/* * Elliptic curves over GF(p): generic functions * * Copyright The Mbed TLS Contributors * SPDX-License-Identifier: Apache-2.0 * * Licensed under the Apache License, Version 2.0 (the "License"); you may * not use this file except in compliance with the License. * You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, WITHOUT * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ /* * References: * * SEC1 http://www.secg.org/index.php?action=secg,docs_secg * GECC = Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone * FIPS 186-3 http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf * RFC 4492 for the related TLS structures and constants * RFC 7748 for the Curve448 and Curve25519 curve definitions * * [Curve25519] http://cr.yp.to/ecdh/curve25519-20060209.pdf * * [2] CORON, Jean-S'ebastien. Resistance against differential power analysis * for elliptic curve cryptosystems. In : Cryptographic Hardware and * Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302. * * * [3] HEDABOU, Mustapha, PINEL, Pierre, et B'EN'ETEAU, Lucien. A comb method to * render ECC resistant against Side Channel Attacks. IACR Cryptology * ePrint Archive, 2004, vol. 2004, p. 342. * */ #include "common.h" /** * \brief Function level alternative implementation. * * The MBEDTLS_ECP_INTERNAL_ALT macro enables alternative implementations to * replace certain functions in this module. The alternative implementations are * typically hardware accelerators and need to activate the hardware before the * computation starts and deactivate it after it finishes. The * mbedtls_internal_ecp_init() and mbedtls_internal_ecp_free() functions serve * this purpose. * * To preserve the correct functionality the following conditions must hold: * * - The alternative implementation must be activated by * mbedtls_internal_ecp_init() before any of the replaceable functions is * called. * - mbedtls_internal_ecp_free() must \b only be called when the alternative * implementation is activated. * - mbedtls_internal_ecp_init() must \b not be called when the alternative * implementation is activated. * - Public functions must not return while the alternative implementation is * activated. * - Replaceable functions are guarded by \c MBEDTLS_ECP_XXX_ALT macros and * before calling them an \code if( mbedtls_internal_ecp_grp_capable( grp ) ) * \endcode ensures that the alternative implementation supports the current * group. */ #if defined(MBEDTLS_ECP_INTERNAL_ALT) #endif #if defined(MBEDTLS_ECP_C) #include "mbedtls/ecp.h" #include "mbedtls/threading.h" #include "mbedtls/platform_util.h" #include "mbedtls/error.h" #include "bn_mul.h" #include "ecp_invasive.h" #include #if !defined(MBEDTLS_ECP_ALT) #include "mbedtls/platform.h" #include "ecp_internal_alt.h" #if defined(MBEDTLS_SELF_TEST) /* * Counts of point addition and doubling, and field multiplications. * Used to test resistance of point multiplication to simple timing attacks. */ static unsigned long add_count, dbl_count, mul_count; #endif #if defined(MBEDTLS_ECP_RESTARTABLE) /* * Maximum number of "basic operations" to be done in a row. * * Default value 0 means that ECC operations will not yield. * Note that regardless of the value of ecp_max_ops, always at * least one step is performed before yielding. * * Setting ecp_max_ops=1 can be suitable for testing purposes * as it will interrupt computation at all possible points. */ static unsigned ecp_max_ops = 0; /* * Set ecp_max_ops */ void mbedtls_ecp_set_max_ops(unsigned max_ops) { ecp_max_ops = max_ops; } /* * Check if restart is enabled */ int mbedtls_ecp_restart_is_enabled(void) { return ecp_max_ops != 0; } /* * Restart sub-context for ecp_mul_comb() */ struct mbedtls_ecp_restart_mul { mbedtls_ecp_point R; /* current intermediate result */ size_t i; /* current index in various loops, 0 outside */ mbedtls_ecp_point *T; /* table for precomputed points */ unsigned char T_size; /* number of points in table T */ enum { /* what were we doing last time we returned? */ ecp_rsm_init = 0, /* nothing so far, dummy initial state */ ecp_rsm_pre_dbl, /* precompute 2^n multiples */ ecp_rsm_pre_norm_dbl, /* normalize precomputed 2^n multiples */ ecp_rsm_pre_add, /* precompute remaining points by adding */ ecp_rsm_pre_norm_add, /* normalize all precomputed points */ ecp_rsm_comb_core, /* ecp_mul_comb_core() */ ecp_rsm_final_norm, /* do the final normalization */ } state; }; /* * Init restart_mul sub-context */ static void ecp_restart_rsm_init(mbedtls_ecp_restart_mul_ctx *ctx) { mbedtls_ecp_point_init(&ctx->R); ctx->i = 0; ctx->T = NULL; ctx->T_size = 0; ctx->state = ecp_rsm_init; } /* * Free the components of a restart_mul sub-context */ static void ecp_restart_rsm_free(mbedtls_ecp_restart_mul_ctx *ctx) { unsigned char i; if (ctx == NULL) { return; } mbedtls_ecp_point_free(&ctx->R); if (ctx->T != NULL) { for (i = 0; i < ctx->T_size; i++) { mbedtls_ecp_point_free(ctx->T + i); } mbedtls_free(ctx->T); } ecp_restart_rsm_init(ctx); } /* * Restart context for ecp_muladd() */ struct mbedtls_ecp_restart_muladd { mbedtls_ecp_point mP; /* mP value */ mbedtls_ecp_point R; /* R intermediate result */ enum { /* what should we do next? */ ecp_rsma_mul1 = 0, /* first multiplication */ ecp_rsma_mul2, /* second multiplication */ ecp_rsma_add, /* addition */ ecp_rsma_norm, /* normalization */ } state; }; /* * Init restart_muladd sub-context */ static void ecp_restart_ma_init(mbedtls_ecp_restart_muladd_ctx *ctx) { mbedtls_ecp_point_init(&ctx->mP); mbedtls_ecp_point_init(&ctx->R); ctx->state = ecp_rsma_mul1; } /* * Free the components of a restart_muladd sub-context */ static void ecp_restart_ma_free(mbedtls_ecp_restart_muladd_ctx *ctx) { if (ctx == NULL) { return; } mbedtls_ecp_point_free(&ctx->mP); mbedtls_ecp_point_free(&ctx->R); ecp_restart_ma_init(ctx); } /* * Initialize a restart context */ void mbedtls_ecp_restart_init(mbedtls_ecp_restart_ctx *ctx) { ctx->ops_done = 0; ctx->depth = 0; ctx->rsm = NULL; ctx->ma = NULL; } /* * Free the components of a restart context */ void mbedtls_ecp_restart_free(mbedtls_ecp_restart_ctx *ctx) { if (ctx == NULL) { return; } ecp_restart_rsm_free(ctx->rsm); mbedtls_free(ctx->rsm); ecp_restart_ma_free(ctx->ma); mbedtls_free(ctx->ma); mbedtls_ecp_restart_init(ctx); } /* * Check if we can do the next step */ int mbedtls_ecp_check_budget(const mbedtls_ecp_group *grp, mbedtls_ecp_restart_ctx *rs_ctx, unsigned ops) { if (rs_ctx != NULL && ecp_max_ops != 0) { /* scale depending on curve size: the chosen reference is 256-bit, * and multiplication is quadratic. Round to the closest integer. */ if (grp->pbits >= 512) { ops *= 4; } else if (grp->pbits >= 384) { ops *= 2; } /* Avoid infinite loops: always allow first step. * Because of that, however, it's not generally true * that ops_done <= ecp_max_ops, so the check * ops_done > ecp_max_ops below is mandatory. */ if ((rs_ctx->ops_done != 0) && (rs_ctx->ops_done > ecp_max_ops || ops > ecp_max_ops - rs_ctx->ops_done)) { return MBEDTLS_ERR_ECP_IN_PROGRESS; } /* update running count */ rs_ctx->ops_done += ops; } return 0; } /* Call this when entering a function that needs its own sub-context */ #define ECP_RS_ENTER(SUB) do { \ /* reset ops count for this call if top-level */ \ if (rs_ctx != NULL && rs_ctx->depth++ == 0) \ rs_ctx->ops_done = 0; \ \ /* set up our own sub-context if needed */ \ if (mbedtls_ecp_restart_is_enabled() && \ rs_ctx != NULL && rs_ctx->SUB == NULL) \ { \ rs_ctx->SUB = mbedtls_calloc(1, sizeof(*rs_ctx->SUB)); \ if (rs_ctx->SUB == NULL) \ return MBEDTLS_ERR_ECP_ALLOC_FAILED; \ \ ecp_restart_## SUB ##_init(rs_ctx->SUB); \ } \ } while (0) /* Call this when leaving a function that needs its own sub-context */ #define ECP_RS_LEAVE(SUB) do { \ /* clear our sub-context when not in progress (done or error) */ \ if (rs_ctx != NULL && rs_ctx->SUB != NULL && \ ret != MBEDTLS_ERR_ECP_IN_PROGRESS) \ { \ ecp_restart_## SUB ##_free(rs_ctx->SUB); \ mbedtls_free(rs_ctx->SUB); \ rs_ctx->SUB = NULL; \ } \ \ if (rs_ctx != NULL) \ rs_ctx->depth--; \ } while (0) #else /* MBEDTLS_ECP_RESTARTABLE */ #define ECP_RS_ENTER(sub) (void) rs_ctx; #define ECP_RS_LEAVE(sub) (void) rs_ctx; #endif /* MBEDTLS_ECP_RESTARTABLE */ static void mpi_init_many(mbedtls_mpi *arr, size_t size) { while (size--) { mbedtls_mpi_init(arr++); } } static void mpi_free_many(mbedtls_mpi *arr, size_t size) { while (size--) { mbedtls_mpi_free(arr++); } } /* * List of supported curves: * - internal ID * - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2, RFC 8446 sec. 4.2.7) * - size in bits * - readable name * * Curves are listed in order: largest curves first, and for a given size, * fastest curves first. * * Reminder: update profiles in x509_crt.c and ssl_tls.c when adding a new curve! */ static const mbedtls_ecp_curve_info ecp_supported_curves[] = { #if defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED) { MBEDTLS_ECP_DP_SECP521R1, 25, 521, "secp521r1" }, #endif #if defined(MBEDTLS_ECP_DP_BP512R1_ENABLED) { MBEDTLS_ECP_DP_BP512R1, 28, 512, "brainpoolP512r1" }, #endif #if defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED) { MBEDTLS_ECP_DP_SECP384R1, 24, 384, "secp384r1" }, #endif #if defined(MBEDTLS_ECP_DP_BP384R1_ENABLED) { MBEDTLS_ECP_DP_BP384R1, 27, 384, "brainpoolP384r1" }, #endif #if defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED) { MBEDTLS_ECP_DP_SECP256R1, 23, 256, "secp256r1" }, #endif #if defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED) { MBEDTLS_ECP_DP_SECP256K1, 22, 256, "secp256k1" }, #endif #if defined(MBEDTLS_ECP_DP_BP256R1_ENABLED) { MBEDTLS_ECP_DP_BP256R1, 26, 256, "brainpoolP256r1" }, #endif #if defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED) { MBEDTLS_ECP_DP_SECP224R1, 21, 224, "secp224r1" }, #endif #if defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED) { MBEDTLS_ECP_DP_SECP224K1, 20, 224, "secp224k1" }, #endif #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) { MBEDTLS_ECP_DP_SECP192R1, 19, 192, "secp192r1" }, #endif #if defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED) { MBEDTLS_ECP_DP_SECP192K1, 18, 192, "secp192k1" }, #endif #if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) { MBEDTLS_ECP_DP_CURVE25519, 29, 256, "x25519" }, #endif #if defined(MBEDTLS_ECP_DP_CURVE448_ENABLED) { MBEDTLS_ECP_DP_CURVE448, 30, 448, "x448" }, #endif { MBEDTLS_ECP_DP_NONE, 0, 0, NULL }, }; #define ECP_NB_CURVES sizeof(ecp_supported_curves) / \ sizeof(ecp_supported_curves[0]) static mbedtls_ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES]; /* * List of supported curves and associated info */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list(void) { return ecp_supported_curves; } /* * List of supported curves, group ID only */ const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list(void) { static int init_done = 0; if (!init_done) { size_t i = 0; const mbedtls_ecp_curve_info *curve_info; for (curve_info = mbedtls_ecp_curve_list(); curve_info->grp_id != MBEDTLS_ECP_DP_NONE; curve_info++) { ecp_supported_grp_id[i++] = curve_info->grp_id; } ecp_supported_grp_id[i] = MBEDTLS_ECP_DP_NONE; init_done = 1; } return ecp_supported_grp_id; } /* * Get the curve info for the internal identifier */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id(mbedtls_ecp_group_id grp_id) { const mbedtls_ecp_curve_info *curve_info; for (curve_info = mbedtls_ecp_curve_list(); curve_info->grp_id != MBEDTLS_ECP_DP_NONE; curve_info++) { if (curve_info->grp_id == grp_id) { return curve_info; } } return NULL; } /* * Get the curve info from the TLS identifier */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id(uint16_t tls_id) { const mbedtls_ecp_curve_info *curve_info; for (curve_info = mbedtls_ecp_curve_list(); curve_info->grp_id != MBEDTLS_ECP_DP_NONE; curve_info++) { if (curve_info->tls_id == tls_id) { return curve_info; } } return NULL; } /* * Get the curve info from the name */ const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name(const char *name) { const mbedtls_ecp_curve_info *curve_info; if (name == NULL) { return NULL; } for (curve_info = mbedtls_ecp_curve_list(); curve_info->grp_id != MBEDTLS_ECP_DP_NONE; curve_info++) { if (strcmp(curve_info->name, name) == 0) { return curve_info; } } return NULL; } /* * Get the type of a curve */ mbedtls_ecp_curve_type mbedtls_ecp_get_type(const mbedtls_ecp_group *grp) { if (grp->G.X.p == NULL) { return MBEDTLS_ECP_TYPE_NONE; } if (grp->G.Y.p == NULL) { return MBEDTLS_ECP_TYPE_MONTGOMERY; } else { return MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS; } } /* * Initialize (the components of) a point */ void mbedtls_ecp_point_init(mbedtls_ecp_point *pt) { mbedtls_mpi_init(&pt->X); mbedtls_mpi_init(&pt->Y); mbedtls_mpi_init(&pt->Z); } /* * Initialize (the components of) a group */ void mbedtls_ecp_group_init(mbedtls_ecp_group *grp) { grp->id = MBEDTLS_ECP_DP_NONE; mbedtls_mpi_init(&grp->P); mbedtls_mpi_init(&grp->A); mbedtls_mpi_init(&grp->B); mbedtls_ecp_point_init(&grp->G); mbedtls_mpi_init(&grp->N); grp->pbits = 0; grp->nbits = 0; grp->h = 0; grp->modp = NULL; grp->t_pre = NULL; grp->t_post = NULL; grp->t_data = NULL; grp->T = NULL; grp->T_size = 0; } /* * Initialize (the components of) a key pair */ void mbedtls_ecp_keypair_init(mbedtls_ecp_keypair *key) { mbedtls_ecp_group_init(&key->grp); mbedtls_mpi_init(&key->d); mbedtls_ecp_point_init(&key->Q); } /* * Unallocate (the components of) a point */ void mbedtls_ecp_point_free(mbedtls_ecp_point *pt) { if (pt == NULL) { return; } mbedtls_mpi_free(&(pt->X)); mbedtls_mpi_free(&(pt->Y)); mbedtls_mpi_free(&(pt->Z)); } /* * Check that the comb table (grp->T) is static initialized. */ static int ecp_group_is_static_comb_table(const mbedtls_ecp_group *grp) { #if MBEDTLS_ECP_FIXED_POINT_OPTIM == 1 return grp->T != NULL && grp->T_size == 0; #else (void) grp; return 0; #endif } /* * Unallocate (the components of) a group */ void mbedtls_ecp_group_free(mbedtls_ecp_group *grp) { size_t i; if (grp == NULL) { return; } if (grp->h != 1) { mbedtls_mpi_free(&grp->A); mbedtls_mpi_free(&grp->B); mbedtls_ecp_point_free(&grp->G); } if (!ecp_group_is_static_comb_table(grp) && grp->T != NULL) { for (i = 0; i < grp->T_size; i++) { mbedtls_ecp_point_free(&grp->T[i]); } mbedtls_free(grp->T); } mbedtls_platform_zeroize(grp, sizeof(mbedtls_ecp_group)); } /* * Unallocate (the components of) a key pair */ void mbedtls_ecp_keypair_free(mbedtls_ecp_keypair *key) { if (key == NULL) { return; } mbedtls_ecp_group_free(&key->grp); mbedtls_mpi_free(&key->d); mbedtls_ecp_point_free(&key->Q); } /* * Copy the contents of a point */ int mbedtls_ecp_copy(mbedtls_ecp_point *P, const mbedtls_ecp_point *Q) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->X, &Q->X)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->Y, &Q->Y)); MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->Z, &Q->Z)); cleanup: return ret; } /* * Copy the contents of a group object */ int mbedtls_ecp_group_copy(mbedtls_ecp_group *dst, const mbedtls_ecp_group *src) { return mbedtls_ecp_group_load(dst, src->id); } /* * Set point to zero */ int mbedtls_ecp_set_zero(mbedtls_ecp_point *pt) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->X, 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Y, 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 0)); cleanup: return ret; } /* * Tell if a point is zero */ int mbedtls_ecp_is_zero(mbedtls_ecp_point *pt) { return mbedtls_mpi_cmp_int(&pt->Z, 0) == 0; } /* * Compare two points lazily */ int mbedtls_ecp_point_cmp(const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q) { if (mbedtls_mpi_cmp_mpi(&P->X, &Q->X) == 0 && mbedtls_mpi_cmp_mpi(&P->Y, &Q->Y) == 0 && mbedtls_mpi_cmp_mpi(&P->Z, &Q->Z) == 0) { return 0; } return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * Import a non-zero point from ASCII strings */ int mbedtls_ecp_point_read_string(mbedtls_ecp_point *P, int radix, const char *x, const char *y) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&P->X, radix, x)); MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&P->Y, radix, y)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&P->Z, 1)); cleanup: return ret; } /* * Export a point into unsigned binary data (SEC1 2.3.3 and RFC7748) */ int mbedtls_ecp_point_write_binary(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *P, int format, size_t *olen, unsigned char *buf, size_t buflen) { int ret = MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; size_t plen; if (format != MBEDTLS_ECP_PF_UNCOMPRESSED && format != MBEDTLS_ECP_PF_COMPRESSED) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } plen = mbedtls_mpi_size(&grp->P); #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) (void) format; /* Montgomery curves always use the same point format */ if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { *olen = plen; if (buflen < *olen) { return MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL; } MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary_le(&P->X, buf, plen)); } #endif #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { /* * Common case: P == 0 */ if (mbedtls_mpi_cmp_int(&P->Z, 0) == 0) { if (buflen < 1) { return MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL; } buf[0] = 0x00; *olen = 1; return 0; } if (format == MBEDTLS_ECP_PF_UNCOMPRESSED) { *olen = 2 * plen + 1; if (buflen < *olen) { return MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL; } buf[0] = 0x04; MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->X, buf + 1, plen)); MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->Y, buf + 1 + plen, plen)); } else if (format == MBEDTLS_ECP_PF_COMPRESSED) { *olen = plen + 1; if (buflen < *olen) { return MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL; } buf[0] = 0x02 + mbedtls_mpi_get_bit(&P->Y, 0); MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->X, buf + 1, plen)); } } #endif cleanup: return ret; } #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) static int mbedtls_ecp_sw_derive_y(const mbedtls_ecp_group *grp, const mbedtls_mpi *X, mbedtls_mpi *Y, int parity_bit); #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ /* * Import a point from unsigned binary data (SEC1 2.3.4 and RFC7748) */ int mbedtls_ecp_point_read_binary(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt, const unsigned char *buf, size_t ilen) { int ret = MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; size_t plen; if (ilen < 1) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } plen = mbedtls_mpi_size(&grp->P); #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { if (plen != ilen) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary_le(&pt->X, buf, plen)); mbedtls_mpi_free(&pt->Y); if (grp->id == MBEDTLS_ECP_DP_CURVE25519) { /* Set most significant bit to 0 as prescribed in RFC7748 ยง5 */ MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(&pt->X, plen * 8 - 1, 0)); } MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 1)); } #endif #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { if (buf[0] == 0x00) { if (ilen == 1) { return mbedtls_ecp_set_zero(pt); } else { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } } if (ilen < 1 + plen) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary(&pt->X, buf + 1, plen)); MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 1)); if (buf[0] == 0x04) { /* format == MBEDTLS_ECP_PF_UNCOMPRESSED */ if (ilen != 1 + plen * 2) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } return mbedtls_mpi_read_binary(&pt->Y, buf + 1 + plen, plen); } else if (buf[0] == 0x02 || buf[0] == 0x03) { /* format == MBEDTLS_ECP_PF_COMPRESSED */ if (ilen != 1 + plen) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } return mbedtls_ecp_sw_derive_y(grp, &pt->X, &pt->Y, (buf[0] & 1)); } else { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } } #endif cleanup: return ret; } /* * Import a point from a TLS ECPoint record (RFC 4492) * struct { * opaque point <1..2^8-1>; * } ECPoint; */ int mbedtls_ecp_tls_read_point(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt, const unsigned char **buf, size_t buf_len) { unsigned char data_len; const unsigned char *buf_start; /* * We must have at least two bytes (1 for length, at least one for data) */ if (buf_len < 2) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } data_len = *(*buf)++; if (data_len < 1 || data_len > buf_len - 1) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * Save buffer start for read_binary and update buf */ buf_start = *buf; *buf += data_len; return mbedtls_ecp_point_read_binary(grp, pt, buf_start, data_len); } /* * Export a point as a TLS ECPoint record (RFC 4492) * struct { * opaque point <1..2^8-1>; * } ECPoint; */ int mbedtls_ecp_tls_write_point(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt, int format, size_t *olen, unsigned char *buf, size_t blen) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; if (format != MBEDTLS_ECP_PF_UNCOMPRESSED && format != MBEDTLS_ECP_PF_COMPRESSED) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * buffer length must be at least one, for our length byte */ if (blen < 1) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } if ((ret = mbedtls_ecp_point_write_binary(grp, pt, format, olen, buf + 1, blen - 1)) != 0) { return ret; } /* * write length to the first byte and update total length */ buf[0] = (unsigned char) *olen; ++*olen; return 0; } /* * Set a group from an ECParameters record (RFC 4492) */ int mbedtls_ecp_tls_read_group(mbedtls_ecp_group *grp, const unsigned char **buf, size_t len) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_ecp_group_id grp_id; if ((ret = mbedtls_ecp_tls_read_group_id(&grp_id, buf, len)) != 0) { return ret; } return mbedtls_ecp_group_load(grp, grp_id); } /* * Read a group id from an ECParameters record (RFC 4492) and convert it to * mbedtls_ecp_group_id. */ int mbedtls_ecp_tls_read_group_id(mbedtls_ecp_group_id *grp, const unsigned char **buf, size_t len) { uint16_t tls_id; const mbedtls_ecp_curve_info *curve_info; /* * We expect at least three bytes (see below) */ if (len < 3) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * First byte is curve_type; only named_curve is handled */ if (*(*buf)++ != MBEDTLS_ECP_TLS_NAMED_CURVE) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * Next two bytes are the namedcurve value */ tls_id = *(*buf)++; tls_id <<= 8; tls_id |= *(*buf)++; if ((curve_info = mbedtls_ecp_curve_info_from_tls_id(tls_id)) == NULL) { return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; } *grp = curve_info->grp_id; return 0; } /* * Write the ECParameters record corresponding to a group (RFC 4492) */ int mbedtls_ecp_tls_write_group(const mbedtls_ecp_group *grp, size_t *olen, unsigned char *buf, size_t blen) { const mbedtls_ecp_curve_info *curve_info; if ((curve_info = mbedtls_ecp_curve_info_from_grp_id(grp->id)) == NULL) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * We are going to write 3 bytes (see below) */ *olen = 3; if (blen < *olen) { return MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL; } /* * First byte is curve_type, always named_curve */ *buf++ = MBEDTLS_ECP_TLS_NAMED_CURVE; /* * Next two bytes are the namedcurve value */ MBEDTLS_PUT_UINT16_BE(curve_info->tls_id, buf, 0); return 0; } /* * Wrapper around fast quasi-modp functions, with fall-back to mbedtls_mpi_mod_mpi. * See the documentation of struct mbedtls_ecp_group. * * This function is in the critial loop for mbedtls_ecp_mul, so pay attention to perf. */ static int ecp_modp(mbedtls_mpi *N, const mbedtls_ecp_group *grp) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; if (grp->modp == NULL) { return mbedtls_mpi_mod_mpi(N, N, &grp->P); } /* N->s < 0 is a much faster test, which fails only if N is 0 */ if ((N->s < 0 && mbedtls_mpi_cmp_int(N, 0) != 0) || mbedtls_mpi_bitlen(N) > 2 * grp->pbits) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } MBEDTLS_MPI_CHK(grp->modp(N)); /* N->s < 0 is a much faster test, which fails only if N is 0 */ while (N->s < 0 && mbedtls_mpi_cmp_int(N, 0) != 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(N, N, &grp->P)); } while (mbedtls_mpi_cmp_mpi(N, &grp->P) >= 0) { /* we known P, N and the result are positive */ MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(N, N, &grp->P)); } cleanup: return ret; } /* * Fast mod-p functions expect their argument to be in the 0..p^2 range. * * In order to guarantee that, we need to ensure that operands of * mbedtls_mpi_mul_mpi are in the 0..p range. So, after each operation we will * bring the result back to this range. * * The following macros are shortcuts for doing that. */ /* * Reduce a mbedtls_mpi mod p in-place, general case, to use after mbedtls_mpi_mul_mpi */ #if defined(MBEDTLS_SELF_TEST) #define INC_MUL_COUNT mul_count++; #else #define INC_MUL_COUNT #endif #define MOD_MUL(N) \ do \ { \ MBEDTLS_MPI_CHK(ecp_modp(&(N), grp)); \ INC_MUL_COUNT \ } while (0) static inline int mbedtls_mpi_mul_mod(const mbedtls_ecp_group *grp, mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(X, A, B)); MOD_MUL(*X); cleanup: return ret; } /* * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_sub_mpi * N->s < 0 is a very fast test, which fails only if N is 0 */ #define MOD_SUB(N) \ do { \ while ((N)->s < 0 && mbedtls_mpi_cmp_int((N), 0) != 0) \ MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi((N), (N), &grp->P)); \ } while (0) #if (defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) && \ !(defined(MBEDTLS_ECP_NO_FALLBACK) && \ defined(MBEDTLS_ECP_DOUBLE_JAC_ALT) && \ defined(MBEDTLS_ECP_ADD_MIXED_ALT))) || \ (defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) && \ !(defined(MBEDTLS_ECP_NO_FALLBACK) && \ defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT))) static inline int mbedtls_mpi_sub_mod(const mbedtls_ecp_group *grp, mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(X, A, B)); MOD_SUB(X); cleanup: return ret; } #endif /* All functions referencing mbedtls_mpi_sub_mod() are alt-implemented without fallback */ /* * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_add_mpi and mbedtls_mpi_mul_int. * We known P, N and the result are positive, so sub_abs is correct, and * a bit faster. */ #define MOD_ADD(N) \ while (mbedtls_mpi_cmp_mpi((N), &grp->P) >= 0) \ MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs((N), (N), &grp->P)) static inline int mbedtls_mpi_add_mod(const mbedtls_ecp_group *grp, mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(X, A, B)); MOD_ADD(X); cleanup: return ret; } static inline int mbedtls_mpi_mul_int_mod(const mbedtls_ecp_group *grp, mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_uint c) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(X, A, c)); MOD_ADD(X); cleanup: return ret; } static inline int mbedtls_mpi_sub_int_mod(const mbedtls_ecp_group *grp, mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_uint c) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(X, A, c)); MOD_SUB(X); cleanup: return ret; } #define MPI_ECP_SUB_INT(X, A, c) \ MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int_mod(grp, X, A, c)) #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) && \ !(defined(MBEDTLS_ECP_NO_FALLBACK) && \ defined(MBEDTLS_ECP_DOUBLE_JAC_ALT) && \ defined(MBEDTLS_ECP_ADD_MIXED_ALT)) static inline int mbedtls_mpi_shift_l_mod(const mbedtls_ecp_group *grp, mbedtls_mpi *X, size_t count) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(X, count)); MOD_ADD(X); cleanup: return ret; } #endif \ /* All functions referencing mbedtls_mpi_shift_l_mod() are alt-implemented without fallback */ /* * Macro wrappers around ECP modular arithmetic * * Currently, these wrappers are defined via the bignum module. */ #define MPI_ECP_ADD(X, A, B) \ MBEDTLS_MPI_CHK(mbedtls_mpi_add_mod(grp, X, A, B)) #define MPI_ECP_SUB(X, A, B) \ MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mod(grp, X, A, B)) #define MPI_ECP_MUL(X, A, B) \ MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mod(grp, X, A, B)) #define MPI_ECP_SQR(X, A) \ MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mod(grp, X, A, A)) #define MPI_ECP_MUL_INT(X, A, c) \ MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int_mod(grp, X, A, c)) #define MPI_ECP_INV(dst, src) \ MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod((dst), (src), &grp->P)) #define MPI_ECP_MOV(X, A) \ MBEDTLS_MPI_CHK(mbedtls_mpi_copy(X, A)) #define MPI_ECP_SHIFT_L(X, count) \ MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l_mod(grp, X, count)) #define MPI_ECP_LSET(X, c) \ MBEDTLS_MPI_CHK(mbedtls_mpi_lset(X, c)) #define MPI_ECP_CMP_INT(X, c) \ mbedtls_mpi_cmp_int(X, c) #define MPI_ECP_CMP(X, Y) \ mbedtls_mpi_cmp_mpi(X, Y) /* Needs f_rng, p_rng to be defined. */ #define MPI_ECP_RAND(X) \ MBEDTLS_MPI_CHK(mbedtls_mpi_random((X), 2, &grp->P, f_rng, p_rng)) /* Conditional negation * Needs grp and a temporary MPI tmp to be defined. */ #define MPI_ECP_COND_NEG(X, cond) \ do \ { \ unsigned char nonzero = mbedtls_mpi_cmp_int((X), 0) != 0; \ MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&tmp, &grp->P, (X))); \ MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign((X), &tmp, \ nonzero & cond)); \ } while (0) #define MPI_ECP_NEG(X) MPI_ECP_COND_NEG((X), 1) #define MPI_ECP_VALID(X) \ ((X)->p != NULL) #define MPI_ECP_COND_ASSIGN(X, Y, cond) \ MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign((X), (Y), (cond))) #define MPI_ECP_COND_SWAP(X, Y, cond) \ MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap((X), (Y), (cond))) #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) /* * Computes the right-hand side of the Short Weierstrass equation * RHS = X^3 + A X + B */ static int ecp_sw_rhs(const mbedtls_ecp_group *grp, mbedtls_mpi *rhs, const mbedtls_mpi *X) { int ret; /* Compute X^3 + A X + B as X (X^2 + A) + B */ MPI_ECP_SQR(rhs, X); /* Special case for A = -3 */ if (grp->A.p == NULL) { MPI_ECP_SUB_INT(rhs, rhs, 3); } else { MPI_ECP_ADD(rhs, rhs, &grp->A); } MPI_ECP_MUL(rhs, rhs, X); MPI_ECP_ADD(rhs, rhs, &grp->B); cleanup: return ret; } /* * Derive Y from X and a parity bit */ static int mbedtls_ecp_sw_derive_y(const mbedtls_ecp_group *grp, const mbedtls_mpi *X, mbedtls_mpi *Y, int parity_bit) { /* w = y^2 = x^3 + ax + b * y = sqrt(w) = w^((p+1)/4) mod p (for prime p where p = 3 mod 4) * * Note: this method for extracting square root does not validate that w * was indeed a square so this function will return garbage in Y if X * does not correspond to a point on the curve. */ /* Check prerequisite p = 3 mod 4 */ if (mbedtls_mpi_get_bit(&grp->P, 0) != 1 || mbedtls_mpi_get_bit(&grp->P, 1) != 1) { return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; } int ret; mbedtls_mpi exp; mbedtls_mpi_init(&exp); /* use Y to store intermediate result, actually w above */ MBEDTLS_MPI_CHK(ecp_sw_rhs(grp, Y, X)); /* w = y^2 */ /* Y contains y^2 intermediate result */ /* exp = ((p+1)/4) */ MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&exp, &grp->P, 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&exp, 2)); /* sqrt(w) = w^((p+1)/4) mod p (for prime p where p = 3 mod 4) */ MBEDTLS_MPI_CHK(mbedtls_mpi_exp_mod(Y, Y /*y^2*/, &exp, &grp->P, NULL)); /* check parity bit match or else invert Y */ /* This quick inversion implementation is valid because Y != 0 for all * Short Weierstrass curves supported by mbedtls, as each supported curve * has an order that is a large prime, so each supported curve does not * have any point of order 2, and a point with Y == 0 would be of order 2 */ if (mbedtls_mpi_get_bit(Y, 0) != parity_bit) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(Y, &grp->P, Y)); } cleanup: mbedtls_mpi_free(&exp); return ret; } /* * For curves in short Weierstrass form, we do all the internal operations in * Jacobian coordinates. * * For multiplication, we'll use a comb method with countermeasures against * SPA, hence timing attacks. */ /* * Normalize jacobian coordinates so that Z == 0 || Z == 1 (GECC 3.2.1) * Cost: 1N := 1I + 3M + 1S */ static int ecp_normalize_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt) { if (MPI_ECP_CMP_INT(&pt->Z, 0) == 0) { return 0; } #if defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_normalize_jac(grp, pt); } #endif /* MBEDTLS_ECP_NORMALIZE_JAC_ALT */ #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi T; mbedtls_mpi_init(&T); MPI_ECP_INV(&T, &pt->Z); /* T <- 1 / Z */ MPI_ECP_MUL(&pt->Y, &pt->Y, &T); /* Y' <- Y*T = Y / Z */ MPI_ECP_SQR(&T, &T); /* T <- T^2 = 1 / Z^2 */ MPI_ECP_MUL(&pt->X, &pt->X, &T); /* X <- X * T = X / Z^2 */ MPI_ECP_MUL(&pt->Y, &pt->Y, &T); /* Y'' <- Y' * T = Y / Z^3 */ MPI_ECP_LSET(&pt->Z, 1); cleanup: mbedtls_mpi_free(&T); return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT) */ } /* * Normalize jacobian coordinates of an array of (pointers to) points, * using Montgomery's trick to perform only one inversion mod P. * (See for example Cohen's "A Course in Computational Algebraic Number * Theory", Algorithm 10.3.4.) * * Warning: fails (returning an error) if one of the points is zero! * This should never happen, see choice of w in ecp_mul_comb(). * * Cost: 1N(t) := 1I + (6t - 3)M + 1S */ static int ecp_normalize_jac_many(const mbedtls_ecp_group *grp, mbedtls_ecp_point *T[], size_t T_size) { if (T_size < 2) { return ecp_normalize_jac(grp, *T); } #if defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_normalize_jac_many(grp, T, T_size); } #endif #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i; mbedtls_mpi *c, t; if ((c = mbedtls_calloc(T_size, sizeof(mbedtls_mpi))) == NULL) { return MBEDTLS_ERR_ECP_ALLOC_FAILED; } mbedtls_mpi_init(&t); mpi_init_many(c, T_size); /* * c[i] = Z_0 * ... * Z_i, i = 0,..,n := T_size-1 */ MPI_ECP_MOV(&c[0], &T[0]->Z); for (i = 1; i < T_size; i++) { MPI_ECP_MUL(&c[i], &c[i-1], &T[i]->Z); } /* * c[n] = 1 / (Z_0 * ... * Z_n) mod P */ MPI_ECP_INV(&c[T_size-1], &c[T_size-1]); for (i = T_size - 1;; i--) { /* At the start of iteration i (note that i decrements), we have * - c[j] = Z_0 * .... * Z_j for j < i, * - c[j] = 1 / (Z_0 * .... * Z_j) for j == i, * * This is maintained via * - c[i-1] <- c[i] * Z_i * * We also derive 1/Z_i = c[i] * c[i-1] for i>0 and use that * to do the actual normalization. For i==0, we already have * c[0] = 1 / Z_0. */ if (i > 0) { /* Compute 1/Z_i and establish invariant for the next iteration. */ MPI_ECP_MUL(&t, &c[i], &c[i-1]); MPI_ECP_MUL(&c[i-1], &c[i], &T[i]->Z); } else { MPI_ECP_MOV(&t, &c[0]); } /* Now t holds 1 / Z_i; normalize as in ecp_normalize_jac() */ MPI_ECP_MUL(&T[i]->Y, &T[i]->Y, &t); MPI_ECP_SQR(&t, &t); MPI_ECP_MUL(&T[i]->X, &T[i]->X, &t); MPI_ECP_MUL(&T[i]->Y, &T[i]->Y, &t); /* * Post-precessing: reclaim some memory by shrinking coordinates * - not storing Z (always 1) * - shrinking other coordinates, but still keeping the same number of * limbs as P, as otherwise it will too likely be regrown too fast. */ MBEDTLS_MPI_CHK(mbedtls_mpi_shrink(&T[i]->X, grp->P.n)); MBEDTLS_MPI_CHK(mbedtls_mpi_shrink(&T[i]->Y, grp->P.n)); MPI_ECP_LSET(&T[i]->Z, 1); if (i == 0) { break; } } cleanup: mbedtls_mpi_free(&t); mpi_free_many(c, T_size); mbedtls_free(c); return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT) */ } /* * Conditional point inversion: Q -> -Q = (Q.X, -Q.Y, Q.Z) without leak. * "inv" must be 0 (don't invert) or 1 (invert) or the result will be invalid */ static int ecp_safe_invert_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *Q, unsigned char inv) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi tmp; mbedtls_mpi_init(&tmp); MPI_ECP_COND_NEG(&Q->Y, inv); cleanup: mbedtls_mpi_free(&tmp); return ret; } /* * Point doubling R = 2 P, Jacobian coordinates * * Based on http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-1998-cmo-2 . * * We follow the variable naming fairly closely. The formula variations that trade a MUL for a SQR * (plus a few ADDs) aren't useful as our bignum implementation doesn't distinguish squaring. * * Standard optimizations are applied when curve parameter A is one of { 0, -3 }. * * Cost: 1D := 3M + 4S (A == 0) * 4M + 4S (A == -3) * 3M + 6S + 1a otherwise */ static int ecp_double_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_ecp_point *P, mbedtls_mpi tmp[4]) { #if defined(MBEDTLS_SELF_TEST) dbl_count++; #endif #if defined(MBEDTLS_ECP_DOUBLE_JAC_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_double_jac(grp, R, P); } #endif /* MBEDTLS_ECP_DOUBLE_JAC_ALT */ #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_DOUBLE_JAC_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; /* Special case for A = -3 */ if (grp->A.p == NULL) { /* tmp[0] <- M = 3(X + Z^2)(X - Z^2) */ MPI_ECP_SQR(&tmp[1], &P->Z); MPI_ECP_ADD(&tmp[2], &P->X, &tmp[1]); MPI_ECP_SUB(&tmp[3], &P->X, &tmp[1]); MPI_ECP_MUL(&tmp[1], &tmp[2], &tmp[3]); MPI_ECP_MUL_INT(&tmp[0], &tmp[1], 3); } else { /* tmp[0] <- M = 3.X^2 + A.Z^4 */ MPI_ECP_SQR(&tmp[1], &P->X); MPI_ECP_MUL_INT(&tmp[0], &tmp[1], 3); /* Optimize away for "koblitz" curves with A = 0 */ if (MPI_ECP_CMP_INT(&grp->A, 0) != 0) { /* M += A.Z^4 */ MPI_ECP_SQR(&tmp[1], &P->Z); MPI_ECP_SQR(&tmp[2], &tmp[1]); MPI_ECP_MUL(&tmp[1], &tmp[2], &grp->A); MPI_ECP_ADD(&tmp[0], &tmp[0], &tmp[1]); } } /* tmp[1] <- S = 4.X.Y^2 */ MPI_ECP_SQR(&tmp[2], &P->Y); MPI_ECP_SHIFT_L(&tmp[2], 1); MPI_ECP_MUL(&tmp[1], &P->X, &tmp[2]); MPI_ECP_SHIFT_L(&tmp[1], 1); /* tmp[3] <- U = 8.Y^4 */ MPI_ECP_SQR(&tmp[3], &tmp[2]); MPI_ECP_SHIFT_L(&tmp[3], 1); /* tmp[2] <- T = M^2 - 2.S */ MPI_ECP_SQR(&tmp[2], &tmp[0]); MPI_ECP_SUB(&tmp[2], &tmp[2], &tmp[1]); MPI_ECP_SUB(&tmp[2], &tmp[2], &tmp[1]); /* tmp[1] <- S = M(S - T) - U */ MPI_ECP_SUB(&tmp[1], &tmp[1], &tmp[2]); MPI_ECP_MUL(&tmp[1], &tmp[1], &tmp[0]); MPI_ECP_SUB(&tmp[1], &tmp[1], &tmp[3]); /* tmp[3] <- U = 2.Y.Z */ MPI_ECP_MUL(&tmp[3], &P->Y, &P->Z); MPI_ECP_SHIFT_L(&tmp[3], 1); /* Store results */ MPI_ECP_MOV(&R->X, &tmp[2]); MPI_ECP_MOV(&R->Y, &tmp[1]); MPI_ECP_MOV(&R->Z, &tmp[3]); cleanup: return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_DOUBLE_JAC_ALT) */ } /* * Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22) * * The coordinates of Q must be normalized (= affine), * but those of P don't need to. R is not normalized. * * P,Q,R may alias, but only at the level of EC points: they must be either * equal as pointers, or disjoint (including the coordinate data buffers). * Fine-grained aliasing at the level of coordinates is not supported. * * Special cases: (1) P or Q is zero, (2) R is zero, (3) P == Q. * None of these cases can happen as intermediate step in ecp_mul_comb(): * - at each step, P, Q and R are multiples of the base point, the factor * being less than its order, so none of them is zero; * - Q is an odd multiple of the base point, P an even multiple, * due to the choice of precomputed points in the modified comb method. * So branches for these cases do not leak secret information. * * Cost: 1A := 8M + 3S */ static int ecp_add_mixed(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q, mbedtls_mpi tmp[4]) { #if defined(MBEDTLS_SELF_TEST) add_count++; #endif #if defined(MBEDTLS_ECP_ADD_MIXED_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_add_mixed(grp, R, P, Q); } #endif /* MBEDTLS_ECP_ADD_MIXED_ALT */ #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_ADD_MIXED_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; /* NOTE: Aliasing between input and output is allowed, so one has to make * sure that at the point X,Y,Z are written, {P,Q}->{X,Y,Z} are no * longer read from. */ mbedtls_mpi * const X = &R->X; mbedtls_mpi * const Y = &R->Y; mbedtls_mpi * const Z = &R->Z; if (!MPI_ECP_VALID(&Q->Z)) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * Trivial cases: P == 0 or Q == 0 (case 1) */ if (MPI_ECP_CMP_INT(&P->Z, 0) == 0) { return mbedtls_ecp_copy(R, Q); } if (MPI_ECP_CMP_INT(&Q->Z, 0) == 0) { return mbedtls_ecp_copy(R, P); } /* * Make sure Q coordinates are normalized */ if (MPI_ECP_CMP_INT(&Q->Z, 1) != 0) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } MPI_ECP_SQR(&tmp[0], &P->Z); MPI_ECP_MUL(&tmp[1], &tmp[0], &P->Z); MPI_ECP_MUL(&tmp[0], &tmp[0], &Q->X); MPI_ECP_MUL(&tmp[1], &tmp[1], &Q->Y); MPI_ECP_SUB(&tmp[0], &tmp[0], &P->X); MPI_ECP_SUB(&tmp[1], &tmp[1], &P->Y); /* Special cases (2) and (3) */ if (MPI_ECP_CMP_INT(&tmp[0], 0) == 0) { if (MPI_ECP_CMP_INT(&tmp[1], 0) == 0) { ret = ecp_double_jac(grp, R, P, tmp); goto cleanup; } else { ret = mbedtls_ecp_set_zero(R); goto cleanup; } } /* {P,Q}->Z no longer used, so OK to write to Z even if there's aliasing. */ MPI_ECP_MUL(Z, &P->Z, &tmp[0]); MPI_ECP_SQR(&tmp[2], &tmp[0]); MPI_ECP_MUL(&tmp[3], &tmp[2], &tmp[0]); MPI_ECP_MUL(&tmp[2], &tmp[2], &P->X); MPI_ECP_MOV(&tmp[0], &tmp[2]); MPI_ECP_SHIFT_L(&tmp[0], 1); /* {P,Q}->X no longer used, so OK to write to X even if there's aliasing. */ MPI_ECP_SQR(X, &tmp[1]); MPI_ECP_SUB(X, X, &tmp[0]); MPI_ECP_SUB(X, X, &tmp[3]); MPI_ECP_SUB(&tmp[2], &tmp[2], X); MPI_ECP_MUL(&tmp[2], &tmp[2], &tmp[1]); MPI_ECP_MUL(&tmp[3], &tmp[3], &P->Y); /* {P,Q}->Y no longer used, so OK to write to Y even if there's aliasing. */ MPI_ECP_SUB(Y, &tmp[2], &tmp[3]); cleanup: return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_ADD_MIXED_ALT) */ } /* * Randomize jacobian coordinates: * (X, Y, Z) -> (l^2 X, l^3 Y, l Z) for random l * This is sort of the reverse operation of ecp_normalize_jac(). * * This countermeasure was first suggested in [2]. */ static int ecp_randomize_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { #if defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_randomize_jac(grp, pt, f_rng, p_rng); } #endif /* MBEDTLS_ECP_RANDOMIZE_JAC_ALT */ #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi l; mbedtls_mpi_init(&l); /* Generate l such that 1 < l < p */ MPI_ECP_RAND(&l); /* Z' = l * Z */ MPI_ECP_MUL(&pt->Z, &pt->Z, &l); /* Y' = l * Y */ MPI_ECP_MUL(&pt->Y, &pt->Y, &l); /* X' = l^2 * X */ MPI_ECP_SQR(&l, &l); MPI_ECP_MUL(&pt->X, &pt->X, &l); /* Y'' = l^2 * Y' = l^3 * Y */ MPI_ECP_MUL(&pt->Y, &pt->Y, &l); cleanup: mbedtls_mpi_free(&l); if (ret == MBEDTLS_ERR_MPI_NOT_ACCEPTABLE) { ret = MBEDTLS_ERR_ECP_RANDOM_FAILED; } return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT) */ } /* * Check and define parameters used by the comb method (see below for details) */ #if MBEDTLS_ECP_WINDOW_SIZE < 2 || MBEDTLS_ECP_WINDOW_SIZE > 7 #error "MBEDTLS_ECP_WINDOW_SIZE out of bounds" #endif /* d = ceil( n / w ) */ #define COMB_MAX_D (MBEDTLS_ECP_MAX_BITS + 1) / 2 /* number of precomputed points */ #define COMB_MAX_PRE (1 << (MBEDTLS_ECP_WINDOW_SIZE - 1)) /* * Compute the representation of m that will be used with our comb method. * * The basic comb method is described in GECC 3.44 for example. We use a * modified version that provides resistance to SPA by avoiding zero * digits in the representation as in [3]. We modify the method further by * requiring that all K_i be odd, which has the small cost that our * representation uses one more K_i, due to carries, but saves on the size of * the precomputed table. * * Summary of the comb method and its modifications: * * - The goal is to compute m*P for some w*d-bit integer m. * * - The basic comb method splits m into the w-bit integers * x[0] .. x[d-1] where x[i] consists of the bits in m whose * index has residue i modulo d, and computes m * P as * S[x[0]] + 2 * S[x[1]] + .. + 2^(d-1) S[x[d-1]], where * S[i_{w-1} .. i_0] := i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + i_0 P. * * - If it happens that, say, x[i+1]=0 (=> S[x[i+1]]=0), one can replace the sum by * .. + 2^{i-1} S[x[i-1]] - 2^i S[x[i]] + 2^{i+1} S[x[i]] + 2^{i+2} S[x[i+2]] .., * thereby successively converting it into a form where all summands * are nonzero, at the cost of negative summands. This is the basic idea of [3]. * * - More generally, even if x[i+1] != 0, we can first transform the sum as * .. - 2^i S[x[i]] + 2^{i+1} ( S[x[i]] + S[x[i+1]] ) + 2^{i+2} S[x[i+2]] .., * and then replace S[x[i]] + S[x[i+1]] = S[x[i] ^ x[i+1]] + 2 S[x[i] & x[i+1]]. * Performing and iterating this procedure for those x[i] that are even * (keeping track of carry), we can transform the original sum into one of the form * S[x'[0]] +- 2 S[x'[1]] +- .. +- 2^{d-1} S[x'[d-1]] + 2^d S[x'[d]] * with all x'[i] odd. It is therefore only necessary to know S at odd indices, * which is why we are only computing half of it in the first place in * ecp_precompute_comb and accessing it with index abs(i) / 2 in ecp_select_comb. * * - For the sake of compactness, only the seven low-order bits of x[i] * are used to represent its absolute value (K_i in the paper), and the msb * of x[i] encodes the sign (s_i in the paper): it is set if and only if * if s_i == -1; * * Calling conventions: * - x is an array of size d + 1 * - w is the size, ie number of teeth, of the comb, and must be between * 2 and 7 (in practice, between 2 and MBEDTLS_ECP_WINDOW_SIZE) * - m is the MPI, expected to be odd and such that bitlength(m) <= w * d * (the result will be incorrect if these assumptions are not satisfied) */ static void ecp_comb_recode_core(unsigned char x[], size_t d, unsigned char w, const mbedtls_mpi *m) { size_t i, j; unsigned char c, cc, adjust; memset(x, 0, d+1); /* First get the classical comb values (except for x_d = 0) */ for (i = 0; i < d; i++) { for (j = 0; j < w; j++) { x[i] |= mbedtls_mpi_get_bit(m, i + d * j) << j; } } /* Now make sure x_1 .. x_d are odd */ c = 0; for (i = 1; i <= d; i++) { /* Add carry and update it */ cc = x[i] & c; x[i] = x[i] ^ c; c = cc; /* Adjust if needed, avoiding branches */ adjust = 1 - (x[i] & 0x01); c |= x[i] & (x[i-1] * adjust); x[i] = x[i] ^ (x[i-1] * adjust); x[i-1] |= adjust << 7; } } /* * Precompute points for the adapted comb method * * Assumption: T must be able to hold 2^{w - 1} elements. * * Operation: If i = i_{w-1} ... i_1 is the binary representation of i, * sets T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P. * * Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1) * * Note: Even comb values (those where P would be omitted from the * sum defining T[i] above) are not needed in our adaption * the comb method. See ecp_comb_recode_core(). * * This function currently works in four steps: * (1) [dbl] Computation of intermediate T[i] for 2-power values of i * (2) [norm_dbl] Normalization of coordinates of these T[i] * (3) [add] Computation of all T[i] * (4) [norm_add] Normalization of all T[i] * * Step 1 can be interrupted but not the others; together with the final * coordinate normalization they are the largest steps done at once, depending * on the window size. Here are operation counts for P-256: * * step (2) (3) (4) * w = 5 142 165 208 * w = 4 136 77 160 * w = 3 130 33 136 * w = 2 124 11 124 * * So if ECC operations are blocking for too long even with a low max_ops * value, it's useful to set MBEDTLS_ECP_WINDOW_SIZE to a lower value in order * to minimize maximum blocking time. */ static int ecp_precompute_comb(const mbedtls_ecp_group *grp, mbedtls_ecp_point T[], const mbedtls_ecp_point *P, unsigned char w, size_t d, mbedtls_ecp_restart_ctx *rs_ctx) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; unsigned char i; size_t j = 0; const unsigned char T_size = 1U << (w - 1); mbedtls_ecp_point *cur, *TT[COMB_MAX_PRE - 1] = { NULL }; mbedtls_mpi tmp[4]; mpi_init_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { if (rs_ctx->rsm->state == ecp_rsm_pre_dbl) { goto dbl; } if (rs_ctx->rsm->state == ecp_rsm_pre_norm_dbl) { goto norm_dbl; } if (rs_ctx->rsm->state == ecp_rsm_pre_add) { goto add; } if (rs_ctx->rsm->state == ecp_rsm_pre_norm_add) { goto norm_add; } } #else (void) rs_ctx; #endif #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { rs_ctx->rsm->state = ecp_rsm_pre_dbl; /* initial state for the loop */ rs_ctx->rsm->i = 0; } dbl: #endif /* * Set T[0] = P and * T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value) */ MBEDTLS_MPI_CHK(mbedtls_ecp_copy(&T[0], P)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL && rs_ctx->rsm->i != 0) { j = rs_ctx->rsm->i; } else #endif j = 0; for (; j < d * (w - 1); j++) { MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_DBL); i = 1U << (j / d); cur = T + i; if (j % d == 0) { MBEDTLS_MPI_CHK(mbedtls_ecp_copy(cur, T + (i >> 1))); } MBEDTLS_MPI_CHK(ecp_double_jac(grp, cur, cur, tmp)); } #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { rs_ctx->rsm->state = ecp_rsm_pre_norm_dbl; } norm_dbl: #endif /* * Normalize current elements in T to allow them to be used in * ecp_add_mixed() below, which requires one normalized input. * * As T has holes, use an auxiliary array of pointers to elements in T. * */ j = 0; for (i = 1; i < T_size; i <<= 1) { TT[j++] = T + i; } MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_INV + 6 * j - 2); MBEDTLS_MPI_CHK(ecp_normalize_jac_many(grp, TT, j)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { rs_ctx->rsm->state = ecp_rsm_pre_add; } add: #endif /* * Compute the remaining ones using the minimal number of additions * Be careful to update T[2^l] only after using it! */ MBEDTLS_ECP_BUDGET((T_size - 1) * MBEDTLS_ECP_OPS_ADD); for (i = 1; i < T_size; i <<= 1) { j = i; while (j--) { MBEDTLS_MPI_CHK(ecp_add_mixed(grp, &T[i + j], &T[j], &T[i], tmp)); } } #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { rs_ctx->rsm->state = ecp_rsm_pre_norm_add; } norm_add: #endif /* * Normalize final elements in T. Even though there are no holes now, we * still need the auxiliary array for homogeneity with the previous * call. Also, skip T[0] which is already normalised, being a copy of P. */ for (j = 0; j + 1 < T_size; j++) { TT[j] = T + j + 1; } MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_INV + 6 * j - 2); MBEDTLS_MPI_CHK(ecp_normalize_jac_many(grp, TT, j)); /* Free Z coordinate (=1 after normalization) to save RAM. * This makes T[i] invalid as mbedtls_ecp_points, but this is OK * since from this point onwards, they are only accessed indirectly * via the getter function ecp_select_comb() which does set the * target's Z coordinate to 1. */ for (i = 0; i < T_size; i++) { mbedtls_mpi_free(&T[i].Z); } cleanup: mpi_free_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL && ret == MBEDTLS_ERR_ECP_IN_PROGRESS) { if (rs_ctx->rsm->state == ecp_rsm_pre_dbl) { rs_ctx->rsm->i = j; } } #endif return ret; } /* * Select precomputed point: R = sign(i) * T[ abs(i) / 2 ] * * See ecp_comb_recode_core() for background */ static int ecp_select_comb(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_ecp_point T[], unsigned char T_size, unsigned char i) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; unsigned char ii, j; /* Ignore the "sign" bit and scale down */ ii = (i & 0x7Fu) >> 1; /* Read the whole table to thwart cache-based timing attacks */ for (j = 0; j < T_size; j++) { MPI_ECP_COND_ASSIGN(&R->X, &T[j].X, j == ii); MPI_ECP_COND_ASSIGN(&R->Y, &T[j].Y, j == ii); } /* Safely invert result if i is "negative" */ MBEDTLS_MPI_CHK(ecp_safe_invert_jac(grp, R, i >> 7)); MPI_ECP_LSET(&R->Z, 1); cleanup: return ret; } /* * Core multiplication algorithm for the (modified) comb method. * This part is actually common with the basic comb method (GECC 3.44) * * Cost: d A + d D + 1 R */ static int ecp_mul_comb_core(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_ecp_point T[], unsigned char T_size, const unsigned char x[], size_t d, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng, mbedtls_ecp_restart_ctx *rs_ctx) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_ecp_point Txi; mbedtls_mpi tmp[4]; size_t i; mbedtls_ecp_point_init(&Txi); mpi_init_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); #if !defined(MBEDTLS_ECP_RESTARTABLE) (void) rs_ctx; #endif #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL && rs_ctx->rsm->state != ecp_rsm_comb_core) { rs_ctx->rsm->i = 0; rs_ctx->rsm->state = ecp_rsm_comb_core; } /* new 'if' instead of nested for the sake of the 'else' branch */ if (rs_ctx != NULL && rs_ctx->rsm != NULL && rs_ctx->rsm->i != 0) { /* restore current index (R already pointing to rs_ctx->rsm->R) */ i = rs_ctx->rsm->i; } else #endif { /* Start with a non-zero point and randomize its coordinates */ i = d; MBEDTLS_MPI_CHK(ecp_select_comb(grp, R, T, T_size, x[i])); if (f_rng != 0) { MBEDTLS_MPI_CHK(ecp_randomize_jac(grp, R, f_rng, p_rng)); } } while (i != 0) { MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_DBL + MBEDTLS_ECP_OPS_ADD); --i; MBEDTLS_MPI_CHK(ecp_double_jac(grp, R, R, tmp)); MBEDTLS_MPI_CHK(ecp_select_comb(grp, &Txi, T, T_size, x[i])); MBEDTLS_MPI_CHK(ecp_add_mixed(grp, R, R, &Txi, tmp)); } cleanup: mbedtls_ecp_point_free(&Txi); mpi_free_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL && ret == MBEDTLS_ERR_ECP_IN_PROGRESS) { rs_ctx->rsm->i = i; /* no need to save R, already pointing to rs_ctx->rsm->R */ } #endif return ret; } /* * Recode the scalar to get constant-time comb multiplication * * As the actual scalar recoding needs an odd scalar as a starting point, * this wrapper ensures that by replacing m by N - m if necessary, and * informs the caller that the result of multiplication will be negated. * * This works because we only support large prime order for Short Weierstrass * curves, so N is always odd hence either m or N - m is. * * See ecp_comb_recode_core() for background. */ static int ecp_comb_recode_scalar(const mbedtls_ecp_group *grp, const mbedtls_mpi *m, unsigned char k[COMB_MAX_D + 1], size_t d, unsigned char w, unsigned char *parity_trick) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi M, mm; mbedtls_mpi_init(&M); mbedtls_mpi_init(&mm); /* N is always odd (see above), just make extra sure */ if (mbedtls_mpi_get_bit(&grp->N, 0) != 1) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* do we need the parity trick? */ *parity_trick = (mbedtls_mpi_get_bit(m, 0) == 0); /* execute parity fix in constant time */ MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&M, m)); MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&mm, &grp->N, m)); MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&M, &mm, *parity_trick)); /* actual scalar recoding */ ecp_comb_recode_core(k, d, w, &M); cleanup: mbedtls_mpi_free(&mm); mbedtls_mpi_free(&M); return ret; } /* * Perform comb multiplication (for short Weierstrass curves) * once the auxiliary table has been pre-computed. * * Scalar recoding may use a parity trick that makes us compute -m * P, * if that is the case we'll need to recover m * P at the end. */ static int ecp_mul_comb_after_precomp(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *T, unsigned char T_size, unsigned char w, size_t d, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng, mbedtls_ecp_restart_ctx *rs_ctx) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; unsigned char parity_trick; unsigned char k[COMB_MAX_D + 1]; mbedtls_ecp_point *RR = R; #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { RR = &rs_ctx->rsm->R; if (rs_ctx->rsm->state == ecp_rsm_final_norm) { goto final_norm; } } #endif MBEDTLS_MPI_CHK(ecp_comb_recode_scalar(grp, m, k, d, w, &parity_trick)); MBEDTLS_MPI_CHK(ecp_mul_comb_core(grp, RR, T, T_size, k, d, f_rng, p_rng, rs_ctx)); MBEDTLS_MPI_CHK(ecp_safe_invert_jac(grp, RR, parity_trick)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { rs_ctx->rsm->state = ecp_rsm_final_norm; } final_norm: MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_INV); #endif /* * Knowledge of the jacobian coordinates may leak the last few bits of the * scalar [1], and since our MPI implementation isn't constant-flow, * inversion (used for coordinate normalization) may leak the full value * of its input via side-channels [2]. * * [1] https://eprint.iacr.org/2003/191 * [2] https://eprint.iacr.org/2020/055 * * Avoid the leak by randomizing coordinates before we normalize them. */ if (f_rng != 0) { MBEDTLS_MPI_CHK(ecp_randomize_jac(grp, RR, f_rng, p_rng)); } MBEDTLS_MPI_CHK(ecp_normalize_jac(grp, RR)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL) { MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, RR)); } #endif cleanup: return ret; } /* * Pick window size based on curve size and whether we optimize for base point */ static unsigned char ecp_pick_window_size(const mbedtls_ecp_group *grp, unsigned char p_eq_g) { unsigned char w; /* * Minimize the number of multiplications, that is minimize * 10 * d * w + 18 * 2^(w-1) + 11 * d + 7 * w, with d = ceil( nbits / w ) * (see costs of the various parts, with 1S = 1M) */ w = grp->nbits >= 384 ? 5 : 4; /* * If P == G, pre-compute a bit more, since this may be re-used later. * Just adding one avoids upping the cost of the first mul too much, * and the memory cost too. */ if (p_eq_g) { w++; } /* * If static comb table may not be used (!p_eq_g) or static comb table does * not exists, make sure w is within bounds. * (The last test is useful only for very small curves in the test suite.) * * The user reduces MBEDTLS_ECP_WINDOW_SIZE does not changes the size of * static comb table, because the size of static comb table is fixed when * it is generated. */ #if (MBEDTLS_ECP_WINDOW_SIZE < 6) if ((!p_eq_g || !ecp_group_is_static_comb_table(grp)) && w > MBEDTLS_ECP_WINDOW_SIZE) { w = MBEDTLS_ECP_WINDOW_SIZE; } #endif if (w >= grp->nbits) { w = 2; } return w; } /* * Multiplication using the comb method - for curves in short Weierstrass form * * This function is mainly responsible for administrative work: * - managing the restart context if enabled * - managing the table of precomputed points (passed between the below two * functions): allocation, computation, ownership transfer, freeing. * * It delegates the actual arithmetic work to: * ecp_precompute_comb() and ecp_mul_comb_with_precomp() * * See comments on ecp_comb_recode_core() regarding the computation strategy. */ static int ecp_mul_comb(mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng, mbedtls_ecp_restart_ctx *rs_ctx) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; unsigned char w, p_eq_g, i; size_t d; unsigned char T_size = 0, T_ok = 0; mbedtls_ecp_point *T = NULL; ECP_RS_ENTER(rsm); /* Is P the base point ? */ #if MBEDTLS_ECP_FIXED_POINT_OPTIM == 1 p_eq_g = (MPI_ECP_CMP(&P->Y, &grp->G.Y) == 0 && MPI_ECP_CMP(&P->X, &grp->G.X) == 0); #else p_eq_g = 0; #endif /* Pick window size and deduce related sizes */ w = ecp_pick_window_size(grp, p_eq_g); T_size = 1U << (w - 1); d = (grp->nbits + w - 1) / w; /* Pre-computed table: do we have it already for the base point? */ if (p_eq_g && grp->T != NULL) { /* second pointer to the same table, will be deleted on exit */ T = grp->T; T_ok = 1; } else #if defined(MBEDTLS_ECP_RESTARTABLE) /* Pre-computed table: do we have one in progress? complete? */ if (rs_ctx != NULL && rs_ctx->rsm != NULL && rs_ctx->rsm->T != NULL) { /* transfer ownership of T from rsm to local function */ T = rs_ctx->rsm->T; rs_ctx->rsm->T = NULL; rs_ctx->rsm->T_size = 0; /* This effectively jumps to the call to mul_comb_after_precomp() */ T_ok = rs_ctx->rsm->state >= ecp_rsm_comb_core; } else #endif /* Allocate table if we didn't have any */ { T = mbedtls_calloc(T_size, sizeof(mbedtls_ecp_point)); if (T == NULL) { ret = MBEDTLS_ERR_ECP_ALLOC_FAILED; goto cleanup; } for (i = 0; i < T_size; i++) { mbedtls_ecp_point_init(&T[i]); } T_ok = 0; } /* Compute table (or finish computing it) if not done already */ if (!T_ok) { MBEDTLS_MPI_CHK(ecp_precompute_comb(grp, T, P, w, d, rs_ctx)); if (p_eq_g) { /* almost transfer ownership of T to the group, but keep a copy of * the pointer to use for calling the next function more easily */ grp->T = T; grp->T_size = T_size; } } /* Actual comb multiplication using precomputed points */ MBEDTLS_MPI_CHK(ecp_mul_comb_after_precomp(grp, R, m, T, T_size, w, d, f_rng, p_rng, rs_ctx)); cleanup: /* does T belong to the group? */ if (T == grp->T) { T = NULL; } /* does T belong to the restart context? */ #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->rsm != NULL && ret == MBEDTLS_ERR_ECP_IN_PROGRESS && T != NULL) { /* transfer ownership of T from local function to rsm */ rs_ctx->rsm->T_size = T_size; rs_ctx->rsm->T = T; T = NULL; } #endif /* did T belong to us? then let's destroy it! */ if (T != NULL) { for (i = 0; i < T_size; i++) { mbedtls_ecp_point_free(&T[i]); } mbedtls_free(T); } /* prevent caller from using invalid value */ int should_free_R = (ret != 0); #if defined(MBEDTLS_ECP_RESTARTABLE) /* don't free R while in progress in case R == P */ if (ret == MBEDTLS_ERR_ECP_IN_PROGRESS) { should_free_R = 0; } #endif if (should_free_R) { mbedtls_ecp_point_free(R); } ECP_RS_LEAVE(rsm); return ret; } #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) /* * For Montgomery curves, we do all the internal arithmetic in projective * coordinates. Import/export of points uses only the x coordinates, which is * internally represented as X / Z. * * For scalar multiplication, we'll use a Montgomery ladder. */ /* * Normalize Montgomery x/z coordinates: X = X/Z, Z = 1 * Cost: 1M + 1I */ static int ecp_normalize_mxz(const mbedtls_ecp_group *grp, mbedtls_ecp_point *P) { #if defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_normalize_mxz(grp, P); } #endif /* MBEDTLS_ECP_NORMALIZE_MXZ_ALT */ #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MPI_ECP_INV(&P->Z, &P->Z); MPI_ECP_MUL(&P->X, &P->X, &P->Z); MPI_ECP_LSET(&P->Z, 1); cleanup: return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT) */ } /* * Randomize projective x/z coordinates: * (X, Z) -> (l X, l Z) for random l * This is sort of the reverse operation of ecp_normalize_mxz(). * * This countermeasure was first suggested in [2]. * Cost: 2M */ static int ecp_randomize_mxz(const mbedtls_ecp_group *grp, mbedtls_ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { #if defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_randomize_mxz(grp, P, f_rng, p_rng); } #endif /* MBEDTLS_ECP_RANDOMIZE_MXZ_ALT */ #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi l; mbedtls_mpi_init(&l); /* Generate l such that 1 < l < p */ MPI_ECP_RAND(&l); MPI_ECP_MUL(&P->X, &P->X, &l); MPI_ECP_MUL(&P->Z, &P->Z, &l); cleanup: mbedtls_mpi_free(&l); if (ret == MBEDTLS_ERR_MPI_NOT_ACCEPTABLE) { ret = MBEDTLS_ERR_ECP_RANDOM_FAILED; } return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT) */ } /* * Double-and-add: R = 2P, S = P + Q, with d = X(P - Q), * for Montgomery curves in x/z coordinates. * * http://www.hyperelliptic.org/EFD/g1p/auto-code/montgom/xz/ladder/mladd-1987-m.op3 * with * d = X1 * P = (X2, Z2) * Q = (X3, Z3) * R = (X4, Z4) * S = (X5, Z5) * and eliminating temporary variables tO, ..., t4. * * Cost: 5M + 4S */ static int ecp_double_add_mxz(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R, mbedtls_ecp_point *S, const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q, const mbedtls_mpi *d, mbedtls_mpi T[4]) { #if defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT) if (mbedtls_internal_ecp_grp_capable(grp)) { return mbedtls_internal_ecp_double_add_mxz(grp, R, S, P, Q, d); } #endif /* MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT */ #if defined(MBEDTLS_ECP_NO_FALLBACK) && defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT) return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #else int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MPI_ECP_ADD(&T[0], &P->X, &P->Z); /* Pp := PX + PZ */ MPI_ECP_SUB(&T[1], &P->X, &P->Z); /* Pm := PX - PZ */ MPI_ECP_ADD(&T[2], &Q->X, &Q->Z); /* Qp := QX + XZ */ MPI_ECP_SUB(&T[3], &Q->X, &Q->Z); /* Qm := QX - QZ */ MPI_ECP_MUL(&T[3], &T[3], &T[0]); /* Qm * Pp */ MPI_ECP_MUL(&T[2], &T[2], &T[1]); /* Qp * Pm */ MPI_ECP_SQR(&T[0], &T[0]); /* Pp^2 */ MPI_ECP_SQR(&T[1], &T[1]); /* Pm^2 */ MPI_ECP_MUL(&R->X, &T[0], &T[1]); /* Pp^2 * Pm^2 */ MPI_ECP_SUB(&T[0], &T[0], &T[1]); /* Pp^2 - Pm^2 */ MPI_ECP_MUL(&R->Z, &grp->A, &T[0]); /* A * (Pp^2 - Pm^2) */ MPI_ECP_ADD(&R->Z, &T[1], &R->Z); /* [ A * (Pp^2-Pm^2) ] + Pm^2 */ MPI_ECP_ADD(&S->X, &T[3], &T[2]); /* Qm*Pp + Qp*Pm */ MPI_ECP_SQR(&S->X, &S->X); /* (Qm*Pp + Qp*Pm)^2 */ MPI_ECP_SUB(&S->Z, &T[3], &T[2]); /* Qm*Pp - Qp*Pm */ MPI_ECP_SQR(&S->Z, &S->Z); /* (Qm*Pp - Qp*Pm)^2 */ MPI_ECP_MUL(&S->Z, d, &S->Z); /* d * ( Qm*Pp - Qp*Pm )^2 */ MPI_ECP_MUL(&R->Z, &T[0], &R->Z); /* [A*(Pp^2-Pm^2)+Pm^2]*(Pp^2-Pm^2) */ cleanup: return ret; #endif /* !defined(MBEDTLS_ECP_NO_FALLBACK) || !defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT) */ } /* * Multiplication with Montgomery ladder in x/z coordinates, * for curves in Montgomery form */ static int ecp_mul_mxz(mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i; unsigned char b; mbedtls_ecp_point RP; mbedtls_mpi PX; mbedtls_mpi tmp[4]; mbedtls_ecp_point_init(&RP); mbedtls_mpi_init(&PX); mpi_init_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); if (f_rng == NULL) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* Save PX and read from P before writing to R, in case P == R */ MPI_ECP_MOV(&PX, &P->X); MBEDTLS_MPI_CHK(mbedtls_ecp_copy(&RP, P)); /* Set R to zero in modified x/z coordinates */ MPI_ECP_LSET(&R->X, 1); MPI_ECP_LSET(&R->Z, 0); mbedtls_mpi_free(&R->Y); /* RP.X might be slightly larger than P, so reduce it */ MOD_ADD(&RP.X); /* Randomize coordinates of the starting point */ MBEDTLS_MPI_CHK(ecp_randomize_mxz(grp, &RP, f_rng, p_rng)); /* Loop invariant: R = result so far, RP = R + P */ i = grp->nbits + 1; /* one past the (zero-based) required msb for private keys */ while (i-- > 0) { b = mbedtls_mpi_get_bit(m, i); /* * if (b) R = 2R + P else R = 2R, * which is: * if (b) double_add( RP, R, RP, R ) * else double_add( R, RP, R, RP ) * but using safe conditional swaps to avoid leaks */ MPI_ECP_COND_SWAP(&R->X, &RP.X, b); MPI_ECP_COND_SWAP(&R->Z, &RP.Z, b); MBEDTLS_MPI_CHK(ecp_double_add_mxz(grp, R, &RP, R, &RP, &PX, tmp)); MPI_ECP_COND_SWAP(&R->X, &RP.X, b); MPI_ECP_COND_SWAP(&R->Z, &RP.Z, b); } /* * Knowledge of the projective coordinates may leak the last few bits of the * scalar [1], and since our MPI implementation isn't constant-flow, * inversion (used for coordinate normalization) may leak the full value * of its input via side-channels [2]. * * [1] https://eprint.iacr.org/2003/191 * [2] https://eprint.iacr.org/2020/055 * * Avoid the leak by randomizing coordinates before we normalize them. */ MBEDTLS_MPI_CHK(ecp_randomize_mxz(grp, R, f_rng, p_rng)); MBEDTLS_MPI_CHK(ecp_normalize_mxz(grp, R)); cleanup: mbedtls_ecp_point_free(&RP); mbedtls_mpi_free(&PX); mpi_free_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); return ret; } #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */ /* * Restartable multiplication R = m * P * * This internal function can be called without an RNG in case where we know * the inputs are not sensitive. */ static int ecp_mul_restartable_internal(mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng, mbedtls_ecp_restart_ctx *rs_ctx) { int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA; #if defined(MBEDTLS_ECP_INTERNAL_ALT) char is_grp_capable = 0; #endif #if defined(MBEDTLS_ECP_RESTARTABLE) /* reset ops count for this call if top-level */ if (rs_ctx != NULL && rs_ctx->depth++ == 0) { rs_ctx->ops_done = 0; } #else (void) rs_ctx; #endif #if defined(MBEDTLS_ECP_INTERNAL_ALT) if ((is_grp_capable = mbedtls_internal_ecp_grp_capable(grp))) { MBEDTLS_MPI_CHK(mbedtls_internal_ecp_init(grp)); } #endif /* MBEDTLS_ECP_INTERNAL_ALT */ int restarting = 0; #if defined(MBEDTLS_ECP_RESTARTABLE) restarting = (rs_ctx != NULL && rs_ctx->rsm != NULL); #endif /* skip argument check when restarting */ if (!restarting) { /* check_privkey is free */ MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_CHK); /* Common sanity checks */ MBEDTLS_MPI_CHK(mbedtls_ecp_check_privkey(grp, m)); MBEDTLS_MPI_CHK(mbedtls_ecp_check_pubkey(grp, P)); } ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA; #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { MBEDTLS_MPI_CHK(ecp_mul_mxz(grp, R, m, P, f_rng, p_rng)); } #endif #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { MBEDTLS_MPI_CHK(ecp_mul_comb(grp, R, m, P, f_rng, p_rng, rs_ctx)); } #endif cleanup: #if defined(MBEDTLS_ECP_INTERNAL_ALT) if (is_grp_capable) { mbedtls_internal_ecp_free(grp); } #endif /* MBEDTLS_ECP_INTERNAL_ALT */ #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL) { rs_ctx->depth--; } #endif return ret; } /* * Restartable multiplication R = m * P */ int mbedtls_ecp_mul_restartable(mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng, mbedtls_ecp_restart_ctx *rs_ctx) { if (f_rng == NULL) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } return ecp_mul_restartable_internal(grp, R, m, P, f_rng, p_rng, rs_ctx); } /* * Multiplication R = m * P */ int mbedtls_ecp_mul(mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { return mbedtls_ecp_mul_restartable(grp, R, m, P, f_rng, p_rng, NULL); } #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) /* * Check that an affine point is valid as a public key, * short weierstrass curves (SEC1 3.2.3.1) */ static int ecp_check_pubkey_sw(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi YY, RHS; /* pt coordinates must be normalized for our checks */ if (mbedtls_mpi_cmp_int(&pt->X, 0) < 0 || mbedtls_mpi_cmp_int(&pt->Y, 0) < 0 || mbedtls_mpi_cmp_mpi(&pt->X, &grp->P) >= 0 || mbedtls_mpi_cmp_mpi(&pt->Y, &grp->P) >= 0) { return MBEDTLS_ERR_ECP_INVALID_KEY; } mbedtls_mpi_init(&YY); mbedtls_mpi_init(&RHS); /* * YY = Y^2 * RHS = X^3 + A X + B */ MPI_ECP_SQR(&YY, &pt->Y); MBEDTLS_MPI_CHK(ecp_sw_rhs(grp, &RHS, &pt->X)); if (MPI_ECP_CMP(&YY, &RHS) != 0) { ret = MBEDTLS_ERR_ECP_INVALID_KEY; } cleanup: mbedtls_mpi_free(&YY); mbedtls_mpi_free(&RHS); return ret; } #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) /* * R = m * P with shortcuts for m == 0, m == 1 and m == -1 * NOT constant-time - ONLY for short Weierstrass! */ static int mbedtls_ecp_mul_shortcuts(mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, mbedtls_ecp_restart_ctx *rs_ctx) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi tmp; mbedtls_mpi_init(&tmp); if (mbedtls_mpi_cmp_int(m, 0) == 0) { MBEDTLS_MPI_CHK(mbedtls_ecp_check_pubkey(grp, P)); MBEDTLS_MPI_CHK(mbedtls_ecp_set_zero(R)); } else if (mbedtls_mpi_cmp_int(m, 1) == 0) { MBEDTLS_MPI_CHK(mbedtls_ecp_check_pubkey(grp, P)); MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, P)); } else if (mbedtls_mpi_cmp_int(m, -1) == 0) { MBEDTLS_MPI_CHK(mbedtls_ecp_check_pubkey(grp, P)); MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, P)); MPI_ECP_NEG(&R->Y); } else { MBEDTLS_MPI_CHK(ecp_mul_restartable_internal(grp, R, m, P, NULL, NULL, rs_ctx)); } cleanup: mbedtls_mpi_free(&tmp); return ret; } /* * Restartable linear combination * NOT constant-time */ int mbedtls_ecp_muladd_restartable( mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, const mbedtls_mpi *n, const mbedtls_ecp_point *Q, mbedtls_ecp_restart_ctx *rs_ctx) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_ecp_point mP; mbedtls_ecp_point *pmP = &mP; mbedtls_ecp_point *pR = R; mbedtls_mpi tmp[4]; #if defined(MBEDTLS_ECP_INTERNAL_ALT) char is_grp_capable = 0; #endif if (mbedtls_ecp_get_type(grp) != MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { return MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; } mbedtls_ecp_point_init(&mP); mpi_init_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); ECP_RS_ENTER(ma); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->ma != NULL) { /* redirect intermediate results to restart context */ pmP = &rs_ctx->ma->mP; pR = &rs_ctx->ma->R; /* jump to next operation */ if (rs_ctx->ma->state == ecp_rsma_mul2) { goto mul2; } if (rs_ctx->ma->state == ecp_rsma_add) { goto add; } if (rs_ctx->ma->state == ecp_rsma_norm) { goto norm; } } #endif /* MBEDTLS_ECP_RESTARTABLE */ MBEDTLS_MPI_CHK(mbedtls_ecp_mul_shortcuts(grp, pmP, m, P, rs_ctx)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->ma != NULL) { rs_ctx->ma->state = ecp_rsma_mul2; } mul2: #endif MBEDTLS_MPI_CHK(mbedtls_ecp_mul_shortcuts(grp, pR, n, Q, rs_ctx)); #if defined(MBEDTLS_ECP_INTERNAL_ALT) if ((is_grp_capable = mbedtls_internal_ecp_grp_capable(grp))) { MBEDTLS_MPI_CHK(mbedtls_internal_ecp_init(grp)); } #endif /* MBEDTLS_ECP_INTERNAL_ALT */ #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->ma != NULL) { rs_ctx->ma->state = ecp_rsma_add; } add: #endif MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_ADD); MBEDTLS_MPI_CHK(ecp_add_mixed(grp, pR, pmP, pR, tmp)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->ma != NULL) { rs_ctx->ma->state = ecp_rsma_norm; } norm: #endif MBEDTLS_ECP_BUDGET(MBEDTLS_ECP_OPS_INV); MBEDTLS_MPI_CHK(ecp_normalize_jac(grp, pR)); #if defined(MBEDTLS_ECP_RESTARTABLE) if (rs_ctx != NULL && rs_ctx->ma != NULL) { MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, pR)); } #endif cleanup: mpi_free_many(tmp, sizeof(tmp) / sizeof(mbedtls_mpi)); #if defined(MBEDTLS_ECP_INTERNAL_ALT) if (is_grp_capable) { mbedtls_internal_ecp_free(grp); } #endif /* MBEDTLS_ECP_INTERNAL_ALT */ mbedtls_ecp_point_free(&mP); ECP_RS_LEAVE(ma); return ret; } /* * Linear combination * NOT constant-time */ int mbedtls_ecp_muladd(mbedtls_ecp_group *grp, mbedtls_ecp_point *R, const mbedtls_mpi *m, const mbedtls_ecp_point *P, const mbedtls_mpi *n, const mbedtls_ecp_point *Q) { return mbedtls_ecp_muladd_restartable(grp, R, m, P, n, Q, NULL); } #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) #if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) #define ECP_MPI_INIT(s, n, p) { s, (n), (mbedtls_mpi_uint *) (p) } #define ECP_MPI_INIT_ARRAY(x) \ ECP_MPI_INIT(1, sizeof(x) / sizeof(mbedtls_mpi_uint), x) /* * Constants for the two points other than 0, 1, -1 (mod p) in * https://cr.yp.to/ecdh.html#validate * See ecp_check_pubkey_x25519(). */ static const mbedtls_mpi_uint x25519_bad_point_1[] = { MBEDTLS_BYTES_TO_T_UINT_8(0xe0, 0xeb, 0x7a, 0x7c, 0x3b, 0x41, 0xb8, 0xae), MBEDTLS_BYTES_TO_T_UINT_8(0x16, 0x56, 0xe3, 0xfa, 0xf1, 0x9f, 0xc4, 0x6a), MBEDTLS_BYTES_TO_T_UINT_8(0xda, 0x09, 0x8d, 0xeb, 0x9c, 0x32, 0xb1, 0xfd), MBEDTLS_BYTES_TO_T_UINT_8(0x86, 0x62, 0x05, 0x16, 0x5f, 0x49, 0xb8, 0x00), }; static const mbedtls_mpi_uint x25519_bad_point_2[] = { MBEDTLS_BYTES_TO_T_UINT_8(0x5f, 0x9c, 0x95, 0xbc, 0xa3, 0x50, 0x8c, 0x24), MBEDTLS_BYTES_TO_T_UINT_8(0xb1, 0xd0, 0xb1, 0x55, 0x9c, 0x83, 0xef, 0x5b), MBEDTLS_BYTES_TO_T_UINT_8(0x04, 0x44, 0x5c, 0xc4, 0x58, 0x1c, 0x8e, 0x86), MBEDTLS_BYTES_TO_T_UINT_8(0xd8, 0x22, 0x4e, 0xdd, 0xd0, 0x9f, 0x11, 0x57), }; static const mbedtls_mpi ecp_x25519_bad_point_1 = ECP_MPI_INIT_ARRAY( x25519_bad_point_1); static const mbedtls_mpi ecp_x25519_bad_point_2 = ECP_MPI_INIT_ARRAY( x25519_bad_point_2); #endif /* MBEDTLS_ECP_DP_CURVE25519_ENABLED */ /* * Check that the input point is not one of the low-order points. * This is recommended by the "May the Fourth" paper: * https://eprint.iacr.org/2017/806.pdf * Those points are never sent by an honest peer. */ static int ecp_check_bad_points_mx(const mbedtls_mpi *X, const mbedtls_mpi *P, const mbedtls_ecp_group_id grp_id) { int ret; mbedtls_mpi XmP; mbedtls_mpi_init(&XmP); /* Reduce X mod P so that we only need to check values less than P. * We know X < 2^256 so we can proceed by subtraction. */ MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&XmP, X)); while (mbedtls_mpi_cmp_mpi(&XmP, P) >= 0) { MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&XmP, &XmP, P)); } /* Check against the known bad values that are less than P. For Curve448 * these are 0, 1 and -1. For Curve25519 we check the values less than P * from the following list: https://cr.yp.to/ecdh.html#validate */ if (mbedtls_mpi_cmp_int(&XmP, 1) <= 0) { /* takes care of 0 and 1 */ ret = MBEDTLS_ERR_ECP_INVALID_KEY; goto cleanup; } #if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) if (grp_id == MBEDTLS_ECP_DP_CURVE25519) { if (mbedtls_mpi_cmp_mpi(&XmP, &ecp_x25519_bad_point_1) == 0) { ret = MBEDTLS_ERR_ECP_INVALID_KEY; goto cleanup; } if (mbedtls_mpi_cmp_mpi(&XmP, &ecp_x25519_bad_point_2) == 0) { ret = MBEDTLS_ERR_ECP_INVALID_KEY; goto cleanup; } } #else (void) grp_id; #endif /* Final check: check if XmP + 1 is P (final because it changes XmP!) */ MBEDTLS_MPI_CHK(mbedtls_mpi_add_int(&XmP, &XmP, 1)); if (mbedtls_mpi_cmp_mpi(&XmP, P) == 0) { ret = MBEDTLS_ERR_ECP_INVALID_KEY; goto cleanup; } ret = 0; cleanup: mbedtls_mpi_free(&XmP); return ret; } /* * Check validity of a public key for Montgomery curves with x-only schemes */ static int ecp_check_pubkey_mx(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) { /* [Curve25519 p. 5] Just check X is the correct number of bytes */ /* Allow any public value, if it's too big then we'll just reduce it mod p * (RFC 7748 sec. 5 para. 3). */ if (mbedtls_mpi_size(&pt->X) > (grp->nbits + 7) / 8) { return MBEDTLS_ERR_ECP_INVALID_KEY; } /* Implicit in all standards (as they don't consider negative numbers): * X must be non-negative. This is normally ensured by the way it's * encoded for transmission, but let's be extra sure. */ if (mbedtls_mpi_cmp_int(&pt->X, 0) < 0) { return MBEDTLS_ERR_ECP_INVALID_KEY; } return ecp_check_bad_points_mx(&pt->X, &grp->P, grp->id); } #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */ /* * Check that a point is valid as a public key */ int mbedtls_ecp_check_pubkey(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) { /* Must use affine coordinates */ if (mbedtls_mpi_cmp_int(&pt->Z, 1) != 0) { return MBEDTLS_ERR_ECP_INVALID_KEY; } #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { return ecp_check_pubkey_mx(grp, pt); } #endif #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { return ecp_check_pubkey_sw(grp, pt); } #endif return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * Check that an mbedtls_mpi is valid as a private key */ int mbedtls_ecp_check_privkey(const mbedtls_ecp_group *grp, const mbedtls_mpi *d) { #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { /* see RFC 7748 sec. 5 para. 5 */ if (mbedtls_mpi_get_bit(d, 0) != 0 || mbedtls_mpi_get_bit(d, 1) != 0 || mbedtls_mpi_bitlen(d) - 1 != grp->nbits) { /* mbedtls_mpi_bitlen is one-based! */ return MBEDTLS_ERR_ECP_INVALID_KEY; } /* see [Curve25519] page 5 */ if (grp->nbits == 254 && mbedtls_mpi_get_bit(d, 2) != 0) { return MBEDTLS_ERR_ECP_INVALID_KEY; } return 0; } #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */ #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { /* see SEC1 3.2 */ if (mbedtls_mpi_cmp_int(d, 1) < 0 || mbedtls_mpi_cmp_mpi(d, &grp->N) >= 0) { return MBEDTLS_ERR_ECP_INVALID_KEY; } else { return 0; } } #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) MBEDTLS_STATIC_TESTABLE int mbedtls_ecp_gen_privkey_mx(size_t high_bit, mbedtls_mpi *d, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA; size_t n_random_bytes = high_bit / 8 + 1; /* [Curve25519] page 5 */ /* Generate a (high_bit+1)-bit random number by generating just enough * random bytes, then shifting out extra bits from the top (necessary * when (high_bit+1) is not a multiple of 8). */ MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(d, n_random_bytes, f_rng, p_rng)); MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(d, 8 * n_random_bytes - high_bit - 1)); MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, high_bit, 1)); /* Make sure the last two bits are unset for Curve448, three bits for Curve25519 */ MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 0, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 1, 0)); if (high_bit == 254) { MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 2, 0)); } cleanup: return ret; } #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */ #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) static int mbedtls_ecp_gen_privkey_sw( const mbedtls_mpi *N, mbedtls_mpi *d, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = mbedtls_mpi_random(d, 1, N, f_rng, p_rng); switch (ret) { case MBEDTLS_ERR_MPI_NOT_ACCEPTABLE: return MBEDTLS_ERR_ECP_RANDOM_FAILED; default: return ret; } } #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ /* * Generate a private key */ int mbedtls_ecp_gen_privkey(const mbedtls_ecp_group *grp, mbedtls_mpi *d, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { return mbedtls_ecp_gen_privkey_mx(grp->nbits, d, f_rng, p_rng); } #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */ #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { return mbedtls_ecp_gen_privkey_sw(&grp->N, d, f_rng, p_rng); } #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } /* * Generate a keypair with configurable base point */ int mbedtls_ecp_gen_keypair_base(mbedtls_ecp_group *grp, const mbedtls_ecp_point *G, mbedtls_mpi *d, mbedtls_ecp_point *Q, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MBEDTLS_MPI_CHK(mbedtls_ecp_gen_privkey(grp, d, f_rng, p_rng)); MBEDTLS_MPI_CHK(mbedtls_ecp_mul(grp, Q, d, G, f_rng, p_rng)); cleanup: return ret; } /* * Generate key pair, wrapper for conventional base point */ int mbedtls_ecp_gen_keypair(mbedtls_ecp_group *grp, mbedtls_mpi *d, mbedtls_ecp_point *Q, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { return mbedtls_ecp_gen_keypair_base(grp, &grp->G, d, Q, f_rng, p_rng); } /* * Generate a keypair, prettier wrapper */ int mbedtls_ecp_gen_key(mbedtls_ecp_group_id grp_id, mbedtls_ecp_keypair *key, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; if ((ret = mbedtls_ecp_group_load(&key->grp, grp_id)) != 0) { return ret; } return mbedtls_ecp_gen_keypair(&key->grp, &key->d, &key->Q, f_rng, p_rng); } #define ECP_CURVE25519_KEY_SIZE 32 #define ECP_CURVE448_KEY_SIZE 56 /* * Read a private key. */ int mbedtls_ecp_read_key(mbedtls_ecp_group_id grp_id, mbedtls_ecp_keypair *key, const unsigned char *buf, size_t buflen) { int ret = 0; if ((ret = mbedtls_ecp_group_load(&key->grp, grp_id)) != 0) { return ret; } ret = MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (mbedtls_ecp_get_type(&key->grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { /* * Mask the key as mandated by RFC7748 for Curve25519 and Curve448. */ if (grp_id == MBEDTLS_ECP_DP_CURVE25519) { if (buflen != ECP_CURVE25519_KEY_SIZE) { return MBEDTLS_ERR_ECP_INVALID_KEY; } MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary_le(&key->d, buf, buflen)); /* Set the three least significant bits to 0 */ MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(&key->d, 0, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(&key->d, 1, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(&key->d, 2, 0)); /* Set the most significant bit to 0 */ MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit(&key->d, ECP_CURVE25519_KEY_SIZE * 8 - 1, 0) ); /* Set the second most significant bit to 1 */ MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit(&key->d, ECP_CURVE25519_KEY_SIZE * 8 - 2, 1) ); } else if (grp_id == MBEDTLS_ECP_DP_CURVE448) { if (buflen != ECP_CURVE448_KEY_SIZE) { return MBEDTLS_ERR_ECP_INVALID_KEY; } MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary_le(&key->d, buf, buflen)); /* Set the two least significant bits to 0 */ MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(&key->d, 0, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(&key->d, 1, 0)); /* Set the most significant bit to 1 */ MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit(&key->d, ECP_CURVE448_KEY_SIZE * 8 - 1, 1) ); } } #endif #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(&key->grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary(&key->d, buf, buflen)); MBEDTLS_MPI_CHK(mbedtls_ecp_check_privkey(&key->grp, &key->d)); } #endif cleanup: if (ret != 0) { mbedtls_mpi_free(&key->d); } return ret; } /* * Write a private key. */ int mbedtls_ecp_write_key(mbedtls_ecp_keypair *key, unsigned char *buf, size_t buflen) { int ret = MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE; #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (mbedtls_ecp_get_type(&key->grp) == MBEDTLS_ECP_TYPE_MONTGOMERY) { if (key->grp.id == MBEDTLS_ECP_DP_CURVE25519) { if (buflen < ECP_CURVE25519_KEY_SIZE) { return MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL; } } else if (key->grp.id == MBEDTLS_ECP_DP_CURVE448) { if (buflen < ECP_CURVE448_KEY_SIZE) { return MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL; } } MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary_le(&key->d, buf, buflen)); } #endif #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) if (mbedtls_ecp_get_type(&key->grp) == MBEDTLS_ECP_TYPE_SHORT_WEIERSTRASS) { MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&key->d, buf, buflen)); } #endif cleanup: return ret; } /* * Check a public-private key pair */ int mbedtls_ecp_check_pub_priv( const mbedtls_ecp_keypair *pub, const mbedtls_ecp_keypair *prv, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_ecp_point Q; mbedtls_ecp_group grp; if (pub->grp.id == MBEDTLS_ECP_DP_NONE || pub->grp.id != prv->grp.id || mbedtls_mpi_cmp_mpi(&pub->Q.X, &prv->Q.X) || mbedtls_mpi_cmp_mpi(&pub->Q.Y, &prv->Q.Y) || mbedtls_mpi_cmp_mpi(&pub->Q.Z, &prv->Q.Z)) { return MBEDTLS_ERR_ECP_BAD_INPUT_DATA; } mbedtls_ecp_point_init(&Q); mbedtls_ecp_group_init(&grp); /* mbedtls_ecp_mul() needs a non-const group... */ mbedtls_ecp_group_copy(&grp, &prv->grp); /* Also checks d is valid */ MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &Q, &prv->d, &prv->grp.G, f_rng, p_rng)); if (mbedtls_mpi_cmp_mpi(&Q.X, &prv->Q.X) || mbedtls_mpi_cmp_mpi(&Q.Y, &prv->Q.Y) || mbedtls_mpi_cmp_mpi(&Q.Z, &prv->Q.Z)) { ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA; goto cleanup; } cleanup: mbedtls_ecp_point_free(&Q); mbedtls_ecp_group_free(&grp); return ret; } /* * Export generic key-pair parameters. */ int mbedtls_ecp_export(const mbedtls_ecp_keypair *key, mbedtls_ecp_group *grp, mbedtls_mpi *d, mbedtls_ecp_point *Q) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; if ((ret = mbedtls_ecp_group_copy(grp, &key->grp)) != 0) { return ret; } if ((ret = mbedtls_mpi_copy(d, &key->d)) != 0) { return ret; } if ((ret = mbedtls_ecp_copy(Q, &key->Q)) != 0) { return ret; } return 0; } #if defined(MBEDTLS_SELF_TEST) /* * PRNG for test - !!!INSECURE NEVER USE IN PRODUCTION!!! * * This is the linear congruential generator from numerical recipes, * except we only use the low byte as the output. See * https://en.wikipedia.org/wiki/Linear_congruential_generator#Parameters_in_common_use */ static int self_test_rng(void *ctx, unsigned char *out, size_t len) { static uint32_t state = 42; (void) ctx; for (size_t i = 0; i < len; i++) { state = state * 1664525u + 1013904223u; out[i] = (unsigned char) state; } return 0; } /* Adjust the exponent to be a valid private point for the specified curve. * This is sometimes necessary because we use a single set of exponents * for all curves but the validity of values depends on the curve. */ static int self_test_adjust_exponent(const mbedtls_ecp_group *grp, mbedtls_mpi *m) { int ret = 0; switch (grp->id) { /* If Curve25519 is available, then that's what we use for the * Montgomery test, so we don't need the adjustment code. */ #if !defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) #if defined(MBEDTLS_ECP_DP_CURVE448_ENABLED) case MBEDTLS_ECP_DP_CURVE448: /* Move highest bit from 254 to N-1. Setting bit N-1 is * necessary to enforce the highest-bit-set constraint. */ MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(m, 254, 0)); MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(m, grp->nbits, 1)); /* Copy second-highest bit from 253 to N-2. This is not * necessary but improves the test variety a bit. */ MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit(m, grp->nbits - 1, mbedtls_mpi_get_bit(m, 253))); break; #endif #endif /* ! defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) */ default: /* Non-Montgomery curves and Curve25519 need no adjustment. */ (void) grp; (void) m; goto cleanup; } cleanup: return ret; } /* Calculate R = m.P for each m in exponents. Check that the number of * basic operations doesn't depend on the value of m. */ static int self_test_point(int verbose, mbedtls_ecp_group *grp, mbedtls_ecp_point *R, mbedtls_mpi *m, const mbedtls_ecp_point *P, const char *const *exponents, size_t n_exponents) { int ret = 0; size_t i = 0; unsigned long add_c_prev, dbl_c_prev, mul_c_prev; add_count = 0; dbl_count = 0; mul_count = 0; MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(m, 16, exponents[0])); MBEDTLS_MPI_CHK(self_test_adjust_exponent(grp, m)); MBEDTLS_MPI_CHK(mbedtls_ecp_mul(grp, R, m, P, self_test_rng, NULL)); for (i = 1; i < n_exponents; i++) { add_c_prev = add_count; dbl_c_prev = dbl_count; mul_c_prev = mul_count; add_count = 0; dbl_count = 0; mul_count = 0; MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(m, 16, exponents[i])); MBEDTLS_MPI_CHK(self_test_adjust_exponent(grp, m)); MBEDTLS_MPI_CHK(mbedtls_ecp_mul(grp, R, m, P, self_test_rng, NULL)); if (add_count != add_c_prev || dbl_count != dbl_c_prev || mul_count != mul_c_prev) { ret = 1; break; } } cleanup: if (verbose != 0) { if (ret != 0) { mbedtls_printf("failed (%u)\n", (unsigned int) i); } else { mbedtls_printf("passed\n"); } } return ret; } /* * Checkup routine */ int mbedtls_ecp_self_test(int verbose) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_ecp_group grp; mbedtls_ecp_point R, P; mbedtls_mpi m; #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) /* Exponents especially adapted for secp192k1, which has the lowest * order n of all supported curves (secp192r1 is in a slightly larger * field but the order of its base point is slightly smaller). */ const char *sw_exponents[] = { "000000000000000000000000000000000000000000000001", /* one */ "FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8C", /* n - 1 */ "5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */ "400000000000000000000000000000000000000000000000", /* one and zeros */ "7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF", /* all ones */ "555555555555555555555555555555555555555555555555", /* 101010... */ }; #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) const char *m_exponents[] = { /* Valid private values for Curve25519. In a build with Curve448 * but not Curve25519, they will be adjusted in * self_test_adjust_exponent(). */ "4000000000000000000000000000000000000000000000000000000000000000", "5C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C3C30", "5715ECCE24583F7A7023C24164390586842E816D7280A49EF6DF4EAE6B280BF8", "41A2B017516F6D254E1F002BCCBADD54BE30F8CEC737A0E912B4963B6BA74460", "5555555555555555555555555555555555555555555555555555555555555550", "7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF8", }; #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */ mbedtls_ecp_group_init(&grp); mbedtls_ecp_point_init(&R); mbedtls_ecp_point_init(&P); mbedtls_mpi_init(&m); #if defined(MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED) /* Use secp192r1 if available, or any available curve */ #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, MBEDTLS_ECP_DP_SECP192R1)); #else MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, mbedtls_ecp_curve_list()->grp_id)); #endif if (verbose != 0) { mbedtls_printf(" ECP SW test #1 (constant op_count, base point G): "); } /* Do a dummy multiplication first to trigger precomputation */ MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&m, 2)); MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &P, &m, &grp.G, self_test_rng, NULL)); ret = self_test_point(verbose, &grp, &R, &m, &grp.G, sw_exponents, sizeof(sw_exponents) / sizeof(sw_exponents[0])); if (ret != 0) { goto cleanup; } if (verbose != 0) { mbedtls_printf(" ECP SW test #2 (constant op_count, other point): "); } /* We computed P = 2G last time, use it */ ret = self_test_point(verbose, &grp, &R, &m, &P, sw_exponents, sizeof(sw_exponents) / sizeof(sw_exponents[0])); if (ret != 0) { goto cleanup; } mbedtls_ecp_group_free(&grp); mbedtls_ecp_point_free(&R); #endif /* MBEDTLS_ECP_SHORT_WEIERSTRASS_ENABLED */ #if defined(MBEDTLS_ECP_MONTGOMERY_ENABLED) if (verbose != 0) { mbedtls_printf(" ECP Montgomery test (constant op_count): "); } #if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, MBEDTLS_ECP_DP_CURVE25519)); #elif defined(MBEDTLS_ECP_DP_CURVE448_ENABLED) MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, MBEDTLS_ECP_DP_CURVE448)); #else #error "MBEDTLS_ECP_MONTGOMERY_ENABLED is defined, but no curve is supported for self-test" #endif ret = self_test_point(verbose, &grp, &R, &m, &grp.G, m_exponents, sizeof(m_exponents) / sizeof(m_exponents[0])); if (ret != 0) { goto cleanup; } #endif /* MBEDTLS_ECP_MONTGOMERY_ENABLED */ cleanup: if (ret < 0 && verbose != 0) { mbedtls_printf("Unexpected error, return code = %08X\n", (unsigned int) ret); } mbedtls_ecp_group_free(&grp); mbedtls_ecp_point_free(&R); mbedtls_ecp_point_free(&P); mbedtls_mpi_free(&m); if (verbose != 0) { mbedtls_printf("\n"); } return ret; } #endif /* MBEDTLS_SELF_TEST */ #endif /* !MBEDTLS_ECP_ALT */ #endif /* MBEDTLS_ECP_C */