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);
}
}
/**
* Precomputes the square root of z.
*/
static void find_srz() {
int i;
fb_set_dig(fb_srz, 2);
for (i = 1; i < FB_BITS; i++) {
fb_sqr(fb_srz, fb_srz);
}
#ifdef FB_PRECO
for (i = 0; i <= 255; i++) {
fb_mul_dig(fb_tab_srz[i], fb_srz, i);
}
#endif
}
/**
* Precomputes the square root of z.
*/
static void find_srz() {
ctx_t *ctx = core_get();
fb_set_dig(ctx->fb_srz, 2);
for (int i = 1; i < FB_BITS; i++) {
fb_sqr(ctx->fb_srz, ctx->fb_srz);
}
#ifdef FB_PRECO
for (int i = 0; i <= 255; i++) {
fb_mul_dig(ctx->fb_tab_srz[i], ctx->fb_srz, i);
}
#endif
}
/**
* 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);
}
}
/**
* 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);
}
}
/**
* 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);
}
}