/** * Core bignum functions * * This interface should only be used by the legacy bignum module (bignum.h) * and the modular bignum modules (bignum_mod.c, bignum_mod_raw.c). All other * modules should use the high-level modular bignum interface (bignum_mod.h) * or the legacy bignum interface (bignum.h). * * This module is about processing non-negative integers with a fixed upper * bound that's of the form 2^n-1 where n is a multiple of #biL. * These can be thought of integers written in base 2^#biL with a fixed * number of digits. Digits in this base are called *limbs*. * Many operations treat these numbers as the principal representation of * a number modulo 2^n or a smaller bound. * * The functions in this module obey the following conventions unless * explicitly indicated otherwise: * * - **Overflow**: some functions indicate overflow from the range * [0, 2^n-1] by returning carry parameters, while others operate * modulo and so cannot overflow. This should be clear from the function * documentation. * - **Bignum parameters**: Bignums are passed as pointers to an array of * limbs. A limb has the type #mbedtls_mpi_uint. Unless otherwise specified: * - Bignum parameters called \p A, \p B, ... are inputs, and are * not modified by the function. * - For operations modulo some number, the modulus is called \p N * and is input-only. * - Bignum parameters called \p X, \p Y are outputs or input-output. * The initial content of output-only parameters is ignored. * - Some functions use different names that reflect traditional * naming of operands of certain operations (e.g. * divisor/dividend/quotient/remainder). * - \p T is a temporary storage area. The initial content of such * parameter is ignored and the final content is unspecified. * - **Bignum sizes**: bignum sizes are always expressed in limbs. * Most functions work on bignums of a given size and take a single * \p limbs parameter that applies to all parameters that are limb arrays. * All bignum sizes must be at least 1 and must be significantly less than * #SIZE_MAX. The behavior if a size is 0 is undefined. The behavior if the * total size of all parameters overflows #SIZE_MAX is undefined. * - **Parameter ordering**: for bignum parameters, outputs come before inputs. * Temporaries come last. * - **Aliasing**: in general, output bignums may be aliased to one or more * inputs. As an exception, parameters that are documented as a modulus value * may not be aliased to an output. Outputs may not be aliased to one another. * Temporaries may not be aliased to any other parameter. * - **Overlap**: apart from aliasing of limb array pointers (where two * arguments are equal pointers), overlap is not supported and may result * in undefined behavior. * - **Error handling**: This is a low-level module. Functions generally do not * try to protect against invalid arguments such as nonsensical sizes or * null pointers. Note that some functions that operate on bignums of * different sizes have constraints about their size, and violating those * constraints may lead to buffer overflows. * - **Modular representatives**: functions that operate modulo \p N expect * all modular inputs to be in the range [0, \p N - 1] and guarantee outputs * in the range [0, \p N - 1]. If an input is out of range, outputs are * fully unspecified, though bignum values out of range should not cause * buffer overflows (beware that this is not extensively tested). */ /* * 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. */ #ifndef MBEDTLS_BIGNUM_CORE_H #define MBEDTLS_BIGNUM_CORE_H #include "common.h" #if defined(MBEDTLS_BIGNUM_C) #include "mbedtls/bignum.h" #endif #define ciL ( sizeof(mbedtls_mpi_uint) ) /** chars in limb */ #define biL ( ciL << 3 ) /** bits in limb */ #define biH ( ciL << 2 ) /** half limb size */ /* * Convert between bits/chars and number of limbs * Divide first in order to avoid potential overflows */ #define BITS_TO_LIMBS(i) ( (i) / biL + ( (i) % biL != 0 ) ) #define CHARS_TO_LIMBS(i) ( (i) / ciL + ( (i) % ciL != 0 ) ) /* Get a specific byte, without range checks. */ #define GET_BYTE( X, i ) \ ( ( (X)[(i) / ciL] >> ( ( (i) % ciL ) * 8 ) ) & 0xff ) /** Count leading zero bits in a given integer. * * \param a Integer to count leading zero bits. * * \return The number of leading zero bits in \p a. */ size_t mbedtls_mpi_core_clz( mbedtls_mpi_uint a ); /** Return the minimum number of bits required to represent the value held * in the MPI. * * \note This function returns 0 if all the limbs of \p A are 0. * * \param[in] A The address of the MPI. * \param A_limbs The number of limbs of \p A. * * \return The number of bits in \p A. */ size_t mbedtls_mpi_core_bitlen( const mbedtls_mpi_uint *A, size_t A_limbs ); /** Convert a big-endian byte array aligned to the size of mbedtls_mpi_uint * into the storage form used by mbedtls_mpi. * * \param[in,out] A The address of the MPI. * \param A_limbs The number of limbs of \p A. */ void mbedtls_mpi_core_bigendian_to_host( mbedtls_mpi_uint *A, size_t A_limbs ); /** * \brief Perform a safe conditional copy of an MPI which doesn't reveal * whether assignment was done or not. * * \param[out] X The address of the destination MPI. * This must be initialized. Must have enough limbs to * store the full value of \p A. * \param[in] A The address of the source MPI. This must be initialized. * \param limbs The number of limbs of \p A. * \param assign The condition deciding whether to perform the * assignment or not. Must be either 0 or 1: * * \c 1: Perform the assignment `X = A`. * * \c 0: Keep the original value of \p X. * * \note This function avoids leaking any information about whether * the assignment was done or not. * * \warning If \p assign is neither 0 nor 1, the result of this function * is indeterminate, and the resulting value in \p X might be * neither its original value nor the value in \p A. */ void mbedtls_mpi_core_cond_assign( mbedtls_mpi_uint *X, const mbedtls_mpi_uint *A, size_t limbs, unsigned char assign ); /** * \brief Perform a safe conditional swap of two MPIs which doesn't reveal * whether the swap was done or not. * * \param[in,out] X The address of the first MPI. * This must be initialized. * \param[in,out] Y The address of the second MPI. * This must be initialized. * \param limbs The number of limbs of \p X and \p Y. * \param swap The condition deciding whether to perform * the swap or not. Must be either 0 or 1: * * \c 1: Swap the values of \p X and \p Y. * * \c 0: Keep the original values of \p X and \p Y. * * \note This function avoids leaking any information about whether * the swap was done or not. * * \warning If \p swap is neither 0 nor 1, the result of this function * is indeterminate, and both \p X and \p Y might end up with * values different to either of the original ones. */ void mbedtls_mpi_core_cond_swap( mbedtls_mpi_uint *X, mbedtls_mpi_uint *Y, size_t limbs, unsigned char swap ); /** Import X from unsigned binary data, little-endian. * * The MPI needs to have enough limbs to store the full value (including any * most significant zero bytes in the input). * * \param[out] X The address of the MPI. * \param X_limbs The number of limbs of \p X. * \param[in] input The input buffer to import from. * \param input_length The length bytes of \p input. * * \return \c 0 if successful. * \return #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p X isn't * large enough to hold the value in \p input. */ int mbedtls_mpi_core_read_le( mbedtls_mpi_uint *X, size_t X_limbs, const unsigned char *input, size_t input_length ); /** Import X from unsigned binary data, big-endian. * * The MPI needs to have enough limbs to store the full value (including any * most significant zero bytes in the input). * * \param[out] X The address of the MPI. * May only be #NULL if \X_limbs is 0 and \p input_length * is 0. * \param X_limbs The number of limbs of \p X. * \param[in] input The input buffer to import from. * May only be #NULL if \p input_length is 0. * \param input_length The length in bytes of \p input. * * \return \c 0 if successful. * \return #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p X isn't * large enough to hold the value in \p input. */ int mbedtls_mpi_core_read_be( mbedtls_mpi_uint *X, size_t X_limbs, const unsigned char *input, size_t input_length ); /** Export A into unsigned binary data, little-endian. * * \note If \p output is shorter than \p A the export is still successful if the * value held in \p A fits in the buffer (that is, if enough of the most * significant bytes of \p A are 0). * * \param[in] A The address of the MPI. * \param A_limbs The number of limbs of \p A. * \param[out] output The output buffer to export to. * \param output_length The length in bytes of \p output. * * \return \c 0 if successful. * \return #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p output isn't * large enough to hold the value of \p A. */ int mbedtls_mpi_core_write_le( const mbedtls_mpi_uint *A, size_t A_limbs, unsigned char *output, size_t output_length ); /** Export A into unsigned binary data, big-endian. * * \note If \p output is shorter than \p A the export is still successful if the * value held in \p A fits in the buffer (that is, if enough of the most * significant bytes of \p A are 0). * * \param[in] A The address of the MPI. * \param A_limbs The number of limbs of \p A. * \param[out] output The output buffer to export to. * \param output_length The length in bytes of \p output. * * \return \c 0 if successful. * \return #MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL if \p output isn't * large enough to hold the value of \p A. */ int mbedtls_mpi_core_write_be( const mbedtls_mpi_uint *A, size_t A_limbs, unsigned char *output, size_t output_length ); /** \brief Shift an MPI right in place by a number of bits. * * Shifting by more bits than there are bit positions * in \p X is valid and results in setting \p X to 0. * * This function's execution time depends on the value * of \p count (and of course \p limbs). * * \param[in,out] X The number to shift. * \param limbs The number of limbs of \p X. This must be at least 1. * \param count The number of bits to shift by. */ void mbedtls_mpi_core_shift_r( mbedtls_mpi_uint *X, size_t limbs, size_t count ); /** * \brief Conditional addition of two fixed-size large unsigned integers, * returning the carry. * * Functionally equivalent to * * ``` * if( cond ) * X += A; * return carry; * ``` * * This function operates modulo `2^(biL*limbs)`. * * \param[in,out] X The pointer to the (little-endian) array * representing the bignum to accumulate onto. * \param[in] A The pointer to the (little-endian) array * representing the bignum to conditionally add * to \p X. This may be aliased to \p X but may not * overlap otherwise. * \param limbs Number of limbs of \p X and \p A. * \param cond Condition bit dictating whether addition should * happen or not. This must be \c 0 or \c 1. * * \warning If \p cond is neither 0 nor 1, the result of this function * is unspecified, and the resulting value in \p X might be * neither its original value nor \p X + \p A. * * \return 1 if `X + cond * A >= 2^(biL*limbs)`, 0 otherwise. */ mbedtls_mpi_uint mbedtls_mpi_core_add_if( mbedtls_mpi_uint *X, const mbedtls_mpi_uint *A, size_t limbs, unsigned cond ); /** * \brief Subtract two fixed-size large unsigned integers, returning the borrow. * * Calculate `A - B` where \p A and \p B have the same size. * This function operates modulo `2^(biL*limbs)` and returns the carry * (1 if there was a wraparound, i.e. if `A < B`, and 0 otherwise). * * \p X may be aliased to \p A or \p B, or even both, but may not overlap * either otherwise. * * \param[out] X The result of the subtraction. * \param[in] A Little-endian presentation of left operand. * \param[in] B Little-endian presentation of right operand. * \param limbs Number of limbs of \p X, \p A and \p B. * * \return 1 if `A < B`. * 0 if `A >= B`. */ mbedtls_mpi_uint mbedtls_mpi_core_sub( mbedtls_mpi_uint *X, const mbedtls_mpi_uint *A, const mbedtls_mpi_uint *B, size_t limbs ); /** * \brief Perform a fixed-size multiply accumulate operation: X += b * A * * \p X may be aliased to \p A (when \p X_limbs == \p A_limbs), but may not * otherwise overlap. * * This function operates modulo `2^(biL*X_limbs)`. * * \param[in,out] X The pointer to the (little-endian) array * representing the bignum to accumulate onto. * \param X_limbs The number of limbs of \p X. This must be * at least \p A_limbs. * \param[in] A The pointer to the (little-endian) array * representing the bignum to multiply with. * This may be aliased to \p X but may not overlap * otherwise. * \param A_limbs The number of limbs of \p A. * \param b X scalar to multiply with. * * \return The carry at the end of the operation. */ mbedtls_mpi_uint mbedtls_mpi_core_mla( mbedtls_mpi_uint *X, size_t X_limbs, const mbedtls_mpi_uint *A, size_t A_limbs, mbedtls_mpi_uint b ); /** * \brief Calculate initialisation value for fast Montgomery modular * multiplication * * \param[in] N Little-endian presentation of the modulus. This must have * at least one limb. * * \return The initialisation value for fast Montgomery modular multiplication */ mbedtls_mpi_uint mbedtls_mpi_core_montmul_init( const mbedtls_mpi_uint *N ); /** * \brief Montgomery multiplication: X = A * B * R^-1 mod N (HAC 14.36) * * \p A and \p B must be in canonical form. That is, < \p N. * * \p X may be aliased to \p A or \p N, or even \p B (if \p AN_limbs == * \p B_limbs) but may not overlap any parameters otherwise. * * \p A and \p B may alias each other, if \p AN_limbs == \p B_limbs. They may * not alias \p N (since they must be in canonical form, they cannot == \p N). * * \param[out] X The destination MPI, as a little-endian array of * length \p AN_limbs. * On successful completion, X contains the result of * the multiplication `A * B * R^-1` mod N where * `R = 2^(biL*AN_limbs)`. * \param[in] A Little-endian presentation of first operand. * Must have the same number of limbs as \p N. * \param[in] B Little-endian presentation of second operand. * \param[in] B_limbs The number of limbs in \p B. * Must be <= \p AN_limbs. * \param[in] N Little-endian presentation of the modulus. * This must be odd, and have exactly the same number * of limbs as \p A. * It may alias \p X, but must not alias or otherwise * overlap any of the other parameters. * \param[in] AN_limbs The number of limbs in \p X, \p A and \p N. * \param mm The Montgomery constant for \p N: -N^-1 mod 2^biL. * This can be calculated by `mbedtls_mpi_core_montmul_init()`. * \param[in,out] T Temporary storage of size at least 2*AN_limbs+1 limbs. * Its initial content is unused and * its final content is indeterminate. * It must not alias or otherwise overlap any of the * other parameters. */ void mbedtls_mpi_core_montmul( mbedtls_mpi_uint *X, const mbedtls_mpi_uint *A, const mbedtls_mpi_uint *B, size_t B_limbs, const mbedtls_mpi_uint *N, size_t AN_limbs, mbedtls_mpi_uint mm, mbedtls_mpi_uint *T ); #endif /* MBEDTLS_BIGNUM_CORE_H */