diff options
| -rw-r--r-- | ecdsa.c | 386 |
1 files changed, 101 insertions, 285 deletions
@@ -77,17 +77,6 @@ #include "asn1_internal.h" /* - * Whether to use the new mixed Jacobian-affine point addition and - * point scalar multiplication algorithms. In theory these are both - * faster and simpler than the previous approach, so this conditional - * will probably go away soon. - */ - -#ifndef HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION -#define HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION 1 -#endif - -/* * Whether we're using static test vectors instead of the random * number generator. Do NOT enable this in production (doh). */ @@ -142,6 +131,20 @@ typedef struct { * * EC points are stored in Jacobian format such that (x, y, z) => * (x/z**2, y/z**3, 1) when interpretted as affine coordinates. + * + * There are really three different representations in use here: + * + * 1) Plain affine representation (z == 1); + * 2) Montgomery form affine representation (z == curve->mu); and + * 3) Montgomery form Jacobian representation (whatever). + * + * Only form (1) is ever visible outside this module. We perform + * explicit conversions from form (1) to form (2) and from forms (2,3) + * to form (1). Form (3) only occurs as the result of compuation. + * + * In theory, we could shave some microscopic amount of time off of + * signature verification by supporting an explicit conversion from + * form (3) to form (2), but it's not worth the additional complexity. */ typedef struct { @@ -347,6 +350,77 @@ static inline void point_copy(const ec_point_t * const P, ec_point_t *R) } /** + * Convert a point into Montgomery form. + * @param P [in/out] The point to map + * @param curve The curve parameters structure + */ + +static inline hal_error_t point_to_montgomery(ec_point_t *P, + const ecdsa_curve_t * const curve) +{ + assert(P != NULL && curve != NULL); + + if (fp_cmp_d(unconst_fp_int(P->z), 1) != FP_EQ) + return HAL_ERROR_BAD_ARGUMENTS; + + if (fp_mulmod(unconst_fp_int(P->x), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), P->x) != FP_OKAY || + fp_mulmod(unconst_fp_int(P->y), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), P->y) != FP_OKAY) + return HAL_ERROR_IMPOSSIBLE; + + fp_copy(unconst_fp_int(curve->mu), P->z); + return HAL_OK; +} + +/** + * Map a point in projective Jacbobian coordinates back to affine + * space. This also converts back from Montgomery to plain form. + * @param P [in/out] The point to map + * @param curve The curve parameters structure + * + * It's not possible to represent the point at infinity in plain + * affine coordinates, and the calling function will have to handle + * the point at infinity specially in any case, so we declare this to + * be the calling function's problem. + */ + +static inline hal_error_t point_to_affine(ec_point_t *P, + const ecdsa_curve_t * const curve) +{ + assert(P != NULL && curve != NULL); + + if (point_is_infinite(P)) + return HAL_ERROR_IMPOSSIBLE; + + hal_error_t err = HAL_ERROR_IMPOSSIBLE; + + fp_int t1[1]; fp_init(t1); + fp_int t2[1]; fp_init(t2); + + fp_int * const q = unconst_fp_int(curve->q); + + fp_montgomery_reduce(P->z, q, curve->rho); + + if (fp_invmod (P->z, q, t1) != FP_OKAY || /* t1 = 1 / z */ + fp_sqrmod (t1, q, t2) != FP_OKAY || /* t2 = 1 / z**2 */ + fp_mulmod (t1, t2, q, t1) != FP_OKAY) /* t1 = 1 / z**3 */ + goto fail; + + fp_mul (P->x, t2, P->x); /* x = x / z**2 */ + fp_mul (P->y, t1, P->y); /* y = y / z**3 */ + fp_set (P->z, 1); /* z = 1 */ + + fp_montgomery_reduce(P->x, q, curve->rho); + fp_montgomery_reduce(P->y, q, curve->rho); + + err = HAL_OK; + + fail: + fp_zero(t1); + fp_zero(t2); + return err; +} + +/** * Double an EC point. * @param P The point to double * @param R [out] The destination of the double @@ -403,8 +477,6 @@ static inline void point_double(const ec_point_t * const P, fp_zero(alpha); fp_zero(beta); fp_zero(gamma); fp_zero(delta); fp_zero(t1); fp_zero(t2); } -#if HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION - /** * Add two EC points * @param P The point to add @@ -430,7 +502,7 @@ static inline void point_add(const ec_point_t * const P, assert(P != NULL && Q != NULL && R != NULL && curve != NULL); /* - * Q must be in affine form. + * Q must be affine in Montgomery form. */ assert(fp_cmp(unconst_fp_int(Q->z), unconst_fp_int(curve->mu)) == FP_EQ); @@ -517,175 +589,12 @@ static inline void point_add(const ec_point_t * const P, point_set_infinite(R, curve); } -#else /* HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION */ - -/** - * Add two EC points - * @param P The point to add - * @param Q The point to add - * @param R [out] The destination of the double - * @param curve The curve parameters structure - * - * Algorithm is add-2007-bl from - * http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html - * - * The special cases for P == Q and P == -Q are unfortunate, but are - * probably unavoidable for this type of curve. - */ - -static inline void point_add(const ec_point_t * const P, - const ec_point_t * const Q, - ec_point_t *R, - const ecdsa_curve_t * const curve) -{ - assert(P != NULL && Q != NULL && R != NULL && curve != NULL); - - if (point_is_infinite(P)) { - point_copy(Q, R); - return; - } - if (point_is_infinite(Q)) { - point_copy(P, R); - return; - } - - - if (fp_cmp(unconst_fp_int(P->x), unconst_fp_int(Q->x)) == FP_EQ && - fp_cmp(unconst_fp_int(P->z), unconst_fp_int(Q->z)) == FP_EQ) { - - /* - * If P == Q, we have to use the doubling algorithm instead. - */ - - if (fp_cmp(unconst_fp_int(P->y), unconst_fp_int(Q->y)) == FP_EQ) - return point_double(P, R, curve); - - fp_int Qy_neg[1]; - fp_sub(unconst_fp_int(curve->q), unconst_fp_int(Q->y), Qy_neg); - const int zero_sum = fp_cmp(unconst_fp_int(P->y), Qy_neg) == FP_EQ; - fp_zero(Qy_neg); - - /* - * If P == -Q, P + Q is the point at infinity. Which can't be - * expressed in affine coordinates, but that's not this function's - * problem. - */ - - if (zero_sum) - return point_set_infinite(R, curve); - } - - fp_int Z1Z1[1], Z2Z2[1], U1[1], U2[1], S1[1], S2[1], H[1], I[1], J[1], r[1], V[1], t[1]; - - fp_init(Z1Z1), fp_init(Z2Z2), fp_init(U1), fp_init(U2), fp_init(S1), fp_init(S2); - fp_init(H), fp_init(I), fp_init(J), fp_init(r), fp_init(V), fp_init(t); - - ff_sqr (curve, P->z, Z1Z1); /* Z1Z1 = Pz ** 2 */ - - ff_sqr (curve, Q->z, Z2Z2); /* Z2Z1 = Qz ** 2 */ - - ff_mul (curve, P->x, Z2Z2, U1); /* U1 = Px * Z2Z2 */ - - ff_mul (curve, Q->x, Z1Z1, U2); /* U2 = Qx * Z1Z1 */ - - ff_mul (curve, Q->z, Z2Z2, S1); /* S1 = Py * (Qz ** 3) */ - ff_mul (curve, P->y, S1, S1); - - ff_mul (curve, P->z, Z1Z1, S2); /* S2 = Qy * (Pz ** 3) */ - ff_mul (curve, Q->y, S2, S2); - - ff_sub (curve, U2, U1, H); /* H = U2 - U1 */ - - ff_add (curve, H, H, I); /* I = (2 * H) ** 2 */ - ff_sqr (curve, I, I); - - ff_mul (curve, H, I, J); /* J = H * I */ - - ff_sub (curve, S2, S1, r); /* r = 2 * (S2 - S1) */ - ff_add (curve, r, r, r); - - ff_mul (curve, U1, I, V); /* V = U1 * I */ - - ff_sqr (curve, r, R->x); /* Rx = (r ** 2) - J - (2 * V) */ - ff_sub (curve, R->x, J, R->x); - ff_sub (curve, R->x, V, R->x); - ff_sub (curve, R->x, V, R->x); - - ff_sub (curve, V, R->x, R->y); /* Ry = (r * (V - Rx)) - (2 * S1 * J) */ - ff_mul (curve, r, R->y, R->y); - ff_mul (curve, S1, J, t); - ff_sub (curve, R->y, t, R->y); - ff_sub (curve, R->y, t, R->y); - - ff_add (curve, P->z, Q->z, R->z); /* Rz = (((Pz + Qz) ** 2) - Z1Z1 - Z2Z2) * H */ - ff_sqr (curve, R->z, R->z); - ff_sub (curve, R->z, Z1Z1, R->z); - ff_sub (curve, R->z, Z2Z2, R->z); - ff_mul (curve, R->z, H, R->z); - - fp_zero(Z1Z1), fp_zero(Z2Z2), fp_zero(U1), fp_zero(U2), fp_zero(S1), fp_zero(S2); - fp_zero(H), fp_zero(I), fp_zero(J), fp_zero(r), fp_zero(V), fp_zero(t); -} - -#endif /* HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION */ - -/** - * Map a point in projective Jacbobian coordinates back to affine space - * @param P [in/out] The point to map - * @param curve The curve parameters structure - * - * It's not possible to represent the point at infinity in affine - * coordinates, and the calling function will have to handle this - * specially in any case, so we declare this to be the calling - * function's problem. - */ - -static inline hal_error_t point_to_affine(ec_point_t *P, - const ecdsa_curve_t * const curve) -{ - assert(P != NULL && curve != NULL); - - if (point_is_infinite(P)) - return HAL_ERROR_IMPOSSIBLE; - - hal_error_t err = HAL_ERROR_IMPOSSIBLE; - - fp_int t1[1]; fp_init(t1); - fp_int t2[1]; fp_init(t2); - - fp_int * const q = unconst_fp_int(curve->q); - - fp_montgomery_reduce(P->z, q, curve->rho); - - if (fp_invmod (P->z, q, t1) != FP_OKAY || /* t1 = 1 / z */ - fp_sqrmod (t1, q, t2) != FP_OKAY || /* t2 = 1 / z**2 */ - fp_mulmod (t1, t2, q, t1) != FP_OKAY) /* t1 = 1 / z**3 */ - goto fail; - - fp_mul (P->x, t2, P->x); /* x = x / z**2 */ - fp_mul (P->y, t1, P->y); /* y = y / z**3 */ - fp_set (P->z, 1); /* z = 1 */ - - fp_montgomery_reduce(P->x, q, curve->rho); - fp_montgomery_reduce(P->y, q, curve->rho); - - err = HAL_OK; - - fail: - fp_zero(t1); - fp_zero(t2); - return err; -} - -#if HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION - /** * Perform a point multiplication. * @param k The scalar to multiply by * @param P The base point * @param R [out] Destination for kP * @param curve Curve parameters - * @param map Boolean whether to map back to affine (1: map, 0: leave projective) * @return HAL_OK on success * * P must be in affine form. @@ -694,29 +603,27 @@ static inline hal_error_t point_to_affine(ec_point_t *P, static hal_error_t point_scalar_multiply(const fp_int * const k, const ec_point_t * const P_, ec_point_t *R, - const ecdsa_curve_t * const curve, - const int map) + const ecdsa_curve_t * const curve) { assert(k != NULL && P_ != NULL && R != NULL && curve != NULL); if (fp_iszero(k) || fp_cmp_d(unconst_fp_int(P_->z), 1) != FP_EQ) return HAL_ERROR_BAD_ARGUMENTS; + hal_error_t err; + /* * Convert P to Montgomery form. */ ec_point_t P[1]; - memset(P, 0, sizeof(P)); + memcpy(P, P_, sizeof(P)); - if (fp_mulmod(unconst_fp_int(P_->x), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), P->x) != FP_OKAY || - fp_mulmod(unconst_fp_int(P_->y), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), P->y) != FP_OKAY) { + if ((err = point_to_montgomery(P, curve)) != HAL_OK) { memset(P, 0, sizeof(P)); - return HAL_ERROR_IMPOSSIBLE; + return err; } - fp_copy(unconst_fp_int(curve->mu), P->z); - /* * Initialize table. * M[0] is a dummy for constant timing. @@ -750,12 +657,12 @@ static hal_error_t point_scalar_multiply(const fp_int * const k, } /* - * Copy result, map back to affine if requested, then done. + * Copy result, map back to affine, then done. */ point_copy(M[1], R); - hal_error_t err = map ? point_to_affine(R, curve) : HAL_OK; + err = point_to_affine(R, curve); memset(P, 0, sizeof(P)); memset(M, 0, sizeof(M)); @@ -763,95 +670,6 @@ static hal_error_t point_scalar_multiply(const fp_int * const k, return err; } -#else /* HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION */ - -/** - * Perform a point multiplication. - * @param k The scalar to multiply by - * @param P The base point - * @param R [out] Destination for kP - * @param curve Curve parameters - * @param map Boolean whether to map back to affine (1: map, 0: leave projective) - * @return HAL_OK on success - * - * This implementation uses the "Montgomery Ladder" approach, which is - * relatively robust against timing channel attacks if nothing else - * goes wrong, but many other things can indeed go wrong. - */ - -static hal_error_t point_scalar_multiply(const fp_int * const k, - const ec_point_t * const P, - ec_point_t *R, - const ecdsa_curve_t * const curve, - const int map) -{ - assert(k != NULL && P != NULL && R != NULL && curve != NULL); - - if (fp_iszero(k)) - return HAL_ERROR_BAD_ARGUMENTS; - - /* - * Convert to Montgomery form and initialize table. Initial values: - * - * M[0] = 1P - * M[1] = 2P - * M[2] = don't care, only used for timing-attack resistance - */ - - ec_point_t M[3][1]; - memset(M, 0, sizeof(M)); - - if (fp_mulmod(unconst_fp_int(P->x), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), M[0]->x) != FP_OKAY || - fp_mulmod(unconst_fp_int(P->y), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), M[0]->y) != FP_OKAY || - fp_mulmod(unconst_fp_int(P->z), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), M[0]->z) != FP_OKAY) { - memset(M, 0, sizeof(M)); - return HAL_ERROR_IMPOSSIBLE; - } - - point_double(M[0], M[1], curve); - - /* - * Walk down bits of the scalar, performing dummy operations to mask - * timing while hunting for the most significant bit of the scalar. - * - * Note that, in order for this timing protection to work, the - * number of iterations in the loop has to depend on the order of - * the base point rather than on the scalar. - */ - - int dummy_mode = 1; - - for (int bit_index = fp_count_bits(unconst_fp_int(curve->n)) - 1; bit_index >= 0; bit_index--) { - - const int digit_index = bit_index / DIGIT_BIT; - const fp_digit digit = digit_index < k->used ? k->dp[digit_index] : 0; - const fp_digit mask = ((fp_digit) 1) << (bit_index % DIGIT_BIT); - const int bit = (digit & mask) != 0; - - if (dummy_mode) { - point_add (M[0], M[1], M[2], curve); - point_double (M[1], M[2], curve); - dummy_mode = !bit; /* Dummy until we find MSB */ - } - - else { - point_add (M[0], M[1], M[bit^1], curve); - point_double (M[bit], M[bit], curve); - } - } - - /* - * Copy result out, map back to affine if requested, then done. - */ - - point_copy(M[0], R); - hal_error_t err = map ? point_to_affine(R, curve) : HAL_OK; - memset(M, 0, sizeof(M)); - return err; -} - -#endif /* HAL_ECDSA_USE_MIXED_JACOBIAN_AFFINE_ADDITION */ - /* * Testing only: ECDSA key generation and signature both have a * critical dependency on random numbers, but we can't use the random @@ -946,7 +764,7 @@ static hal_error_t point_pick_random(const ecdsa_curve_t * const curve, fp_copy(curve->Gy, P->y); fp_set(P->z, 1); - return point_scalar_multiply(k, P, P, curve, 1); + return point_scalar_multiply(k, P, P, curve); } /* @@ -1752,21 +1570,19 @@ hal_error_t hal_ecdsa_verify(const hal_ecdsa_key_t * const key, fp_copy(unconst_fp_int(curve->Gy), u1G->y); fp_set(u1G->z, 1); - if ((err = point_scalar_multiply(u1, u1G, u1G, curve, 1)) != HAL_OK || - (err = point_scalar_multiply(u2, key->Q, u2Q, curve, 1)) != HAL_OK) + if ((err = point_scalar_multiply(u1, u1G, u1G, curve)) != HAL_OK || + (err = point_scalar_multiply(u2, key->Q, u2Q, curve)) != HAL_OK) return err; if (point_is_infinite(u1G)) point_copy(u2Q, R); else if (point_is_infinite(u2Q)) point_copy(u1G, R); - else if (fp_mulmod(unconst_fp_int(u1G->x), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), u1G->x) != FP_OKAY || - fp_mulmod(unconst_fp_int(u1G->y), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), u1G->y) != FP_OKAY || - fp_mulmod(unconst_fp_int(u2Q->x), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), u2Q->x) != FP_OKAY || - fp_mulmod(unconst_fp_int(u2Q->y), unconst_fp_int(curve->mu), unconst_fp_int(curve->q), u2Q->y) != FP_OKAY) - return HAL_ERROR_IMPOSSIBLE; + else if ((err = point_to_montgomery(u1G, curve)) != HAL_OK || + (err = point_to_montgomery(u2Q, curve)) != HAL_OK) + return err; else - fp_copy(unconst_fp_int(curve->mu), u1G->z), fp_copy(unconst_fp_int(curve->mu), u2Q->z), point_add(u1G, u2Q, R, curve); + point_add(u1G, u2Q, R, curve); /* * Signature is OK if |