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 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); } }
/** * 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 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); } }
/** * 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 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); } }
/** * 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); } }