library/bignum_core.h

/**
 *  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^#biL-1. Many operations treat these numbers
 * as the principal representation of a number modulo 2^#biL 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^#biL-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. 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 );

/** 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 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 */