void eb_pck(eb_t r, const eb_t p) {
if (eb_curve_is_super()) {
/* z3 = y1/c. */
fb_inv(r->z, eb_curve_get_c());
fb_mul(r->z, r->z, p->y);
/* x3 = x1. */
fb_copy(r->x, p->x);
/* y3 = b(y1/c). */
fb_set_dig(r->y, fb_get_bit(r->z, 0));
/* z3 = 1. */
fb_set_dig(r->z, 1);
} else {
/* z3 = y1/x1. */
fb_inv(r->z, p->x);
fb_mul(r->z, r->z, p->y);
/* x3 = x1. */
fb_copy(r->x, p->x);
/* y3 = b(y1/x1). */
fb_set_dig(r->y, fb_get_bit(r->z, 0));
/* z3 = 1. */
fb_set_dig(r->z, 1);
}
r->norm = 1;
}
void fb2_inv(fb2_t c, fb2_t a) {
fb_t a0, a1, m0, m1;
fb_null(a0);
fb_null(a1);
fb_null(m0);
fb_null(m1);
TRY {
fb_new(a0);
fb_new(a1);
fb_new(m0);
fb_new(m1);
fb_add(a0, a[0], a[1]);
fb_sqr(m0, a[0]);
fb_mul(m1, a0, a[1]);
fb_add(a1, m0, m1);
fb_inv(a1, a1);
fb_mul(c[0], a0, a1);
fb_mul(c[1], a[1], a1);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fb_free(a0);
fb_free(a1);
fb_free(m0);
fb_free(m1);
}
}
void eb_hlv(eb_t r, const eb_t p) {
fb_t l, t;
fb_null(l);
fb_null(t);
TRY {
fb_new(l);
fb_new(t);
/* Solve l^2 + l = u + a. */
switch (eb_curve_opt_a()) {
case RLC_ZERO:
fb_copy(t, p->x);
break;
case RLC_ONE:
fb_add_dig(t, p->x, (dig_t)1);
break;
case RLC_TINY:
fb_add_dig(t, p->x, eb_curve_get_a()[0]);
break;
default:
fb_add(t, p->x, eb_curve_get_a());
break;
}
fb_slv(l, t);
if (p->norm == 1) {
/* Compute t = v + u * lambda. */
fb_mul(t, l, p->x);
fb_add(t, t, p->y);
} else {
/* Compute t = u * (u + lambda_P + lambda). */
fb_add(t, l, p->y);
fb_add(t, t, p->x);
fb_mul(t, t, p->x);
}
/* If Tr(t) = 0 then lambda_P = lambda, u = sqrt(t + u). */
if (fb_trc(t) == 0) {
fb_copy(r->y, l);
fb_add(t, t, p->x);
fb_srt(r->x, t);
} else {
/* Else lambda_P = lambda + 1, u = sqrt(t). */
fb_add_dig(r->y, l, 1);
fb_srt(r->x, t);
}
fb_set_dig(r->z, 1);
r->norm = 2;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(l);
fb_free(t);
}
}
void fb2_mul(fb2_t c, fb2_t a, fb2_t b) {
fb_t t0, t1, t2;
fb_null(t0);
fb_null(t1);
fb_null(t2);
TRY {
fb_new(t0);
fb_new(t1);
fb_new(t2);
fb_add(t0, a[0], a[1]);
fb_add(t1, b[0], b[1]);
fb_mul(t0, t0, t1);
fb_mul(t1, a[0], b[0]);
fb_mul(t2, a[1], b[1]);
fb_add(c[0], t1, t2);
fb_add(c[1], t0, t1);
} CATCH_ANY {
THROW(ERR_CAUGHT);
} FINALLY {
fb_free(t0);
fb_free(t1);
fb_free(t2);
}
}
int eb_upk(eb_t r, const eb_t p) {
fb_t t0, t1;
int res = 0;
fb_null(t0);
fb_null(t1);
TRY {
fb_new(t0);
fb_new(t1);
eb_rhs(t1, p);
if (eb_curve_is_super()) {
/* t0 = c^2. */
fb_sqr(t0, eb_curve_get_c());
/* t0 = 1/c^2. */
fb_inv(t0, t0);
/* t0 = t1/c^2. */
fb_mul(t0, t0, t1);
res = (fb_trc(t0) == 0);
/* Solve t1^2 + t1 = t0. */
fb_slv(t1, t0);
/* If this is not the correct solution, try the other. */
if (fb_get_bit(t1, 0) != fb_get_bit(p->y, 0)) {
fb_add_dig(t1, t1, 1);
}
/* x3 = x1, y3 = t1 * c, z3 = 1. */
fb_mul(r->y, t1, eb_curve_get_c());
} else {
fb_sqr(t0, p->x);
/* t0 = 1/x1^2. */
fb_inv(t0, t0);
/* t0 = t1/x1^2. */
fb_mul(t0, t0, t1);
res = (fb_trc(t0) == 0);
/* Solve t1^2 + t1 = t0. */
fb_slv(t1, t0);
/* If this is not the correct solution, try the other. */
if (fb_get_bit(t1, 0) != fb_get_bit(p->y, 0)) {
fb_add_dig(t1, t1, 1);
}
/* x3 = x1, y3 = t1 * x1, z3 = 1. */
fb_mul(r->y, t1, p->x);
}
fb_copy(r->x, p->x);
fb_set_dig(r->z, 1);
r->norm = 1;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
}
return res;
}
void fb_invn_low(dig_t *c, dig_t *a) {
int i, j, x, y, u[11];
fb_t table[11];
int len, *chain = fb_poly_get_chain(&len);
u[0] = 1;
u[1] = 2;
fb_copy(table[0], a);
fb_sqr(table[1], table[0]);
fb_mul(table[1], table[1], table[0]);
u[2] = u[1] + u[0];
fb_sqr(table[2], table[1]);
fb_mul(table[2], table[2], table[0]);
u[3] = u[2] + u[1];
fb_sqr(table[3], table[2]);
for (j = 1; j < u[1]; j++) {
fb_sqr(table[3], table[3]);
}
fb_mul(table[3], table[3], table[1]);
u[4] = 2 * u[3];
fb_sqr(table[4], table[3]);
for (j = 1; j < u[3]; j++) {
fb_sqr(table[4], table[4]);
}
fb_mul(table[4], table[4], table[3]);
u[5] = u[4] + u[3];
fb_sqr(table[5], table[4]);
for (j = 1; j < u[3]; j++) {
fb_sqr(table[5], table[5]);
}
fb_mul(table[5], table[5], table[3]);
u[6] = u[5] + u[4];
fb_itr(table[6], table[5], u[4], inv_tab[4]);
fb_mul(table[6], table[6], table[4]);
u[7] = 2 * u[6];
fb_itr(table[7], table[6], u[6], inv_tab[6]);
fb_mul(table[7], table[7], table[6]);
u[8] = u[7] + u[6];
fb_itr(table[8], table[7], u[6], inv_tab[6]);
fb_mul(table[8], table[8], table[6]);
u[9] = u[8] + u[7];
fb_itr(table[9], table[8], u[8], inv_tab[7]);
fb_mul(table[9], table[9], table[7]);
u[10] = 2 * u[9];
fb_itr(table[10], table[9], u[9], inv_tab[9]);
fb_mul(table[10], table[10], table[9]);
fb_sqr(c, table[10]);
}
void eb_rhs(fb_t rhs, const eb_t p) {
fb_t t0, t1;
fb_null(t0);
fb_null(t1);
TRY {
fb_new(t0);
fb_new(t1);
/* t0 = x1^2. */
fb_sqr(t0, p->x);
/* t1 = x1^3. */
fb_mul(t1, t0, p->x);
/* t1 = x1^3 + a * x1^2 + b. */
switch (eb_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fb_add(t1, t1, t0);
break;
case OPT_DIGIT:
fb_mul_dig(t0, t0, eb_curve_get_a()[0]);
fb_add(t1, t1, t0);
break;
default:
fb_mul(t0, t0, eb_curve_get_a());
fb_add(t1, t1, t0);
break;
}
switch (eb_curve_opt_b()) {
case OPT_ZERO:
break;
case OPT_ONE:
fb_add_dig(t1, t1, 1);
break;
case OPT_DIGIT:
fb_add_dig(t1, t1, eb_curve_get_b()[0]);
break;
default:
fb_add(t1, t1, eb_curve_get_b());
break;
}
fb_copy(rhs, t1);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
}
}
/**
* Normalizes a point represented in projective coordinates.
*
* @param r - the result.
* @param p - the point to normalize.
*/
static void eb_norm_super(eb_t r, const eb_t p, int flag) {
if (!p->norm) {
if (flag) {
fb_copy(r->z, p->z);
} else {
fb_inv(r->z, p->z);
}
fb_mul(r->x, p->x, r->z);
fb_mul(r->y, p->y, r->z);
fb_set_dig(r->z, 1);
}
r->norm = 1;
}
/**
* Normalizes a point represented in projective coordinates.
*
* @param[out] r - the result.
* @param[in] p - the point to normalize.
* @param[in] flag - if the Z coordinate is already inverted.
*/
static void eb_norm_ordin(eb_t r, eb_t p, int flag) {
if (!p->norm) {
if (flag) {
fb_copy(r->z, p->z);
} else {
fb_inv(r->z, p->z);
}
fb_mul(r->x, p->x, r->z);
fb_sqr(r->z, r->z);
fb_mul(r->y, p->y, r->z);
fb_set_dig(r->z, 1);
}
r->norm = 1;
}
int eb_is_valid(const eb_t p) {
eb_t t;
fb_t lhs;
int r = 0;
eb_null(t);
fb_null(lhs);
TRY {
eb_new(t);
fb_new(lhs);
eb_norm(t, p);
fb_mul(lhs, t->x, t->y);
eb_rhs(t->x, t);
fb_sqr(t->y, t->y);
fb_add(lhs, lhs, t->y);
r = (fb_cmp(lhs, t->x) == CMP_EQ) || eb_is_infty(p);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
eb_free(t);
fb_free(lhs);
}
return r;
}
void fb_inv_sim(fb_t *c, fb_t *a, int n) {
int i;
fb_t u, t[n];
for (i = 0; i < n; i++) {
fb_null(t[i]);
}
fb_null(u);
TRY {
for (i = 0; i < n; i++) {
fb_new(t[i]);
}
fb_new(u);
fb_copy(c[0], a[0]);
fb_copy(t[0], a[0]);
for (i = 1; i < n; i++) {
fb_copy(t[i], a[i]);
fb_mul(c[i], c[i - 1], a[i]);
}
fb_inv(u, c[n - 1]);
for (i = n - 1; i > 0; i--) {
fb_mul(c[i], u, c[i - 1]);
fb_mul(u, u, t[i]);
}
fb_copy(c[0], u);
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
for (i = 0; i < n; i++) {
fb_free(t[i]);
}
fb_free(u);
}
}
void eb_pck(eb_t r, const eb_t p) {
/* z3 = y1/x1. */
fb_inv(r->z, p->x);
fb_mul(r->z, r->z, p->y);
/* x3 = x1. */
fb_copy(r->x, p->x);
/* y3 = b(y1/x1). */
fb_set_dig(r->y, fb_get_bit(r->z, 0));
/* z3 = 1. */
fb_set_dig(r->z, 1);
r->norm = 1;
}
void eb_neg_projc(eb_t r, const eb_t p) {
fb_t t;
fb_null(t);
if (eb_is_infty(p)) {
eb_set_infty(r);
return;
}
if (p->norm) {
if (r != p) {
fb_copy(r->x, p->x);
fb_copy(r->z, p->z);
}
fb_add(r->y, p->x, p->y);
r->norm = 1;
return;
}
TRY {
fb_new(t);
fb_mul(t, p->x, p->z);
fb_add(r->y, p->y, t);
if (r != p) {
fb_copy(r->z, p->z);
fb_copy(r->x, p->x);
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t);
}
}
/**
* Adds a point represented in affine coordinates to a point represented in
* projective coordinates.
*
* @param[out] r - the result.
* @param[in] p - the affine point.
* @param[in] q - the projective point.
*/
static void eb_add_projc_mix(eb_t r, const eb_t p, const eb_t q) {
fb_t t0, t1, t2, t3, t4, t5;
fb_null(t0);
fb_null(t1);
fb_null(t2);
fb_null(t3);
fb_null(t4);
fb_null(t5);
TRY {
fb_new(t0);
fb_new(t1);
fb_new(t2);
fb_new(t3);
fb_new(t4);
fb_new(t5);
/* madd-2005-dl formulas: 7M + 4S + 9add + 1*4 + 3*2. */
/* http://www.hyperelliptic.org/EFD/g12o/auto-shortw-lopezdahab-1.html#addition-madd-2005-dl */
if (!p->norm) {
/* A = y1 + y2 * z1^2. */
fb_sqr(t0, p->z);
fb_mul(t0, t0, q->y);
fb_add(t0, t0, p->y);
/* B = x1 + x2 * z1. */
fb_mul(t1, p->z, q->x);
fb_add(t1, t1, p->x);
} else {
/* t0 = A = y1 + y2. */
fb_add(t0, p->y, q->y);
/* t1 = B = x1 + x2. */
fb_add(t1, p->x, q->x);
}
if (fb_is_zero(t1)) {
if (fb_is_zero(t0)) {
/* If t0 = 0 and t1 = 0, p = q, should have doubled! */
eb_dbl_projc(r, p);
} else {
/* If t0 = 0, r is infinity. */
eb_set_infty(r);
}
} else {
if (!p->norm) {
/* t2 = C = B * z1. */
fb_mul(t2, p->z, t1);
/* z3 = C^2. */
fb_sqr(r->z, t2);
/* t1 = B^2. */
fb_sqr(t1, t1);
/* t1 = A + B^2. */
fb_add(t1, t0, t1);
} else {
/* If z1 = 0, t2 = C = B. */
fb_copy(t2, t1);
/* z3 = B^2. */
fb_sqr(r->z, t1);
/* t1 = A + z3. */
fb_add(t1, t0, r->z);
}
/* t3 = D = x2 * z3. */
fb_mul(t3, r->z, q->x);
/* t4 = (y2 + x2). */
fb_add(t4, q->x, q->y);
/* z3 = A^2. */
fb_sqr(r->x, t0);
/* t1 = A + B^2 + a2 * C. */
switch (eb_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fb_add(t1, t1, t2);
break;
case OPT_DIGIT:
/* t5 = a2 * C. */
fb_mul_dig(t5, t2, eb_curve_get_a()[0]);
fb_add(t1, t1, t5);
break;
default:
/* t5 = a2 * C. */
fb_mul(t5, eb_curve_get_a(), t2);
fb_add(t1, t1, t5);
break;
}
/* t1 = C * (A + B^2 + a2 * C). */
fb_mul(t1, t1, t2);
/* x3 = A^2 + C * (A + B^2 + a2 * C). */
fb_add(r->x, r->x, t1);
/* t3 = D + x3. */
fb_add(t3, t3, r->x);
/* t2 = A * B. */
fb_mul(t2, t0, t2);
/* y3 = (D + x3) * (A * B + z3). */
fb_add(r->y, t2, r->z);
fb_mul(r->y, r->y, t3);
/* t0 = z3^2. */
fb_sqr(t0, r->z);
/* t0 = (y2 + x2) * z3^2. */
fb_mul(t0, t0, t4);
/* y3 = (D + x3) * (A * B + z3) + (y2 + x2) * z3^2. */
fb_add(r->y, r->y, t0);
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
fb_free(t2);
fb_free(t3);
fb_free(t4);
fb_free(t5);
}
}
/**
* Adds a point represented in affine coordinates to a point represented in
* projective coordinates.
*
* @param r - the result.
* @param p - the affine point.
* @param q - the projective point.
*/
static void eb_add_projc_ordin_mix(eb_t r, eb_t p, eb_t q) {
fb_t t0, t1, t2, t3, t4, t5;
fb_null(t0);
fb_null(t1);
fb_null(t2);
fb_null(t3);
fb_null(t4);
fb_null(t5);
TRY {
fb_new(t0);
fb_new(t1);
fb_new(t2);
fb_new(t3);
fb_new(t4);
fb_new(t5);
if (!p->norm) {
/* A = y1 + y2 * z1^2. */
fb_sqr(t0, p->z);
fb_mul(t0, t0, q->y);
fb_add(t0, t0, p->y);
/* B = x1 + x2 * z1. */
fb_mul(t1, p->z, q->x);
fb_add(t1, t1, p->x);
} else {
/* t0 = A = y1 + y2. */
fb_add(t0, p->y, q->y);
/* t1 = B = x1 + x2. */
fb_add(t1, p->x, q->x);
}
if (fb_is_zero(t1)) {
if (fb_is_zero(t0)) {
/* If t0 = 0 and t1 = 0, p = q, should have doubled! */
eb_dbl_projc(r, p);
} else {
/* If t0 = 0, r is infinity. */
eb_set_infty(r);
}
} else {
if (!p->norm) {
/* t2 = C = B * z1. */
fb_mul(t2, p->z, t1);
/* z3 = C^2. */
fb_sqr(r->z, t2);
/* t1 = B^2. */
fb_sqr(t1, t1);
/* t1 = A + B^2. */
fb_add(t1, t0, t1);
} else {
/* If z1 = 0, t2 = C = B. */
fb_copy(t2, t1);
/* z3 = B^2. */
fb_sqr(r->z, t1);
/* t1 = A + z3. */
fb_add(t1, t0, r->z);
}
/* t3 = D = x2 * z3. */
fb_mul(t3, r->z, q->x);
/* t4 = (y2 + x2). */
fb_add(t4, q->x, q->y);
/* z3 = A^2. */
fb_sqr(r->x, t0);
/* t1 = A + B^2 + a2 * C. */
switch (eb_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fb_add(t1, t1, t2);
break;
case OPT_DIGIT:
/* t5 = a2 * C. */
fb_mul_dig(t5, t2, eb_curve_get_a()[0]);
fb_add(t1, t1, t5);
break;
default:
/* t5 = a2 * C. */
fb_mul(t5, eb_curve_get_a(), t2);
fb_add(t1, t1, t5);
break;
}
/* t1 = C * (A + B^2 + a2 * C). */
fb_mul(t1, t1, t2);
/* x3 = A^2 + C * (A + B^2 + a2 * C). */
fb_add(r->x, r->x, t1);
/* t3 = D + x3. */
fb_add(t3, t3, r->x);
/* t2 = A * B. */
fb_mul(t2, t0, t2);
/* y3 = (D + x3) * (A * B + z3). */
fb_add(r->y, t2, r->z);
fb_mul(r->y, r->y, t3);
/* t0 = z3^2. */
fb_sqr(t0, r->z);
/* t0 = (y2 + x2) * z3^2. */
fb_mul(t0, t0, t4);
/* y3 = (D + x3) * (A * B + z3) + (y2 + x2) * z3^2. */
fb_add(r->y, r->y, t0);
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
fb_free(t2);
fb_free(t3);
fb_free(t4);
fb_free(t5);
}
}
/**
* Adds two points represented in projective coordinates on an ordinary binary
* elliptic curve.
*
* @param r - the result.
* @param p - the first point to add.
* @param q - the second point to add.
*/
static void eb_add_projc_ordin(eb_t r, eb_t p, eb_t q) {
#if defined(EB_MIXED) && defined(STRIP)
eb_add_projc_ordin_mix(r, p, q);
#else /* General addition. */
fb_t t0, t1, t2, t3, t4, t5, t6, t7;
fb_null(t0);
fb_null(t1);
fb_null(t2);
fb_null(t3);
fb_null(t4);
fb_null(t5);
fb_null(t6);
fb_null(t7);
TRY {
fb_new(t0);
fb_new(t1);
fb_new(t2);
fb_new(t3);
fb_new(t4);
fb_new(t5);
fb_new(t6);
fb_new(t7);
if (q->norm) {
eb_add_projc_ordin_mix(r, p, q);
} else {
/* t0 = B = x2 * z1. */
fb_mul(t0, q->x, p->z);
/* A = x1 * z2 */
fb_mul(t1, p->x, q->z);
/* t2 = E = A + B. */
fb_add(t2, t1, t0);
/* t3 = D = B^2. */
fb_sqr(t3, t0);
/* t4 = C = A^2. */
fb_sqr(t4, t1);
/* t5 = F = C + D. */
fb_add(t5, t3, t4);
/* t6 = H = y2 * z1^2. */
fb_sqr(t6, p->z);
fb_mul(t6, t6, q->y);
/* t7 = G = y1 * z2^2. */
fb_sqr(t7, q->z);
fb_mul(t7, t7, p->y);
/* t3 = D + H. */
fb_add(t3, t3, t6);
/* t4 = C + G. */
fb_add(t4, t4, t7);
/* t6 = I = G + H. */
fb_add(t6, t7, t6);
/* If E is zero. */
if (fb_is_zero(t2)) {
if (fb_is_zero(t6)) {
/* If I is zero, p = q, should have doubled. */
eb_dbl_projc(r, p);
} else {
/* If I is not zero, q = -p, r = infinity. */
eb_set_infty(r);
}
} else {
/* t6 = J = I * E. */
fb_mul(t6, t6, t2);
/* z3 = F * z1 * z2. */
fb_mul(r->z, p->z, q->z);
fb_mul(r->z, t5, r->z);
/* t4 = B * (C + G). */
fb_mul(t4, t0, t4);
/* t2 = A * J. */
fb_mul(t2, t1, t6);
/* x3 = A * (D + H) + B * (C + G). */
fb_mul(r->x, t1, t3);
fb_add(r->x, r->x, t4);
/* t7 = F * G. */
fb_mul(t7, t7, t5);
/* Y3 = (A * J + F * G) * F + (J + z3) * x3. */
fb_add(r->y, t2, t7);
fb_mul(r->y, r->y, t5);
/* t7 = (J + z3) * x3. */
fb_add(t7, t6, r->z);
fb_mul(t7, t7, r->x);
fb_add(r->y, r->y, t7);
}
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
fb_free(t2);
fb_free(t3);
fb_free(t4);
fb_free(t5);
fb_free(t6);
fb_free(t7);
}
#endif
}
/**
* Doubles a point represented in projective coordinates on an ordinary binary
* elliptic curve.
*
* @param[out] r - the result.
* @param[in] p - the point to double.
*/
static void eb_dbl_projc_imp(eb_t r, const eb_t p) {
fb_t t0, t1;
fb_null(t0);
fb_null(t1);
TRY {
fb_new(t0);
fb_new(t1);
/* t0 = B = x1^2. */
fb_sqr(t0, p->x);
/* C = B + y1. */
fb_add(r->y, t0, p->y);
if (!p->norm) {
/* A = x1 * z1. */
fb_mul(t1, p->x, p->z);
/* z3 = A^2. */
fb_sqr(r->z, t1);
} else {
/* if z1 = 1, A = x1. */
fb_copy(t1, p->x);
/* if z1 = 1, z3 = x1^2. */
fb_copy(r->z, t0);
}
/* t1 = D = A * C. */
fb_mul(t1, t1, r->y);
/* C^2 + D. */
fb_sqr(r->y, r->y);
fb_add(r->x, t1, r->y);
/* C^2 + D + a2 * z3. */
switch (eb_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fb_add(r->x, r->z, r->x);
break;
case OPT_DIGIT:
fb_mul_dig(r->y, r->z, eb_curve_get_a()[0]);
fb_add(r->x, r->y, r->x);
break;
default:
fb_mul(r->y, r->z, eb_curve_get_a());
fb_add(r->x, r->y, r->x);
break;
}
/* t1 = (D + z3). */
fb_add(t1, t1, r->z);
/* t0 = B^2. */
fb_sqr(t0, t0);
/* t0 = B^2 * z3. */
fb_mul(t0, t0, r->z);
/* y3 = (D + z3) * r3 + B^2 * z3. */
fb_mul(r->y, t1, r->x);
fb_add(r->y, r->y, t0);
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
}
}
/**
* Doubles a point represented in affine coordinates on an ordinary binary
* elliptic curve.
*
* @param[out] r - the result.
* @param[in] p - the point to double.
*/
static void eb_dbl_basic_imp(eb_t r, const eb_t p) {
fb_t t0, t1, t2;
fb_null(t0);
fb_null(t1);
fb_null(t2);
TRY {
fb_new(t0);
fb_new(t1);
fb_new(t2);
/* t0 = 1/x1. */
fb_inv(t0, p->x);
/* t0 = y1/x1. */
fb_mul(t0, t0, p->y);
/* t0 = lambda = x1 + y1/x1. */
fb_add(t0, t0, p->x);
/* t1 = lambda^2. */
fb_sqr(t1, t0);
/* t2 = lambda^2 + lambda. */
fb_add(t2, t1, t0);
/* t2 = lambda^2 + lambda + a2. */
switch (eb_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fb_add_dig(t2, t2, (dig_t)1);
break;
case OPT_DIGIT:
fb_add_dig(t2, t2, eb_curve_get_a()[0]);
break;
default:
fb_add(t2, t2, eb_curve_get_a());
break;
}
/* t1 = x1 + x3. */
fb_add(t1, t2, p->x);
/* t1 = lambda * (x1 + x3). */
fb_mul(t1, t0, t1);
fb_copy(r->x, t2);
/* y3 = lambda * (x1 + x3) + x3 + y1. */
fb_add(t1, t1, r->x);
fb_add(r->y, t1, p->y);
fb_copy(r->z, p->z);
r->norm = 1;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
fb_free(t2);
}
}
/**
* Normalizes a point represented in lambda-coordinates.
*
* @param[out] r - the result.
* @param[in] p - the point to normalize.
*/
static void eb_norm_halve(eb_t r, const eb_t p) {
fb_add(r->y, p->x, p->y);
fb_mul(r->y, r->y, p->x);
fb_copy(r->x, p->x);
r->norm = 1;
}
/**
* Adds two points represented in affine coordinates on an ordinary binary
* 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 eb_add_basic_imp(eb_t r, const eb_t p, const eb_t q) {
fb_t t0, t1, t2;
fb_null(t0);
fb_null(t1);
fb_null(t2);
TRY {
fb_new(t0);
fb_new(t1);
fb_new(t2);
/* t0 = (y1 + y2). */
fb_add(t0, p->y, q->y);
/* t1 = (x1 + x2). */
fb_add(t1, p->x, q->x);
if (fb_is_zero(t1)) {
if (fb_is_zero(t0)) {
/* If t1 is zero and t0 is zero, p = q, should have doubled. */
eb_dbl_basic(r, p);
} else {
/* If t0 is not zero and t1 is zero, q = -p and r = infinity. */
eb_set_infty(r);
}
} else {
/* t2 = 1/(x1 + x2). */
fb_inv(t2, t1);
/* t0 = lambda = (y1 + y2)/(x1 + x2). */
fb_mul(t0, t0, t2);
/* t2 = lambda^2. */
fb_sqr(t2, t0);
/* t2 = lambda^2 + lambda + x1 + x2 + a. */
fb_add(t2, t2, t0);
fb_add(t2, t2, t1);
switch (eb_curve_opt_a()) {
case OPT_ZERO:
break;
case OPT_ONE:
fb_add_dig(t2, t2, (dig_t)1);
break;
case OPT_DIGIT:
fb_add_dig(t2, t2, eb_curve_get_a()[0]);
break;
default:
fb_add(t2, t2, eb_curve_get_a());
break;
}
/* y3 = lambda*(x3 + x1) + x3 + y1. */
fb_add(t1, t2, p->x);
fb_mul(t1, t1, t0);
fb_add(t1, t1, t2);
fb_add(r->y, p->y, t1);
/* x3 = lambda^2 + lambda + x1 + x2 + a. */
fb_copy(r->x, t2);
fb_copy(r->z, p->z);
r->norm = 1;
}
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t0);
fb_free(t1);
fb_free(t2);
}
}
void eb_neg_projc(eb_t r, const eb_t p) {
fb_t t;
fb_null(t);
if (eb_is_infty(p)) {
eb_set_infty(r);
return;
}
if (p->norm) {
if (r != p) {
fb_copy(r->x, p->x);
fb_copy(r->z, p->z);
}
#if defined(EB_SUPER)
if (eb_curve_is_super()) {
switch (eb_curve_opt_c()) {
case OPT_ZERO:
fb_copy(r->y, p->y);
break;
case OPT_ONE:
fb_add_dig(r->y, p->y, (dig_t)1);
break;
case OPT_DIGIT:
fb_add_dig(r->y, p->y, eb_curve_get_c()[0]);
break;
default:
fb_add(r->y, p->y, eb_curve_get_c());
break;
}
r->norm = 1;
return;
}
#endif
fb_add(r->y, p->x, p->y);
r->norm = 1;
return;
}
#if defined(EB_SUPER)
if (eb_curve_is_super()) {
fb_add(r->y, p->y, p->z);
fb_copy(r->z, p->z);
fb_copy(r->x, p->x);
r->norm = 0;
return;
}
#endif
TRY {
fb_new(t);
fb_mul(t, p->x, p->z);
fb_add(r->y, p->y, t);
if (r != p) {
fb_copy(r->z, p->z);
fb_copy(r->x, p->x);
}
r->norm = 0;
}
CATCH_ANY {
THROW(ERR_CAUGHT);
}
FINALLY {
fb_free(t);
}
}