1 /** 2 * Modular bignum functions 3 * 4 * This module implements operations on integers modulo some fixed modulus. 5 * 6 * The functions in this module obey the following conventions unless 7 * explicitly indicated otherwise: 8 * 9 * - **Modulus parameters**: the modulus is passed as a pointer to a structure 10 * of type #mbedtls_mpi_mod_modulus. The structure must be set up with an 11 * array of limbs storing the bignum value of the modulus. The modulus must 12 * be odd and is assumed to have no leading zeroes. The modulus is usually 13 * named \c N and is usually input-only. Functions which take a parameter 14 * of type \c const #mbedtls_mpi_mod_modulus* must not modify its value. 15 * - **Bignum parameters**: Bignums are passed as pointers to an array of 16 * limbs or to a #mbedtls_mpi_mod_residue structure. A limb has the type 17 * #mbedtls_mpi_uint. Residues must be initialized before use, and must be 18 * associated with the modulus \c N. Unless otherwise specified: 19 * - Bignum parameters called \c A, \c B, ... are inputs and are not 20 * modified by the function. Functions which take a parameter of 21 * type \c const #mbedtls_mpi_mod_residue* must not modify its value. 22 * - Bignum parameters called \c X, \c Y, ... are outputs or input-output. 23 * The initial bignum value of output-only parameters is ignored, but 24 * they must be set up and associated with the modulus \c N. Some 25 * functions (typically constant-flow) require that the limbs in an 26 * output residue are initialized. 27 * - Bignum parameters called \c p are inputs used to set up a modulus or 28 * residue. These must be pointers to an array of limbs. 29 * - \c T is a temporary storage area. The initial content of such a 30 * parameter is ignored and the final content is unspecified. 31 * - Some functions use different names, such as \c r for the residue. 32 * - **Bignum sizes**: bignum sizes are always expressed in limbs. Both 33 * #mbedtls_mpi_mod_modulus and #mbedtls_mpi_mod_residue have a \c limbs 34 * member storing its size. All bignum parameters must have the same 35 * number of limbs as the modulus. All bignum sizes must be at least 1 and 36 * must be significantly less than #SIZE_MAX. The behavior if a size is 0 is 37 * undefined. 38 * - **Bignum representation**: the representation of inputs and outputs is 39 * specified by the \c int_rep field of the modulus. 40 * - **Parameter ordering**: for bignum parameters, outputs come before inputs. 41 * The modulus is passed after residues. Temporaries come last. 42 * - **Aliasing**: in general, output bignums may be aliased to one or more 43 * inputs. Modulus values may not be aliased to any other parameter. Outputs 44 * may not be aliased to one another. Temporaries may not be aliased to any 45 * other parameter. 46 * - **Overlap**: apart from aliasing of residue pointers (where two residue 47 * arguments are equal pointers), overlap is not supported and may result 48 * in undefined behavior. 49 * - **Error handling**: functions generally check compatibility of input 50 * sizes. Most functions will not check that input values are in canonical 51 * form (i.e. that \c A < \c N), this is only checked during setup of a 52 * residue structure. 53 * - **Modular representatives**: all functions expect inputs to be in the 54 * range [0, \c N - 1] and guarantee outputs in the range [0, \c N - 1]. 55 * Residues are set up with an associated modulus, and operations are only 56 * guaranteed to work if the modulus is associated with all residue 57 * parameters. If a residue is passed with a modulus other than the one it 58 * is associated with, then it may be out of range. If an input is out of 59 * range, outputs are fully unspecified, though bignum values out of range 60 * should not cause buffer overflows (beware that this is not extensively 61 * tested). 62 */ 63 64 /* 65 * Copyright The Mbed TLS Contributors 66 * SPDX-License-Identifier: Apache-2.0 OR GPL-2.0-or-later 67 */ 68 69 #ifndef MBEDTLS_BIGNUM_MOD_H 70 #define MBEDTLS_BIGNUM_MOD_H 71 72 #include "common.h" 73 74 #if defined(MBEDTLS_BIGNUM_C) 75 #include "mbedtls/bignum.h" 76 #endif 77 78 /** How residues associated with a modulus are represented. 79 * 80 * This also determines which fields of the modulus structure are valid and 81 * what their contents are (see #mbedtls_mpi_mod_modulus). 82 */ 83 typedef enum { 84 /** Representation not chosen (makes the modulus structure invalid). */ 85 MBEDTLS_MPI_MOD_REP_INVALID = 0, 86 /* Skip 1 as it is slightly easier to accidentally pass to functions. */ 87 /** Montgomery representation. */ 88 MBEDTLS_MPI_MOD_REP_MONTGOMERY = 2, 89 /* Optimised reduction available. This indicates a coordinate modulus (P) 90 * and one or more of the following have been configured: 91 * - A nist curve (MBEDTLS_ECP_DP_SECPXXXR1_ENABLED) & MBEDTLS_ECP_NIST_OPTIM. 92 * - A Kobliz Curve. 93 * - A Fast Reduction Curve CURVE25519 or CURVE448. */ 94 MBEDTLS_MPI_MOD_REP_OPT_RED, 95 } mbedtls_mpi_mod_rep_selector; 96 97 /* Make mbedtls_mpi_mod_rep_selector and mbedtls_mpi_mod_ext_rep disjoint to 98 * make it easier to catch when they are accidentally swapped. */ 99 typedef enum { 100 MBEDTLS_MPI_MOD_EXT_REP_INVALID = 0, 101 MBEDTLS_MPI_MOD_EXT_REP_LE = 8, 102 MBEDTLS_MPI_MOD_EXT_REP_BE 103 } mbedtls_mpi_mod_ext_rep; 104 105 typedef struct { 106 mbedtls_mpi_uint *p; 107 size_t limbs; 108 } mbedtls_mpi_mod_residue; 109 110 typedef struct { 111 mbedtls_mpi_uint const *rr; /* The residue for 2^{2*n*biL} mod N */ 112 mbedtls_mpi_uint mm; /* Montgomery const for -N^{-1} mod 2^{ciL} */ 113 } mbedtls_mpi_mont_struct; 114 115 typedef int (*mbedtls_mpi_modp_fn)(mbedtls_mpi_uint *X, size_t X_limbs); 116 117 typedef struct { 118 mbedtls_mpi_modp_fn modp; /* The optimised reduction function pointer */ 119 } mbedtls_mpi_opt_red_struct; 120 121 typedef struct { 122 const mbedtls_mpi_uint *p; 123 size_t limbs; // number of limbs 124 size_t bits; // bitlen of p 125 mbedtls_mpi_mod_rep_selector int_rep; // selector to signal the active member of the union 126 union rep { 127 /* if int_rep == #MBEDTLS_MPI_MOD_REP_MONTGOMERY */ 128 mbedtls_mpi_mont_struct mont; 129 /* if int_rep == #MBEDTLS_MPI_MOD_REP_OPT_RED */ 130 mbedtls_mpi_opt_red_struct ored; 131 } rep; 132 } mbedtls_mpi_mod_modulus; 133 134 /** Setup a residue structure. 135 * 136 * The residue will be set up with the buffer \p p and modulus \p N. 137 * 138 * The memory pointed to by \p p will be used by the resulting residue structure. 139 * The value at the pointed-to memory will be the initial value of \p r and must 140 * hold a value that is less than the modulus. This value will be used as-is 141 * and interpreted according to the value of the `N->int_rep` field. 142 * 143 * The modulus \p N will be the modulus associated with \p r. The residue \p r 144 * should only be used in operations where the modulus is \p N. 145 * 146 * \param[out] r The address of the residue to setup. 147 * \param[in] N The address of the modulus related to \p r. 148 * \param[in] p The address of the limb array containing the value of \p r. 149 * The memory pointed to by \p p will be used by \p r and must 150 * not be modified in any way until after 151 * mbedtls_mpi_mod_residue_release() is called. The data 152 * pointed to by \p p must be less than the modulus (the value 153 * pointed to by `N->p`) and already in the representation 154 * indicated by `N->int_rep`. 155 * \param p_limbs The number of limbs of \p p. Must be the same as the number 156 * of limbs in the modulus \p N. 157 * 158 * \return \c 0 if successful. 159 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p p_limbs is less than the 160 * limbs in \p N or if \p p is not less than \p N. 161 */ 162 int mbedtls_mpi_mod_residue_setup(mbedtls_mpi_mod_residue *r, 163 const mbedtls_mpi_mod_modulus *N, 164 mbedtls_mpi_uint *p, 165 size_t p_limbs); 166 167 /** Unbind elements of a residue structure. 168 * 169 * This function removes the reference to the limb array that was passed to 170 * mbedtls_mpi_mod_residue_setup() to make it safe to free or use again. 171 * 172 * This function invalidates \p r and it must not be used until after 173 * mbedtls_mpi_mod_residue_setup() is called on it again. 174 * 175 * \param[out] r The address of residue to release. 176 */ 177 void mbedtls_mpi_mod_residue_release(mbedtls_mpi_mod_residue *r); 178 179 /** Initialize a modulus structure. 180 * 181 * \param[out] N The address of the modulus structure to initialize. 182 */ 183 void mbedtls_mpi_mod_modulus_init(mbedtls_mpi_mod_modulus *N); 184 185 /** Setup a modulus structure. 186 * 187 * \param[out] N The address of the modulus structure to populate. 188 * \param[in] p The address of the limb array storing the value of \p N. 189 * The memory pointed to by \p p will be used by \p N and must 190 * not be modified in any way until after 191 * mbedtls_mpi_mod_modulus_free() is called. 192 * \param p_limbs The number of limbs of \p p. 193 * 194 * \return \c 0 if successful. 195 */ 196 int mbedtls_mpi_mod_modulus_setup(mbedtls_mpi_mod_modulus *N, 197 const mbedtls_mpi_uint *p, 198 size_t p_limbs); 199 200 /** Setup an optimised-reduction compatible modulus structure. 201 * 202 * \param[out] N The address of the modulus structure to populate. 203 * \param[in] p The address of the limb array storing the value of \p N. 204 * The memory pointed to by \p p will be used by \p N and must 205 * not be modified in any way until after 206 * mbedtls_mpi_mod_modulus_free() is called. 207 * \param p_limbs The number of limbs of \p p. 208 * \param modp A pointer to the optimised reduction function to use. \p p. 209 * 210 * \return \c 0 if successful. 211 */ 212 int mbedtls_mpi_mod_optred_modulus_setup(mbedtls_mpi_mod_modulus *N, 213 const mbedtls_mpi_uint *p, 214 size_t p_limbs, 215 mbedtls_mpi_modp_fn modp); 216 217 /** Free elements of a modulus structure. 218 * 219 * This function frees any memory allocated by mbedtls_mpi_mod_modulus_setup(). 220 * 221 * \warning This function does not free the limb array passed to 222 * mbedtls_mpi_mod_modulus_setup() only removes the reference to it, 223 * making it safe to free or to use it again. 224 * 225 * \param[in,out] N The address of the modulus structure to free. 226 */ 227 void mbedtls_mpi_mod_modulus_free(mbedtls_mpi_mod_modulus *N); 228 229 /** \brief Multiply two residues, returning the residue modulo the specified 230 * modulus. 231 * 232 * \note Currently handles the case when `N->int_rep` is 233 * MBEDTLS_MPI_MOD_REP_MONTGOMERY. 234 * 235 * The size of the operation is determined by \p N. \p A, \p B and \p X must 236 * all be associated with the modulus \p N and must all have the same number 237 * of limbs as \p N. 238 * 239 * \p X may be aliased to \p A or \p B, or even both, but may not overlap 240 * either otherwise. They may not alias \p N (since they must be in canonical 241 * form, they cannot == \p N). 242 * 243 * \param[out] X The address of the result MPI. Must have the same 244 * number of limbs as \p N. 245 * On successful completion, \p X contains the result of 246 * the multiplication `A * B * R^-1` mod N where 247 * `R = 2^(biL * N->limbs)`. 248 * \param[in] A The address of the first MPI. 249 * \param[in] B The address of the second MPI. 250 * \param[in] N The address of the modulus. Used to perform a modulo 251 * operation on the result of the multiplication. 252 * 253 * \return \c 0 if successful. 254 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if all the parameters do not 255 * have the same number of limbs or \p N is invalid. 256 * \return #MBEDTLS_ERR_MPI_ALLOC_FAILED on memory-allocation failure. 257 */ 258 int mbedtls_mpi_mod_mul(mbedtls_mpi_mod_residue *X, 259 const mbedtls_mpi_mod_residue *A, 260 const mbedtls_mpi_mod_residue *B, 261 const mbedtls_mpi_mod_modulus *N); 262 263 /** 264 * \brief Perform a fixed-size modular subtraction. 265 * 266 * Calculate `A - B modulo N`. 267 * 268 * \p A, \p B and \p X must all have the same number of limbs as \p N. 269 * 270 * \p X may be aliased to \p A or \p B, or even both, but may not overlap 271 * either otherwise. 272 * 273 * \note This function does not check that \p A or \p B are in canonical 274 * form (that is, are < \p N) - that will have been done by 275 * mbedtls_mpi_mod_residue_setup(). 276 * 277 * \param[out] X The address of the result MPI. Must be initialized. 278 * Must have the same number of limbs as the modulus \p N. 279 * \param[in] A The address of the first MPI. 280 * \param[in] B The address of the second MPI. 281 * \param[in] N The address of the modulus. Used to perform a modulo 282 * operation on the result of the subtraction. 283 * 284 * \return \c 0 if successful. 285 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if the given MPIs do not 286 * have the correct number of limbs. 287 */ 288 int mbedtls_mpi_mod_sub(mbedtls_mpi_mod_residue *X, 289 const mbedtls_mpi_mod_residue *A, 290 const mbedtls_mpi_mod_residue *B, 291 const mbedtls_mpi_mod_modulus *N); 292 293 /** 294 * \brief Perform modular inversion of an MPI with respect to a modulus \p N. 295 * 296 * \p A and \p X must be associated with the modulus \p N and will therefore 297 * have the same number of limbs as \p N. 298 * 299 * \p X may be aliased to \p A. 300 * 301 * \warning Currently only supports prime moduli, but does not check for them. 302 * 303 * \param[out] X The modular inverse of \p A with respect to \p N. 304 * \param[in] A The number to calculate the modular inverse of. 305 * Must not be 0. 306 * \param[in] N The modulus to use. 307 * 308 * \return \c 0 if successful. 309 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p A and \p N do not 310 * have the same number of limbs. 311 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p A is zero. 312 * \return #MBEDTLS_ERR_MPI_ALLOC_FAILED if couldn't allocate enough 313 * memory (needed for conversion to and from Mongtomery form 314 * when not in Montgomery form already, and for temporary use 315 * by the inversion calculation itself). 316 */ 317 318 int mbedtls_mpi_mod_inv(mbedtls_mpi_mod_residue *X, 319 const mbedtls_mpi_mod_residue *A, 320 const mbedtls_mpi_mod_modulus *N); 321 /** 322 * \brief Perform a fixed-size modular addition. 323 * 324 * Calculate `A + B modulo N`. 325 * 326 * \p A, \p B and \p X must all be associated with the modulus \p N and must 327 * all have the same number of limbs as \p N. 328 * 329 * \p X may be aliased to \p A or \p B, or even both, but may not overlap 330 * either otherwise. 331 * 332 * \note This function does not check that \p A or \p B are in canonical 333 * form (that is, are < \p N) - that will have been done by 334 * mbedtls_mpi_mod_residue_setup(). 335 * 336 * \param[out] X The address of the result residue. Must be initialized. 337 * Must have the same number of limbs as the modulus \p N. 338 * \param[in] A The address of the first input residue. 339 * \param[in] B The address of the second input residue. 340 * \param[in] N The address of the modulus. Used to perform a modulo 341 * operation on the result of the addition. 342 * 343 * \return \c 0 if successful. 344 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if the given MPIs do not 345 * have the correct number of limbs. 346 */ 347 int mbedtls_mpi_mod_add(mbedtls_mpi_mod_residue *X, 348 const mbedtls_mpi_mod_residue *A, 349 const mbedtls_mpi_mod_residue *B, 350 const mbedtls_mpi_mod_modulus *N); 351 352 /** Generate a random number uniformly in a range. 353 * 354 * This function generates a random number between \p min inclusive and 355 * \p N exclusive. 356 * 357 * The procedure complies with RFC 6979 §3.3 (deterministic ECDSA) 358 * when the RNG is a suitably parametrized instance of HMAC_DRBG 359 * and \p min is \c 1. 360 * 361 * \note There are `N - min` possible outputs. The lower bound 362 * \p min can be reached, but the upper bound \p N cannot. 363 * 364 * \param X The destination residue. 365 * \param min The minimum value to return. It must be strictly smaller 366 * than \b N. 367 * \param N The modulus. 368 * This is the upper bound of the output range, exclusive. 369 * \param f_rng The RNG function to use. This must not be \c NULL. 370 * \param p_rng The RNG parameter to be passed to \p f_rng. 371 * 372 * \return \c 0 if successful. 373 * \return #MBEDTLS_ERR_MPI_NOT_ACCEPTABLE if the implementation was 374 * unable to find a suitable value within a limited number 375 * of attempts. This has a negligible probability if \p N 376 * is significantly larger than \p min, which is the case 377 * for all usual cryptographic applications. 378 */ 379 int mbedtls_mpi_mod_random(mbedtls_mpi_mod_residue *X, 380 mbedtls_mpi_uint min, 381 const mbedtls_mpi_mod_modulus *N, 382 int (*f_rng)(void *, unsigned char *, size_t), 383 void *p_rng); 384 385 /** Read a residue from a byte buffer. 386 * 387 * The residue will be automatically converted to the internal representation 388 * based on the value of the `N->int_rep` field. 389 * 390 * The modulus \p N will be the modulus associated with \p r. The residue \p r 391 * should only be used in operations where the modulus is \p N or a modulus 392 * equivalent to \p N (in the sense that all their fields or memory pointed by 393 * their fields hold the same value). 394 * 395 * \param[out] r The address of the residue. It must have exactly the same 396 * number of limbs as the modulus \p N. 397 * \param[in] N The address of the modulus. 398 * \param[in] buf The input buffer to import from. 399 * \param buflen The length in bytes of \p buf. 400 * \param ext_rep The endianness of the number in the input buffer. 401 * 402 * \return \c 0 if successful. 403 * \return #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p r isn't 404 * large enough to hold the value in \p buf. 405 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p ext_rep 406 * is invalid or the value in the buffer is not less than \p N. 407 */ 408 int mbedtls_mpi_mod_read(mbedtls_mpi_mod_residue *r, 409 const mbedtls_mpi_mod_modulus *N, 410 const unsigned char *buf, 411 size_t buflen, 412 mbedtls_mpi_mod_ext_rep ext_rep); 413 414 /** Write a residue into a byte buffer. 415 * 416 * The modulus \p N must be the modulus associated with \p r (see 417 * mbedtls_mpi_mod_residue_setup() and mbedtls_mpi_mod_read()). 418 * 419 * The residue will be automatically converted from the internal representation 420 * based on the value of `N->int_rep` field. 421 * 422 * \warning If the buffer is smaller than `N->bits`, the number of 423 * leading zeroes is leaked through timing. If \p r is 424 * secret, the caller must ensure that \p buflen is at least 425 * (`N->bits`+7)/8. 426 * 427 * \param[in] r The address of the residue. It must have the same number of 428 * limbs as the modulus \p N. (\p r is an input parameter, but 429 * its value will be modified during execution and restored 430 * before the function returns.) 431 * \param[in] N The address of the modulus associated with \p r. 432 * \param[out] buf The output buffer to export to. 433 * \param buflen The length in bytes of \p buf. 434 * \param ext_rep The endianness in which the number should be written into 435 * the output buffer. 436 * 437 * \return \c 0 if successful. 438 * \return #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p buf isn't 439 * large enough to hold the value of \p r (without leading 440 * zeroes). 441 * \return #MBEDTLS_ERR_MPI_BAD_INPUT_DATA if \p ext_rep is invalid. 442 * \return #MBEDTLS_ERR_MPI_ALLOC_FAILED if couldn't allocate enough 443 * memory for conversion. Can occur only for moduli with 444 * MBEDTLS_MPI_MOD_REP_MONTGOMERY. 445 */ 446 int mbedtls_mpi_mod_write(const mbedtls_mpi_mod_residue *r, 447 const mbedtls_mpi_mod_modulus *N, 448 unsigned char *buf, 449 size_t buflen, 450 mbedtls_mpi_mod_ext_rep ext_rep); 451 452 #endif /* MBEDTLS_BIGNUM_MOD_H */ 453
