2, t2);
/* z3 = (y1 + z1)^2 - gamma - delta. */
fp_add(r->z, p->y, p->z);
fp_sqr(r->z, r->z);
fp_sub(r->z, r->z, t1);
fp_sub(r->z, r->z, t0);
/* l0 = 2 * gamma - alpha * (delta * xq + x1). */
fp_dbl(t1, t1);
fp_mul(t5, t0, q->x);
fp_add(t5, t5, p->x);
fp_mul(t5, t5, t3);
fp_sub(l[0], t1, t5);
/* x3 = alpha^2 - 8 * beta. */
fp_dbl(t5, t2);
fp_sqr(r->x, t3);
fp_sub(r->x, r->x, t5);
/* y3 = alpha * (4 * beta - x3) - 8 * gamma^2. */
fp_sqrn_low(u0, t1);
fp_addc_low(u0, u0, u0);
fp_subm_low(r->y, t2, r->x);
fp_muln_low(u1, r->y, t3);
fp_subc_low(u1, u1, u0);
fp_rdcn_low(r->y, u1);
/* l1 = - z3 * delta * yq. */
fp_mul(l[1], r->z, t0);
fp_mul(l[1], l[1], q->y);
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp_free(t0);
fp_free(t1);
fp_free(t2);
fp_free(t3);
fp_free(t4);
fp_free(t5);
dv_free(u0);
dv_free(u1);
}
}
void fp3_mul_nor(fp3_t c, fp3_t a) {
fp_copy(c[1], a[0]);
fp_dbl(c[0], a[2]);
fp_neg(c[0], c[0]);
fp_copy(c[2], a[1]);
}
/**
* Adds two points represented in projective coordinates on an ordinary prime
* elliptic curve.
*
* @param[out] r - the result.
* @param[in] p - the first point to add.
* @param[in] q - the second point to add.
*/
static void ep_add_projc_imp(ep_t r, const ep_t p, const ep_t q) {
#if defined(EP_MIXED) && defined(STRIP)
/* If code size is a problem, leave only the mixed version. */
ep_add_projc_mix(r, p, q);
#else /* General addition. */
#if defined(EP_MIXED) || !defined(STRIP)
/* Test if z2 = 1 only if mixed coordinates are turned on. */
if (q->norm) {
ep_add_projc_mix(r, p, q);
return;
}
#endif
fp_t t0, t1, t2, t3, t4, t5, t6;
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);
/* add-2007-bl formulas: 11M + 5S + 9add + 4*2 */
/* http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl */
/* t0 = z1^2. */
fp_sqr(t0, p->z);
/* t1 = z2^2. */
fp_sqr(t1, q->z);
/* t2 = U1 = x1 * z2^2. */
fp_mul(t2, p->x, t1);
/* t3 = U2 = x2 * z1^2. */
fp_mul(t3, q->x, t0);
/* t6 = z1^2 + z2^2. */
fp_add(t6, t0, t1);
/* t0 = S2 = y2 * z1^3. */
fp_mul(t0, t0, p->z);
fp_mul(t0, t0, q->y);
/* t1 = S1 = y1 * z2^3. */
fp_mul(t1, t1, q->z);
fp_mul(t1, t1, p->y);
/* t3 = H = U2 - U1. */
fp_sub(t3, t3, t2);
/* t0 = R = 2 * (S2 - S1). */
fp_sub(t0, t0, t1);
fp_dbl(t0, t0);
/* If E is zero. */
if (fp_is_zero(t3)) {
if (fp_is_zero(t0)) {
/* If I is zero, p = q, should have doubled. */
ep_dbl_projc(r, p);
} else {
/* If I is not zero, q = -p, r = infinity. */
ep_set_infty(r);
}
} else {
/* t4 = I = (2*H)^2. */
fp_dbl(t4, t3);
fp_sqr(t4, t4);
/* t5 = J = H * I. */
fp_mul(t5, t3, t4);
/* t4 = V = U1 * I. */
fp_mul(t4, t2, t4);
/* x3 = R^2 - J - 2 * V. */
fp_sqr(r->x, t0);
fp_sub(r->x, r->x, t5);
fp_dbl(t2, t4);
fp_sub(r->x, r->x, t2);
/* y3 = R * (V - x3) - 2 * S1 * J. */
fp_sub(t4, t4, r->x);
fp_mul(t4, t4, t0);
fp_mul(t1, t1, t5);
fp_dbl(t1, t1);
fp_sub(r->y, t4, t1);
/* z3 = ((z1 + z2)^2 - z1^2 - z2^2) * H. */
fp_add(r->z, p->z, q->z);
fp_sqr(r->z, r->z);
fp_sub(r->z, r->z, t6);
fp_mul(r->z, r->z, t3);
}
r->norm = 0;
}
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);
}
#endif
}
/**
* Adds a point represented in affine coordinates to a point represented in
* projective coordinates.
*
* @param[out] r - the result.
* @param[in] p - the projective point.
* @param[in] q - the affine point.
*/
static void ep_add_projc_mix(ep_t r, const ep_t p, const ep_t q) {
fp_t t0, t1, t2, t3, t4, t5, t6;
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);
/* madd-2007-bl formulas: 7M + 4S + 9add + 1*4 + 3*2. */
/* http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-madd-2007-bl */
if (!p->norm) {
/* t0 = z1^2. */
fp_sqr(t0, p->z);
/* t3 = U2 = x2 * z1^2. */
fp_mul(t3, q->x, t0);
/* t1 = S2 = y2 * z1^3. */
fp_mul(t1, t0, p->z);
fp_mul(t1, t1, q->y);
/* t3 = H = U2 - x1. */
fp_sub(t3, t3, p->x);
/* t1 = R = 2 * (S2 - y1). */
fp_sub(t1, t1, p->y);
fp_dbl(t1, t1);
} else {
/* H = x2 - x1. */
fp_sub(t3, q->x, p->x);
/* t1 = R = 2 * (y2 - y1). */
fp_sub(t1, q->y, p->y);
fp_dbl(t1, t1);
}
/* t2 = HH = H^2. */
fp_sqr(t2, t3);
/* If E is zero. */
if (fp_is_zero(t3)) {
if (fp_is_zero(t1)) {
/* If I is zero, p = q, should have doubled. */
ep_dbl_projc(r, p);
} else {
/* If I is not zero, q = -p, r = infinity. */
ep_set_infty(r);
}
} else {
/* t4 = I = 4*HH. */
fp_dbl(t4, t2);
fp_dbl(t4, t4);
/* t5 = J = H * I. */
fp_mul(t5, t3, t4);
/* t4 = V = x1 * I. */
fp_mul(t4, p->x, t4);
/* x3 = R^2 - J - 2 * V. */
fp_sqr(r->x, t1);
fp_sub(r->x, r->x, t5);
fp_dbl(t6, t4);
fp_sub(r->x, r->x, t6);
/* y3 = R * (V - x3) - 2 * Y1 * J. */
fp_sub(t4, t4, r->x);
fp_mul(t4, t4, t1);
fp_mul(t1, p->y, t5);
fp_dbl(t1, t1);
fp_sub(r->y, t4, t1);
if (!p->norm) {
/* z3 = (z1 + H)^2 - z1^2 - HH. */
fp_add(r->z, p->z, t3);
fp_sqr(r->z, r->z);
fp_sub(r->z, r->z, t0);
fp_sub(r->z, r->z, t2);
} else {
/* z3 = 2 * H. */
fp_dbl(r->z, t3);
}
}
r->norm = 0;
}
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);
}
}
