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 pp_map_weilp_k2(fp2_t r, ep_t p, ep_t q) {
ep_t _p[1], _q[1], t0[1], t1[1];
fp2_t r0, r1;
bn_t n;
ep_null(_p[0]);
ep_null(_q[0]);
ep_null(t0[0]);
ep_null(t1[0]);
fp2_null(r0);
fp2_null(r1);
bn_null(n);
TRY {
ep_new(_p[0]);
ep_new(_q[0]);
ep_new(t0[0]);
ep_new(t1[0]);
fp2_new(r0);
fp2_new(r1);
bn_new(n);
ep_norm(_p[0], p);
ep_norm(_q[0], q);
ep_curve_get_ord(n);
/* Since p has order n, we do not have to perform last iteration. */
bn_sub_dig(n, n, 1);
fp2_set_dig(r0, 1);
fp2_set_dig(r1, 1);
if (!ep_is_infty(_p[0]) && !ep_is_infty(_q[0])) {
pp_mil_lit_k2(r0, t0, _p, _q, 1, n);
pp_mil_k2(r1, t1, _q, _p, 1, n);
fp2_inv(r1, r1);
fp2_mul(r0, r0, r1);
fp2_inv(r1, r0);
fp2_inv_uni(r0, r0);
}
fp2_mul(r, r0, r1);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
ep_free(_p[0]);
ep_free(_q[0]);
ep_free(t0[0]);
ep_free(t1[0]);
fp2_free(r0);
fp2_free(r1);
bn_free(n);
}
}
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 pp_dbl_k12_basic(fp12_t l, ep2_t r, ep2_t q, ep_t p) {
fp2_t s;
ep2_t t;
int one = 1, zero = 0;
fp2_null(s);
ep2_null(t);
TRY {
fp2_new(s);
ep2_new(t);
ep2_copy(t, q);
ep2_dbl_slp_basic(r, s, q);
fp12_zero(l);
if (ep2_curve_is_twist() == EP_MTYPE) {
one ^= 1;
zero ^= 1;
}
fp_mul(l[one][zero][0], s[0], p->x);
fp_mul(l[one][zero][1], s[1], p->x);
fp2_mul(l[one][one], s, t->x);
fp2_sub(l[one][one], t->y, l[one][one]);
fp_copy(l[zero][zero][0], p->y);
} CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(s);
ep2_free(t);
}
}
void ep2_map(ep2_t p, uint8_t *msg, int len) {
bn_t x;
fp2_t t0;
uint8_t digest[MD_LEN];
bn_null(x);
fp2_null(t0);
TRY {
bn_new(x);
fp2_new(t0);
md_map(digest, msg, len);
bn_read_bin(x, digest, MIN(FP_BYTES, MD_LEN));
fp_prime_conv(p->x[0], x);
fp_zero(p->x[1]);
fp_set_dig(p->z[0], 1);
fp_zero(p->z[1]);
while (1) {
ep2_rhs(t0, p);
if (fp2_srt(p->y, t0)) {
p->norm = 1;
break;
}
fp_add_dig(p->x[0], p->x[0], 1);
}
switch (ep_param_get()) {
case BN_P158:
case BN_P254:
case BN_P256:
case BN_P638:
ep2_mul_cof_bn(p, p);
break;
case B12_P638:
ep2_mul_cof_b12(p, p);
break;
default:
/* Now, multiply by cofactor to get the correct group. */
ep2_curve_get_cof(x);
if (bn_bits(x) < BN_DIGIT) {
ep2_mul_dig(p, p, x->dp[0]);
} else {
ep2_mul(p, p, x);
}
break;
}
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
bn_free(x);
fp2_free(t0);
}
}
void fp2_write_bin(uint8_t *bin, int len, fp2_t a, int pack) {
fp2_t t;
fp2_null(t);
TRY {
fp2_new(t);
if (pack && fp2_test_uni(a)) {
if (len != FP_BYTES + 1) {
THROW(ERR_NO_BUFFER);
} else {
fp2_pck(t, t);
fp_write_bin(bin, FP_BYTES, a[0]);
bin[FP_BYTES] = fp_get_bit(a[1], 0);
}
} else {
if (len != 2 * FP_BYTES) {
THROW(ERR_NO_BUFFER);
} else {
fp_write_bin(bin, FP_BYTES, a[0]);
fp_write_bin(bin + FP_BYTES, FP_BYTES, a[1]);
}
}
} CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t);
}
}
/**
* 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_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);
}
}
void ep2_curve_init(void) {
ctx_t *ctx = core_get();
#ifdef EP_PRECO
for (int i = 0; i < EP_TABLE; i++) {
ctx->ep2_ptr[i] = &(ctx->ep2_pre[i]);
}
#endif
#if ALLOC == STATIC || ALLOC == DYNAMIC || ALLOC == STACK
ctx->ep2_g.x[0] = ctx->ep2_gx[0];
ctx->ep2_g.x[1] = ctx->ep2_gx[1];
ctx->ep2_g.y[0] = ctx->ep2_gy[0];
ctx->ep2_g.y[1] = ctx->ep2_gy[1];
ctx->ep2_g.z[0] = ctx->ep2_gz[0];
ctx->ep2_g.z[1] = ctx->ep2_gz[1];
#endif
#ifdef EP_PRECO
#if ALLOC == STATIC || ALLOC == DYNAMIC
for (int i = 0; i < EP_TABLE; i++) {
fp2_new(ctx->ep2_pre[i].x);
fp2_new(ctx->ep2_pre[i].y);
fp2_new(ctx->ep2_pre[i].z);
}
#elif ALLOC == STACK
for (int i = 0; i < EP_TABLE; i++) {
ctx->ep2_pre[i].x[0] = ctx->_ep2_pre[3 * i][0];
ctx->ep2_pre[i].x[1] = ctx->_ep2_pre[3 * i][1];
ctx->ep2_pre[i].y[0] = ctx->_ep2_pre[3 * i + 1][0];
ctx->ep2_pre[i].y[1] = ctx->_ep2_pre[3 * i + 1][1];
ctx->ep2_pre[i].z[0] = ctx->_ep2_pre[3 * i + 2][0];
ctx->ep2_pre[i].z[1] = ctx->_ep2_pre[3 * i + 2][1];
}
#endif
#endif
ep2_set_infty(&(ctx->ep2_g));
bn_init(&(ctx->ep2_r), FP_DIGS);
bn_init(&(ctx->ep2_h), FP_DIGS);
}
static void fp4_mul_unr(dv2_t e, dv2_t f, fp2_t a, fp2_t b, fp2_t c, fp2_t d) {
fp2_t t0, t1;
dv2_t u0, u1;
fp2_null(t0);
fp2_null(t1);
dv2_null(u0);
dv2_null(u1);
TRY {
fp2_new(t0);
fp2_new(t1);
dv2_new(u0);
dv2_new(u1);
#ifdef FP_SPACE
fp2_mulc_low(u0, a, c);
fp2_mulc_low(u1, b, d);
fp2_addn_low(t0, c, d);
fp2_addn_low(t1, a, b);
#else
fp2_muln_low(u0, a, c);
fp2_muln_low(u1, b, d);
fp2_addm_low(t0, c, d);
fp2_addm_low(t1, a, b);
#endif
fp2_muln_low(f, t1, t0);
fp2_subc_low(f, f, u0);
fp2_subc_low(f, f, u1);
fp2_norh_low(e, u1);
fp2_addc_low(e, e, u0);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t0);
dv2_free(t1);
dv2_free(u0);
dv2_free(u1);
}
}
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);
}
}
/**
* Compute the Miller loop for pairings of type G_1 x G_2 over the bits of a
* given parameter.
*
* @param[out] r - the result.
* @param[out] t - the resulting point.
* @param[in] p - the first pairing argument in affine coordinates.
* @param[in] q - the second pairing argument in affine coordinates.
* @param[in] a - the loop parameter.
*/
static void pp_mil_lit_k2(fp2_t r, ep_t *t, ep_t *p, ep_t *q, int m, bn_t a) {
fp2_t l, _l;
ep_t _q[m];
int i, j;
fp2_null(_l);
ep_null(_q);
TRY {
fp2_new(_l);
for (j = 0; j < m; j++) {
ep_null(_q[j]);
ep_new(_q[j]);
ep_copy(t[j], p[j]);
ep_neg(_q[j], q[j]);
}
for (i = bn_bits(a) - 2; i >= 0; i--) {
fp2_sqr(r, r);
for (j = 0; j < m; j++) {
pp_dbl_k2(l, t[j], t[j], _q[j]);
fp_copy(_l[0], l[1]);
fp_copy(_l[1], l[0]);
fp2_mul(r, r, _l);
if (bn_get_bit(a, i)) {
pp_add_k2(l, t[j], p[j], q[j]);
fp_copy(_l[0], l[1]);
fp_copy(_l[1], l[0]);
fp2_mul(r, r, _l);
}
}
}
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(_l);
fp2_free(m);
ep_free(_q);
}
}
void fp2_mul_frb(fp2_t c, fp2_t a, int i, int j) {
ctx_t *ctx = core_get();
if (i == 2) {
fp_mul(c[0], a[0], ctx->fp2_p2[j - 1]);
fp_mul(c[1], a[1], ctx->fp2_p2[j - 1]);
} else {
#if ALLOC == AUTO
if (i == 1) {
fp2_mul(c, a, ctx->fp2_p[j - 1]);
} else {
fp2_mul(c, a, ctx->fp2_p3[j - 1]);
}
#else
fp2_t t;
fp2_null(t);
TRY {
fp2_new(t);
if (i == 1) {
fp_copy(t[0], ctx->fp2_p[j - 1][0]);
fp_copy(t[1], ctx->fp2_p[j - 1][1]);
} else {
fp_copy(t[0], ctx->fp2_p3[j - 1][0]);
fp_copy(t[1], ctx->fp2_p3[j - 1][1]);
}
fp2_mul(c, a, t);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(t);
}
#endif
}
}
static void fp4_sqr_unr(dv2_t c, dv2_t d, fp2_t a, fp2_t b) {
fp2_t t;
dv2_t u0, u1;
fp2_null(t);
dv2_null(u0);
dv2_null(u1);
TRY {
fp2_new(t);
dv2_new(u0);
dv2_new(u1);
/* t0 = a^2. */
fp2_sqrn_low(u0, a);
/* t1 = b^2. */
fp2_sqrn_low(u1, b);
fp2_addm_low(t, a, b);
/* c = a^2 + b^2 * E. */
fp2_norh_low(c, u1);
fp2_addc_low(c, c, u0);
/* d = (a + b)^2 - a^2 - b^2 = 2 * a * b. */
fp2_addc_low(u1, u1, u0);
fp2_sqrn_low(d, t);
fp2_subc_low(d, d, u1);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t);
dv2_free(u0);
dv2_free(u1);
}
}
void ep2_curve_set_twist(int type) {
char str[2 * FP_BYTES + 1];
ctx_t *ctx = core_get();
ep2_t g;
fp2_t a;
fp2_t b;
bn_t r;
ep2_null(g);
fp2_null(a);
fp2_null(b);
bn_null(r);
ctx->ep2_is_twist = 0;
if (type == EP_MTYPE || type == EP_DTYPE) {
ctx->ep2_is_twist = type;
} else {
return;
}
TRY {
ep2_new(g);
fp2_new(a);
fp2_new(b);
bn_new(r);
switch (ep_param_get()) {
#if FP_PRIME == 158
case BN_P158:
ASSIGN(BN_P158);
break;
#elif FP_PRIME == 254
case BN_P254:
ASSIGN(BN_P254);
break;
#elif FP_PRIME == 256
case BN_P256:
ASSIGN(BN_P256);
break;
#elif FP_PRIME == 638
case BN_P638:
ASSIGN(BN_P638);
break;
case B12_P638:
ASSIGN(B12_P638);
break;
#endif
default:
(void)str;
THROW(ERR_NO_VALID);
break;
}
fp2_zero(g->z);
fp_set_dig(g->z[0], 1);
g->norm = 1;
ep2_copy(&(ctx->ep2_g), g);
fp_copy(ctx->ep2_a[0], a[0]);
fp_copy(ctx->ep2_a[1], a[1]);
fp_copy(ctx->ep2_b[0], b[0]);
fp_copy(ctx->ep2_b[1], b[1]);
bn_copy(&(ctx->ep2_r), r);
/* I don't have a better place for this. */
fp_prime_calc();
#if defined(EP_PRECO)
ep2_mul_pre((ep2_t *)ep2_curve_get_tab(), &(ctx->ep2_g));
#endif
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
ep2_free(g);
fp2_free(a);
fp2_free(b);
bn_free(r);
}
}
void fp12_sqr_lazyr(fp12_t c, fp12_t a) {
fp2_t t0, t1, t2, t3;
dv2_t u0, u1, u2, u3, u4, u5, u6, u7, u8, u9;
fp2_null(t0);
fp2_null(t1);
fp2_null(t2);
fp2_null(t3);
dv2_null(u0);
dv2_null(u1);
dv2_null(u2);
dv2_null(u3);
dv2_null(u4);
dv2_null(u5);
dv2_null(u6);
dv2_null(u7);
dv2_null(u8);
dv2_null(u9);
TRY {
fp2_new(t0);
fp2_new(t1);
fp2_new(t2);
fp2_new(t3);
dv2_new(u0);
dv2_new(u1);
dv2_new(u2);
dv2_new(u3);
dv2_new(u4);
dv2_new(u5);
dv2_new(u6);
dv2_new(u7);
dv2_new(u8);
dv2_new(u9);
/* a0 = (a00, a11). */
/* a1 = (a10, a02). */
/* a2 = (a01, a12). */
/* (t0,t1) = a0^2 */
fp4_sqr_unr(u0, u1, a[0][0], a[1][1]);
/* (t2,t3) = 2 * a1 * a2 */
fp4_mul_unr(u2, u3, a[1][0], a[0][2], a[0][1], a[1][2]);
fp2_addc_low(u2, u2, u2);
fp2_addc_low(u3, u3, u3);
/* (t4,t5) = a2^2. */
fp4_sqr_unr(u4, u5, a[0][1], a[1][2]);
/* c2 = a0 + a2. */
fp2_addm_low(t2, a[0][0], a[0][1]);
fp2_addm_low(t3, a[1][1], a[1][2]);
/* (t6,t7) = (a0 + a2 + a1)^2. */
fp2_addm_low(t0, t2, a[1][0]);
fp2_addm_low(t1, t3, a[0][2]);
fp4_sqr_unr(u6, u7, t0, t1);
/* c2 = (a0 + a2 - a1)^2. */
fp2_subm_low(t2, t2, a[1][0]);
fp2_subm_low(t3, t3, a[0][2]);
fp4_sqr_unr(u8, u9, t2, t3);
/* c2 = (c2 + (t6,t7))/2. */
#ifdef FP_SPACE
fp2_addd_low(u8, u8, u6);
fp2_addd_low(u9, u9, u7);
#else
fp2_addc_low(u8, u8, u6);
fp2_addc_low(u9, u9, u7);
#endif
fp_hlvd_low(u8[0], u8[0]);
fp_hlvd_low(u8[1], u8[1]);
fp_hlvd_low(u9[0], u9[0]);
fp_hlvd_low(u9[1], u9[1]);
/* (t6,t7) = (t6,t7) - c2 - (t2,t3). */
fp2_subc_low(u6, u6, u8);
fp2_subc_low(u7, u7, u9);
fp2_subc_low(u6, u6, u2);
fp2_subc_low(u7, u7, u3);
/* c2 = c2 - (t0,t1) - (t4,t5). */
fp2_subc_low(u8, u8, u0);
fp2_subc_low(u9, u9, u1);
fp2_subc_low(u8, u8, u4);
fp2_subc_low(u9, u9, u5);
fp2_rdcn_low(c[0][1], u8);
fp2_rdcn_low(c[1][2], u9);
/* c1 = (t6,t7) + (t4,t5) * E. */
fp2_nord_low(u9, u5);
fp2_addc_low(u6, u6, u9);
fp2_addc_low(u7, u7, u4);
fp2_rdcn_low(c[1][0], u6);
fp2_rdcn_low(c[0][2], u7);
/* c0 = (t0,t1) + (t2,t3) * E. */
fp2_nord_low(u9, u3);
fp2_addc_low(u0, u0, u9);
fp2_addc_low(u1, u1, u2);
fp2_rdcn_low(c[0][0], u0);
fp2_rdcn_low(c[1][1], u1);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t0);
fp2_free(t1);
fp2_free(t2);
fp2_free(t3);
dv2_free(u0);
dv2_free(u1);
dv2_free(u2);
dv2_free(u3);
dv2_free(u4);
dv2_free(u5);
dv2_free(u6);
dv2_free(u7);
dv2_free(u8);
dv2_free(u9);
}
}
inline static void fp6_mul_dxs_unr_lazyr(dv6_t c, fp6_t a, fp6_t b) {
dv2_t u0, u1, u2, u3;
fp2_t t0, t1;
dv2_null(u0);
dv2_null(u1);
dv2_null(u2);
dv2_null(u3);
fp2_null(t0);
fp2_null(t1);
TRY {
dv2_new(u0);
dv2_new(u1);
dv2_new(u2);
dv2_new(u3);
fp2_new(t0);
fp2_new(t1);
#ifdef RLC_FP_ROOM
fp2_mulc_low(u0, a[0], b[0]);
fp2_mulc_low(u1, a[1], b[1]);
fp2_addn_low(t0, a[0], a[1]);
fp2_addn_low(t1, b[0], b[1]);
/* c_1 = (a_0 + a_1)(b_0 + b_1) - a_0b_0 - a_1b_1 */
fp2_muln_low(u2, t0, t1);
fp2_subc_low(u2, u2, u0);
fp2_subc_low(c[1], u2, u1);
/* c_0 = a_0b_0 + E a_2b_1 */
fp2_mulc_low(u2, a[2], b[1]);
fp2_norh_low(c[0], u2);
fp2_addc_low(c[0], u0, c[0]);
/* c_2 = a_0b_2 + a_1b_1 */
fp2_mulc_low(u2, a[2], b[0]);
fp2_addc_low(c[2], u1, u2);
#else
fp2_muln_low(u0, a[0], b[0]);
fp2_muln_low(u1, a[1], b[1]);
fp2_addm_low(t0, a[0], a[1]);
fp2_addm_low(t1, b[0], b[1]);
/* c_1 = (a_0 + a_1)(b_0 + b_1) - a_0b_0 - a_1b_1 */
fp2_muln_low(u2, t0, t1);
fp2_subc_low(u2, u2, u0);
fp2_subc_low(c[1], u2, u1);
/* c_0 = a_0b_0 + E a_2b_1 */
fp2_muln_low(u2, a[2], b[1]);
fp2_nord_low(c[0], u2);
fp2_addc_low(c[0], u0, c[0]);
/* c_2 = a_0b_2 + a_1b_1 */
fp2_muln_low(u2, a[2], b[0]);
fp2_addc_low(c[2], u1, u2);
#endif
#ifdef RLC_FP_ROOM
#else
#endif
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
dv2_free(u0);
dv2_free(u1);
dv2_free(u2);
dv2_free(u3);
fp2_free(t0);
fp2_free(t1);
}
}
void pp_map_sim_weilp_k2(fp2_t r, ep_t *p, ep_t *q, int m) {
ep_t _p[m], _q[m], t0[m], t1[m];
fp2_t r0, r1;
bn_t n;
int i, j;
fp2_null(r0);
fp2_null(r1);
bn_null(r);
TRY {
fp2_new(r0);
fp2_new(r1);
bn_new(n);
for (i = 0; i < m; i++) {
ep_null(_p[i]);
ep_null(_q[i]);
ep_null(t0[i]);
ep_null(t1[i]);
ep_new(_p[i]);
ep_new(_q[i]);
ep_new(t0[i]);
ep_new(t1[i]);
}
j = 0;
for (i = 0; i < m; i++) {
if (!ep_is_infty(p[i]) && !ep_is_infty(q[i])) {
ep_norm(_p[j], p[i]);
ep_norm(_q[j++], q[i]);
}
}
ep_curve_get_ord(n);
bn_sub_dig(n, n, 1);
fp2_set_dig(r0, 1);
fp2_set_dig(r1, 1);
if (j > 0) {
pp_mil_lit_k2(r0, t0, _p, _q, j, n);
pp_mil_k2(r1, t1, _q, _p, j, n);
fp2_inv(r1, r1);
fp2_mul(r0, r0, r1);
fp2_inv(r1, r0);
fp2_inv_uni(r0, r0);
}
fp2_mul(r, r0, r1);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fp2_free(r0);
fp2_free(r1);
bn_free(n);
for (i = 0; i < m; i++) {
ep_free(_p[i]);
ep_free(_q[i]);
ep_free(t0[i]);
ep_free(t1[i]);
}
}
}
/**
* 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);
}
}
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 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);
}
}
/**
* 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 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
}
/**
* 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 fp12_sqr_pck_lazyr(fp12_t c, fp12_t a) {
fp2_t t0, t1;
dv2_t u0, u1, u2, u3, u4;
fp2_null(t0);
fp2_null(t1);
dv2_null(u0);
dv2_null(u1);
dv2_null(u2);
dv2_null(u3);
dv2_null(u4);
TRY {
fp2_new(t0);
fp2_new(t1);
dv2_new(u0);
dv2_new(u1);
dv2_new(u2);
dv2_new(u3);
dv2_new(u4);
fp2_sqrn_low(u0, a[0][1]);
fp2_sqrn_low(u1, a[1][2]);
fp2_norh_low(u4, u1);
fp2_addm_low(t0, a[0][1], a[1][2]);
fp2_sqrn_low(u2, t0);
fp2_addd_low(u3, u0, u1);
fp2_subc_low(u3, u2, u3);
fp2_rdcn_low(t0, u3);
fp2_sqrn_low(u2, a[1][0]);
fp2_addm_low(t1, a[1][0], a[0][2]);
fp2_sqrn_low(u3, t1);
fp2_norm_low(t1, t0);
fp2_addm_low(t0, t1, a[1][0]);
fp2_dblm_low(t0, t0);
fp2_addm_low(c[1][0], t0, t1);
fp2_sqrn_low(u1, a[0][2]);
fp2_addc_low(u4, u0, u4);
fp2_rdcn_low(t0, u4);
fp2_subm_low(t1, t0, a[0][2]);
fp2_dblm_low(t1, t1);
fp2_norh_low(u4, u1);
fp2_addm_low(c[0][2], t1, t0);
fp2_addc_low(u4, u2, u4);
fp2_rdcn_low(t0, u4);
fp2_subm_low(t1, t0, a[0][1]);
fp2_dblm_low(t1, t1);
fp2_addd_low(u0, u1, u2);
fp2_addm_low(c[0][1], t1, t0);
fp2_subc_low(u3, u3, u0);
fp2_rdcn_low(t0, u3);
fp2_addm_low(t1, t0, a[1][2]);
fp2_dblm_low(t1, t1);
fp2_addm_low(c[1][2], t1, t0);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fp2_free(t0);
fp2_free(t1);
dv2_free(u0);
dv2_free(u1);
dv2_free(u2);
dv2_free(u3);
dv2_free(u4);
}
}
/**
* 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_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);
}
}
