2180 lines
46 KiB
C
2180 lines
46 KiB
C
/*
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* Multi-precision integer library
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*
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* Copyright (C) 2006-2015, ARM Limited, All Rights Reserved
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* SPDX-License-Identifier: Apache-2.0
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*
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* Licensed under the Apache License, Version 2.0 (the "License"); you may
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* not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
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* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*
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*/
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/*
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* The following sources were referenced in the design of this Multi-precision
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* Integer library:
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*
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* [1] Handbook of Applied Cryptography - 1997
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* Menezes, van Oorschot and Vanstone
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*
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* [2] Multi-Precision Math
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* Tom St Denis
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* https://github.com/libtom/libtommath/blob/develop/tommath.pdf
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*
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* [3] GNU Multi-Precision Arithmetic Library
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* https://gmplib.org/manual/index.html
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*
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*/
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#include <string.h>
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#include <stdlib.h>
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#include "bignum.h"
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#include "esp_crypto.h"
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/* Implementation that should never be optimized out by the compiler */
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//static void bzero( void *v, size_t n ) {
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// volatile unsigned char *p = v; while( n-- ) *p++ = 0;
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//}
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#define ciL (sizeof(esp_mpi_uint)) /* chars in limb */
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#define biL (ciL << 3) /* bits in limb */
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#define biH (ciL << 2) /* half limb size */
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#define MPI_SIZE_T_MAX ( (size_t) -1 ) /* SIZE_T_MAX is not standard */
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/*
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* Convert between bits/chars and number of limbs
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* Divide first in order to avoid potential overflows
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*/
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#define BITS_TO_LIMBS(i) ( (i) / biL + ( (i) % biL != 0 ) )
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#define CHARS_TO_LIMBS(i) ( (i) / ciL + ( (i) % ciL != 0 ) )
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/*
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* Initialize one MPI
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*/
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void esp_mpi_init( mpi *X )
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{
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if( X == NULL )
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return;
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X->s = 1;
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X->n = 0;
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X->p = NULL;
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BIGNUM_LOCK();
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BIGNUM_TAKE();
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ets_bigint_enable();
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BIGNUM_UNLOCK();
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}
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/*
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* Unallocate one MPI
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*/
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void esp_mpi_free( mpi *X )
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{
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if( X == NULL )
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return;
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if( X->p != NULL )
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{
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bzero( X->p, X->n * ciL );
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free( X->p );
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}
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X->s = 1;
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X->n = 0;
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X->p = NULL;
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BIGNUM_LOCK();
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BIGNUM_GIVE();
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if (false == BIGNUM_IS_USED())
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ets_bigint_disable();
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BIGNUM_UNLOCK();
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}
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/*
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* Enlarge to the specified number of limbs
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*/
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int esp_mpi_grow( mpi *X, size_t nblimbs )
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{
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esp_mpi_uint *p;
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if( nblimbs > MPI_MAX_LIMBS )
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return( ERR_MPI_ALLOC_FAILED );
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if( X->n < nblimbs )
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{
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if( ( p = calloc( nblimbs, ciL ) ) == NULL )
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return( ERR_MPI_ALLOC_FAILED );
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if( X->p != NULL )
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{
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memcpy( p, X->p, X->n * ciL );
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bzero( X->p, X->n * ciL );
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free( X->p );
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}
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X->n = nblimbs;
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X->p = p;
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}
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return( 0 );
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}
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/*
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* Resize down as much as possible,
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* while keeping at least the specified number of limbs
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*/
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int esp_mpi_shrink( mpi *X, size_t nblimbs )
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{
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esp_mpi_uint *p;
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size_t i;
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/* Actually resize up in this case */
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if( X->n <= nblimbs )
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return( esp_mpi_grow( X, nblimbs ) );
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for( i = X->n - 1; i > 0; i-- )
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if( X->p[i] != 0 )
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break;
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i++;
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if( i < nblimbs )
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i = nblimbs;
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if( ( p = calloc( i, ciL ) ) == NULL )
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return( ERR_MPI_ALLOC_FAILED );
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if( X->p != NULL )
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{
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memcpy( p, X->p, i * ciL );
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bzero( X->p, X->n * ciL );
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free( X->p );
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}
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X->n = i;
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X->p = p;
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return( 0 );
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}
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/*
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* Copy the contents of Y into X
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*/
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int esp_mpi_copy( mpi *X, const mpi *Y )
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{
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int ret;
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size_t i;
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if( X == Y )
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return( 0 );
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if( Y->p == NULL )
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{
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esp_mpi_free( X );
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return( 0 );
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}
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for( i = Y->n - 1; i > 0; i-- )
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if( Y->p[i] != 0 )
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break;
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i++;
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X->s = Y->s;
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MPI_CHK( esp_mpi_grow( X, i ) );
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memset( X->p, 0, X->n * ciL );
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memcpy( X->p, Y->p, i * ciL );
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cleanup:
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return( ret );
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}
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/*
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* Swap the contents of X and Y
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*/
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void esp_mpi_swap( mpi *X, mpi *Y )
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{
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mpi T;
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memcpy( &T, X, sizeof( mpi ) );
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memcpy( X, Y, sizeof( mpi ) );
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memcpy( Y, &T, sizeof( mpi ) );
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}
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/*
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* Conditionally assign X = Y, without leaking information
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* about whether the assignment was made or not.
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* (Leaking information about the respective sizes of X and Y is ok however.)
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*/
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int esp_mpi_safe_cond_assign( mpi *X, const mpi *Y, unsigned char assign )
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{
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int ret = 0;
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size_t i;
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/* make sure assign is 0 or 1 in a time-constant manner */
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assign = (assign | (unsigned char)-assign) >> 7;
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MPI_CHK( esp_mpi_grow( X, Y->n ) );
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X->s = X->s * ( 1 - assign ) + Y->s * assign;
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for( i = 0; i < Y->n; i++ )
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X->p[i] = X->p[i] * ( 1 - assign ) + Y->p[i] * assign;
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for( ; i < X->n; i++ )
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X->p[i] *= ( 1 - assign );
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cleanup:
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return( ret );
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}
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/*
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* Conditionally swap X and Y, without leaking information
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* about whether the swap was made or not.
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* Here it is not ok to simply swap the pointers, which whould lead to
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* different memory access patterns when X and Y are used afterwards.
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*/
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int esp_mpi_safe_cond_swap( mpi *X, mpi *Y, unsigned char swap )
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{
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int ret, s;
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size_t i;
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esp_mpi_uint tmp;
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if( X == Y )
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return( 0 );
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/* make sure swap is 0 or 1 in a time-constant manner */
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swap = (swap | (unsigned char)-swap) >> 7;
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MPI_CHK( esp_mpi_grow( X, Y->n ) );
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MPI_CHK( esp_mpi_grow( Y, X->n ) );
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s = X->s;
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X->s = X->s * ( 1 - swap ) + Y->s * swap;
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Y->s = Y->s * ( 1 - swap ) + s * swap;
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for( i = 0; i < X->n; i++ )
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{
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tmp = X->p[i];
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X->p[i] = X->p[i] * ( 1 - swap ) + Y->p[i] * swap;
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Y->p[i] = Y->p[i] * ( 1 - swap ) + tmp * swap;
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}
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cleanup:
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return( ret );
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}
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/*
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* Set value from integer
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*/
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int esp_mpi_lset( mpi *X, esp_mpi_sint z )
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{
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int ret;
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MPI_CHK( esp_mpi_grow( X, 1 ) );
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memset( X->p, 0, X->n * ciL );
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X->p[0] = ( z < 0 ) ? -z : z;
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X->s = ( z < 0 ) ? -1 : 1;
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cleanup:
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return( ret );
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}
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/*
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* Get a specific bit
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*/
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int esp_mpi_get_bit( const mpi *X, size_t pos )
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{
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if( X->n * biL <= pos )
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return( 0 );
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return( ( X->p[pos / biL] >> ( pos % biL ) ) & 0x01 );
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}
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/*
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* Set a bit to a specific value of 0 or 1
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*/
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int esp_mpi_set_bit( mpi *X, size_t pos, unsigned char val )
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{
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int ret = 0;
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size_t off = pos / biL;
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size_t idx = pos % biL;
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if( val != 0 && val != 1 )
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return( ERR_MPI_BAD_INPUT_DATA );
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if( X->n * biL <= pos )
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{
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if( val == 0 )
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return( 0 );
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MPI_CHK( esp_mpi_grow( X, off + 1 ) );
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}
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X->p[off] &= ~( (esp_mpi_uint) 0x01 << idx );
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X->p[off] |= (esp_mpi_uint) val << idx;
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cleanup:
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return( ret );
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}
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/*
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* Return the number of less significant zero-bits
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*/
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size_t esp_mpi_lsb( const mpi *X )
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{
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size_t i, j, count = 0;
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for( i = 0; i < X->n; i++ )
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for( j = 0; j < biL; j++, count++ )
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if( ( ( X->p[i] >> j ) & 1 ) != 0 )
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return( count );
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return( 0 );
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}
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/*
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* Count leading zero bits in a given integer
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*/
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static size_t clz( const esp_mpi_uint x )
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{
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size_t j;
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esp_mpi_uint mask = (esp_mpi_uint) 1 << (biL - 1);
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for( j = 0; j < biL; j++ )
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{
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if( x & mask ) break;
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mask >>= 1;
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}
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return j;
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}
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/*
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* Return the number of bits
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*/
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size_t esp_mpi_bitlen( const mpi *X )
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{
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size_t i, j;
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if( X->n == 0 )
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return( 0 );
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for( i = X->n - 1; i > 0; i-- )
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if( X->p[i] != 0 )
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break;
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j = biL - clz( X->p[i] );
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return( ( i * biL ) + j );
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}
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/*
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* Return the total size in bytes
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*/
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size_t esp_mpi_size( const mpi *X )
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{
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return( ( esp_mpi_bitlen( X ) + 7 ) >> 3 );
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}
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/*
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* Convert an ASCII character to digit value
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*/
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static int esp_mpi_get_digit( esp_mpi_uint *d, int radix, char c )
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{
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*d = 255;
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if( c >= 0x30 && c <= 0x39 ) *d = c - 0x30;
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if( c >= 0x41 && c <= 0x46 ) *d = c - 0x37;
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if( c >= 0x61 && c <= 0x66 ) *d = c - 0x57;
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if( *d >= (esp_mpi_uint) radix )
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return( ERR_MPI_INVALID_CHARACTER );
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return( 0 );
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}
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/*
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* Import from an ASCII string
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*/
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int esp_mpi_read_string( mpi *X, int radix, const char *s )
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{
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int ret;
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size_t i, j, slen, n;
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esp_mpi_uint d;
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mpi T;
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if( radix < 2 || radix > 16 )
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return( ERR_MPI_BAD_INPUT_DATA );
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esp_mpi_init( &T );
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slen = strlen( s );
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if( radix == 16 )
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{
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if( slen > MPI_SIZE_T_MAX >> 2 )
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return( ERR_MPI_BAD_INPUT_DATA );
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n = BITS_TO_LIMBS( slen << 2 );
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MPI_CHK( esp_mpi_grow( X, n ) );
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MPI_CHK( esp_mpi_lset( X, 0 ) );
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for( i = slen, j = 0; i > 0; i--, j++ )
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{
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if( i == 1 && s[i - 1] == '-' )
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{
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X->s = -1;
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break;
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}
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MPI_CHK( esp_mpi_get_digit( &d, radix, s[i - 1] ) );
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X->p[j / ( 2 * ciL )] |= d << ( ( j % ( 2 * ciL ) ) << 2 );
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}
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}
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else
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{
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MPI_CHK( esp_mpi_lset( X, 0 ) );
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for( i = 0; i < slen; i++ )
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{
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if( i == 0 && s[i] == '-' )
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{
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X->s = -1;
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continue;
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}
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MPI_CHK( esp_mpi_get_digit( &d, radix, s[i] ) );
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MPI_CHK( esp_mpi_mul_int( &T, X, radix ) );
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if( X->s == 1 )
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{
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MPI_CHK( esp_mpi_add_int( X, &T, d ) );
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}
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else
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{
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MPI_CHK( esp_mpi_sub_int( X, &T, d ) );
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}
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}
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}
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cleanup:
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esp_mpi_free( &T );
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return( ret );
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}
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/*
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* Helper to write the digits high-order first
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*/
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static int esp_mpi_write_hlp( mpi *X, int radix, char **p )
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{
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int ret;
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esp_mpi_uint r;
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if( radix < 2 || radix > 16 )
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return( ERR_MPI_BAD_INPUT_DATA );
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MPI_CHK( esp_mpi_mod_int( &r, X, radix ) );
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MPI_CHK( esp_mpi_div_int( X, NULL, X, radix ) );
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if( esp_mpi_cmp_int( X, 0 ) != 0 )
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MPI_CHK( esp_mpi_write_hlp( X, radix, p ) );
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if( r < 10 )
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*(*p)++ = (char)( r + 0x30 );
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else
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*(*p)++ = (char)( r + 0x37 );
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cleanup:
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return( ret );
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}
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|
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/*
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* Export into an ASCII string
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*/
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int esp_mpi_write_string( const mpi *X, int radix,
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char *buf, size_t buflen, size_t *olen )
|
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{
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int ret = 0;
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size_t n;
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char *p;
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mpi T;
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|
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if( radix < 2 || radix > 16 )
|
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return( ERR_MPI_BAD_INPUT_DATA );
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|
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n = esp_mpi_bitlen( X );
|
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if( radix >= 4 ) n >>= 1;
|
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if( radix >= 16 ) n >>= 1;
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n += 3;
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|
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if( buflen < n )
|
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{
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*olen = n;
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return( ERR_MPI_BUFFER_TOO_SMALL );
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}
|
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|
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p = buf;
|
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esp_mpi_init( &T );
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|
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if( X->s == -1 )
|
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*p++ = '-';
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|
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if( radix == 16 )
|
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{
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int c;
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size_t i, j, k;
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|
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for( i = X->n, k = 0; i > 0; i-- )
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{
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for( j = ciL; j > 0; j-- )
|
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{
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c = ( X->p[i - 1] >> ( ( j - 1 ) << 3) ) & 0xFF;
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if( c == 0 && k == 0 && ( i + j ) != 2 )
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continue;
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|
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*(p++) = "0123456789ABCDEF" [c / 16];
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*(p++) = "0123456789ABCDEF" [c % 16];
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k = 1;
|
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}
|
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}
|
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}
|
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else
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{
|
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MPI_CHK( esp_mpi_copy( &T, X ) );
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|
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if( T.s == -1 )
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T.s = 1;
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|
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MPI_CHK( esp_mpi_write_hlp( &T, radix, &p ) );
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}
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|
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*p++ = '\0';
|
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*olen = p - buf;
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|
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cleanup:
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|
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esp_mpi_free( &T );
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|
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return( ret );
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}
|
|
|
|
/*
|
|
* Import X from unsigned binary data, big endian
|
|
*/
|
|
int esp_mpi_read_binary( mpi *X, const unsigned char *buf, size_t buflen )
|
|
{
|
|
int ret;
|
|
size_t i, j, n;
|
|
|
|
for( n = 0; n < buflen; n++ )
|
|
if( buf[n] != 0 )
|
|
break;
|
|
|
|
MPI_CHK( esp_mpi_grow( X, CHARS_TO_LIMBS( buflen - n ) ) );
|
|
MPI_CHK( esp_mpi_lset( X, 0 ) );
|
|
|
|
for( i = buflen, j = 0; i > n; i--, j++ )
|
|
X->p[j / ciL] |= ((esp_mpi_uint) buf[i - 1]) << ((j % ciL) << 3);
|
|
|
|
cleanup:
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Export X into unsigned binary data, big endian
|
|
*/
|
|
int esp_mpi_write_binary( const mpi *X, unsigned char *buf, size_t buflen )
|
|
{
|
|
size_t i, j, n;
|
|
|
|
n = esp_mpi_size( X );
|
|
|
|
if( buflen < n )
|
|
return( ERR_MPI_BUFFER_TOO_SMALL );
|
|
|
|
memset( buf, 0, buflen );
|
|
|
|
for( i = buflen - 1, j = 0; n > 0; i--, j++, n-- )
|
|
buf[i] = (unsigned char)( X->p[j / ciL] >> ((j % ciL) << 3) );
|
|
|
|
return( 0 );
|
|
}
|
|
|
|
/*
|
|
* Left-shift: X <<= count
|
|
*/
|
|
int esp_mpi_shift_l( mpi *X, size_t count )
|
|
{
|
|
int ret;
|
|
size_t i, v0, t1;
|
|
esp_mpi_uint r0 = 0, r1;
|
|
|
|
v0 = count / (biL );
|
|
t1 = count & (biL - 1);
|
|
|
|
i = esp_mpi_bitlen( X ) + count;
|
|
|
|
if( X->n * biL < i )
|
|
MPI_CHK( esp_mpi_grow( X, BITS_TO_LIMBS( i ) ) );
|
|
|
|
ret = 0;
|
|
|
|
/*
|
|
* shift by count / limb_size
|
|
*/
|
|
if( v0 > 0 )
|
|
{
|
|
for( i = X->n; i > v0; i-- )
|
|
X->p[i - 1] = X->p[i - v0 - 1];
|
|
|
|
for( ; i > 0; i-- )
|
|
X->p[i - 1] = 0;
|
|
}
|
|
|
|
/*
|
|
* shift by count % limb_size
|
|
*/
|
|
if( t1 > 0 )
|
|
{
|
|
for( i = v0; i < X->n; i++ )
|
|
{
|
|
r1 = X->p[i] >> (biL - t1);
|
|
X->p[i] <<= t1;
|
|
X->p[i] |= r0;
|
|
r0 = r1;
|
|
}
|
|
}
|
|
|
|
cleanup:
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Right-shift: X >>= count
|
|
*/
|
|
int esp_mpi_shift_r( mpi *X, size_t count )
|
|
{
|
|
size_t i, v0, v1;
|
|
esp_mpi_uint r0 = 0, r1;
|
|
|
|
v0 = count / biL;
|
|
v1 = count & (biL - 1);
|
|
|
|
if( v0 > X->n || ( v0 == X->n && v1 > 0 ) )
|
|
return esp_mpi_lset( X, 0 );
|
|
|
|
/*
|
|
* shift by count / limb_size
|
|
*/
|
|
if( v0 > 0 )
|
|
{
|
|
for( i = 0; i < X->n - v0; i++ )
|
|
X->p[i] = X->p[i + v0];
|
|
|
|
for( ; i < X->n; i++ )
|
|
X->p[i] = 0;
|
|
}
|
|
|
|
/*
|
|
* shift by count % limb_size
|
|
*/
|
|
if( v1 > 0 )
|
|
{
|
|
for( i = X->n; i > 0; i-- )
|
|
{
|
|
r1 = X->p[i - 1] << (biL - v1);
|
|
X->p[i - 1] >>= v1;
|
|
X->p[i - 1] |= r0;
|
|
r0 = r1;
|
|
}
|
|
}
|
|
|
|
return( 0 );
|
|
}
|
|
|
|
/*
|
|
* Compare unsigned values
|
|
*/
|
|
int esp_mpi_cmp_abs( const mpi *X, const mpi *Y )
|
|
{
|
|
size_t i, j;
|
|
|
|
for( i = X->n; i > 0; i-- )
|
|
if( X->p[i - 1] != 0 )
|
|
break;
|
|
|
|
for( j = Y->n; j > 0; j-- )
|
|
if( Y->p[j - 1] != 0 )
|
|
break;
|
|
|
|
if( i == 0 && j == 0 )
|
|
return( 0 );
|
|
|
|
if( i > j ) return( 1 );
|
|
if( j > i ) return( -1 );
|
|
|
|
for( ; i > 0; i-- )
|
|
{
|
|
if( X->p[i - 1] > Y->p[i - 1] ) return( 1 );
|
|
if( X->p[i - 1] < Y->p[i - 1] ) return( -1 );
|
|
}
|
|
|
|
return( 0 );
|
|
}
|
|
|
|
/*
|
|
* Compare signed values
|
|
*/
|
|
int esp_mpi_cmp_mpi( const mpi *X, const mpi *Y )
|
|
{
|
|
size_t i, j;
|
|
|
|
for( i = X->n; i > 0; i-- )
|
|
if( X->p[i - 1] != 0 )
|
|
break;
|
|
|
|
for( j = Y->n; j > 0; j-- )
|
|
if( Y->p[j - 1] != 0 )
|
|
break;
|
|
|
|
if( i == 0 && j == 0 )
|
|
return( 0 );
|
|
|
|
if( i > j ) return( X->s );
|
|
if( j > i ) return( -Y->s );
|
|
|
|
if( X->s > 0 && Y->s < 0 ) return( 1 );
|
|
if( Y->s > 0 && X->s < 0 ) return( -1 );
|
|
|
|
for( ; i > 0; i-- )
|
|
{
|
|
if( X->p[i - 1] > Y->p[i - 1] ) return( X->s );
|
|
if( X->p[i - 1] < Y->p[i - 1] ) return( -X->s );
|
|
}
|
|
|
|
return( 0 );
|
|
}
|
|
|
|
/*
|
|
* Compare signed values
|
|
*/
|
|
int esp_mpi_cmp_int( const mpi *X, esp_mpi_sint z )
|
|
{
|
|
mpi Y;
|
|
esp_mpi_uint p[1];
|
|
|
|
*p = ( z < 0 ) ? -z : z;
|
|
Y.s = ( z < 0 ) ? -1 : 1;
|
|
Y.n = 1;
|
|
Y.p = p;
|
|
|
|
return( esp_mpi_cmp_mpi( X, &Y ) );
|
|
}
|
|
|
|
/*
|
|
* Unsigned addition: X = |A| + |B| (HAC 14.7)
|
|
*/
|
|
int esp_mpi_add_abs( mpi *X, const mpi *A, const mpi *B )
|
|
{
|
|
int ret;
|
|
size_t i, j;
|
|
esp_mpi_uint *o, *p, c;
|
|
|
|
if( X == B )
|
|
{
|
|
const mpi *T = A; A = X; B = T;
|
|
}
|
|
|
|
if( X != A )
|
|
MPI_CHK( esp_mpi_copy( X, A ) );
|
|
|
|
/*
|
|
* X should always be positive as a result of unsigned additions.
|
|
*/
|
|
X->s = 1;
|
|
|
|
for( j = B->n; j > 0; j-- )
|
|
if( B->p[j - 1] != 0 )
|
|
break;
|
|
|
|
MPI_CHK( esp_mpi_grow( X, j ) );
|
|
|
|
o = B->p; p = X->p; c = 0;
|
|
|
|
for( i = 0; i < j; i++, o++, p++ )
|
|
{
|
|
*p += c; c = ( *p < c );
|
|
*p += *o; c += ( *p < *o );
|
|
}
|
|
|
|
while( c != 0 )
|
|
{
|
|
if( i >= X->n )
|
|
{
|
|
MPI_CHK( esp_mpi_grow( X, i + 1 ) );
|
|
p = X->p + i;
|
|
}
|
|
|
|
*p += c; c = ( *p < c ); i++; p++;
|
|
}
|
|
|
|
cleanup:
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Helper for mpi subtraction
|
|
*/
|
|
static void esp_mpi_sub_hlp( size_t n, esp_mpi_uint *s, esp_mpi_uint *d )
|
|
{
|
|
size_t i;
|
|
esp_mpi_uint c, z;
|
|
|
|
for( i = c = 0; i < n; i++, s++, d++ )
|
|
{
|
|
z = ( *d < c ); *d -= c;
|
|
c = ( *d < *s ) + z; *d -= *s;
|
|
}
|
|
|
|
while( c != 0 )
|
|
{
|
|
z = ( *d < c ); *d -= c;
|
|
c = z; i++; d++;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* Unsigned subtraction: X = |A| - |B| (HAC 14.9)
|
|
*/
|
|
int esp_mpi_sub_abs( mpi *X, const mpi *A, const mpi *B )
|
|
{
|
|
mpi TB;
|
|
int ret;
|
|
size_t n;
|
|
|
|
if( esp_mpi_cmp_abs( A, B ) < 0 )
|
|
return( ERR_MPI_NEGATIVE_VALUE );
|
|
|
|
esp_mpi_init( &TB );
|
|
|
|
if( X == B )
|
|
{
|
|
MPI_CHK( esp_mpi_copy( &TB, B ) );
|
|
B = &TB;
|
|
}
|
|
|
|
if( X != A )
|
|
MPI_CHK( esp_mpi_copy( X, A ) );
|
|
|
|
/*
|
|
* X should always be positive as a result of unsigned subtractions.
|
|
*/
|
|
X->s = 1;
|
|
|
|
ret = 0;
|
|
|
|
for( n = B->n; n > 0; n-- )
|
|
if( B->p[n - 1] != 0 )
|
|
break;
|
|
|
|
esp_mpi_sub_hlp( n, B->p, X->p );
|
|
|
|
cleanup:
|
|
|
|
esp_mpi_free( &TB );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Signed addition: X = A + B
|
|
*/
|
|
int esp_mpi_add_mpi( mpi *X, const mpi *A, const mpi *B )
|
|
{
|
|
int ret, s = A->s;
|
|
|
|
if( A->s * B->s < 0 )
|
|
{
|
|
if( esp_mpi_cmp_abs( A, B ) >= 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_sub_abs( X, A, B ) );
|
|
X->s = s;
|
|
}
|
|
else
|
|
{
|
|
MPI_CHK( esp_mpi_sub_abs( X, B, A ) );
|
|
X->s = -s;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
MPI_CHK( esp_mpi_add_abs( X, A, B ) );
|
|
X->s = s;
|
|
}
|
|
|
|
cleanup:
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Signed subtraction: X = A - B
|
|
*/
|
|
int esp_mpi_sub_mpi( mpi *X, const mpi *A, const mpi *B )
|
|
{
|
|
int ret, s = A->s;
|
|
|
|
if( A->s * B->s > 0 )
|
|
{
|
|
if( esp_mpi_cmp_abs( A, B ) >= 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_sub_abs( X, A, B ) );
|
|
X->s = s;
|
|
}
|
|
else
|
|
{
|
|
MPI_CHK( esp_mpi_sub_abs( X, B, A ) );
|
|
X->s = -s;
|
|
}
|
|
}
|
|
else
|
|
{
|
|
MPI_CHK( esp_mpi_add_abs( X, A, B ) );
|
|
X->s = s;
|
|
}
|
|
|
|
cleanup:
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Signed addition: X = A + b
|
|
*/
|
|
int esp_mpi_add_int( mpi *X, const mpi *A, esp_mpi_sint b )
|
|
{
|
|
mpi _B;
|
|
esp_mpi_uint p[1];
|
|
|
|
p[0] = ( b < 0 ) ? -b : b;
|
|
_B.s = ( b < 0 ) ? -1 : 1;
|
|
_B.n = 1;
|
|
_B.p = p;
|
|
|
|
return( esp_mpi_add_mpi( X, A, &_B ) );
|
|
}
|
|
|
|
/*
|
|
* Signed subtraction: X = A - b
|
|
*/
|
|
int esp_mpi_sub_int( mpi *X, const mpi *A, esp_mpi_sint b )
|
|
{
|
|
mpi _B;
|
|
esp_mpi_uint p[1];
|
|
|
|
p[0] = ( b < 0 ) ? -b : b;
|
|
_B.s = ( b < 0 ) ? -1 : 1;
|
|
_B.n = 1;
|
|
_B.p = p;
|
|
|
|
return( esp_mpi_sub_mpi( X, A, &_B ) );
|
|
}
|
|
|
|
/*
|
|
* Helper for mpi multiplication
|
|
*/
|
|
static void esp_mpi_mul_hlp( size_t i, esp_mpi_uint *s, esp_mpi_uint *d, esp_mpi_uint b )
|
|
{
|
|
|
|
}
|
|
|
|
/*
|
|
* Baseline multiplication: X = A * B (HAC 14.12)
|
|
*/
|
|
|
|
static int mul_pram_alloc( mpi *X, const mpi *A, const mpi *B, char **pA, char **pB, char **pX, size_t *bites)
|
|
{
|
|
char *sa, *sb, *sx;
|
|
// int algn;
|
|
int words, bytes;
|
|
int abytes, bbytes;
|
|
|
|
if (A->n > B->n)
|
|
words = A->n;
|
|
else
|
|
words = B->n;
|
|
|
|
bytes = (words / 16 + ((words % 16) ? 1 : 0 )) * 16 * 4 * 2;
|
|
|
|
abytes = A->n * 4;
|
|
bbytes = B->n * 4;
|
|
|
|
sa = malloc(bytes);
|
|
if (!sa) {
|
|
return -1;
|
|
}
|
|
|
|
sb = malloc(bytes);
|
|
if (!sb) {
|
|
free(sa);
|
|
return -1;
|
|
}
|
|
|
|
sx = malloc(bytes);
|
|
if (!sx) {
|
|
free(sa);
|
|
free(sb);
|
|
return -1;
|
|
}
|
|
|
|
memcpy(sa, A->p, abytes);
|
|
memset(sa + abytes, 0, bytes - abytes);
|
|
|
|
memcpy(sb, B->p, bbytes);
|
|
memset(sb + bbytes, 0, bytes - bbytes);
|
|
|
|
*pA = sa;
|
|
*pB = sb;
|
|
|
|
*pX = sx;
|
|
|
|
*bites = bytes * 4;
|
|
|
|
return 0;
|
|
}
|
|
|
|
void mul_pram_free(char **pA, char **pB, char **pX)
|
|
{
|
|
free(*pA);
|
|
*pA = NULL;
|
|
|
|
free(*pB);
|
|
*pB = NULL;
|
|
|
|
free(*pX);
|
|
*pX = NULL;
|
|
}
|
|
|
|
int esp_mpi_mul_mpi( mpi *X, const mpi *A, const mpi *B )
|
|
{
|
|
int ret = -1;
|
|
size_t i, j;
|
|
char *s1 = NULL, *s2 = NULL, *dest = NULL;
|
|
size_t bites;
|
|
|
|
mpi TA, TB;
|
|
|
|
esp_mpi_init( &TA ); esp_mpi_init( &TB );
|
|
|
|
if( X == A ) { MPI_CHK( esp_mpi_copy( &TA, A ) ); A = &TA; }
|
|
if( X == B ) { MPI_CHK( esp_mpi_copy( &TB, B ) ); B = &TB; }
|
|
|
|
for( i = A->n; i > 0; i-- )
|
|
if( A->p[i - 1] != 0 )
|
|
break;
|
|
|
|
for( j = B->n; j > 0; j-- )
|
|
if( B->p[j - 1] != 0 )
|
|
break;
|
|
|
|
MPI_CHK( esp_mpi_grow( X, i + j ) );
|
|
MPI_CHK( esp_mpi_lset( X, 0 ) );
|
|
|
|
if (mul_pram_alloc(X, A, B, &s1, &s2, &dest, &bites)) {
|
|
goto cleanup;
|
|
}
|
|
|
|
BIGNUM_LOCK();
|
|
if (ets_bigint_mult_prepare((uint32_t *)s1, (uint32_t *)s2, bites)){
|
|
ets_bigint_wait_finish();
|
|
if (ets_bigint_mult_getz((uint32_t *)dest, bites) == true) {
|
|
memcpy(X->p, dest, (i + j) * 4);
|
|
ret = 0;
|
|
} else {
|
|
esp_mpi_printf("ets_bigint_mult_getz failed\n");
|
|
}
|
|
} else{
|
|
esp_mpi_printf("Baseline multiplication failed\n");
|
|
}
|
|
BIGNUM_UNLOCK();
|
|
|
|
X->s = A->s * B->s;
|
|
|
|
mul_pram_free(&s1, &s2, &dest);
|
|
|
|
cleanup:
|
|
|
|
esp_mpi_free( &TB ); esp_mpi_free( &TA );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Baseline multiplication: X = A * b
|
|
*/
|
|
int esp_mpi_mul_int( mpi *X, const mpi *A, esp_mpi_uint b )
|
|
{
|
|
mpi _B;
|
|
esp_mpi_uint p[1];
|
|
|
|
_B.s = 1;
|
|
_B.n = 1;
|
|
_B.p = p;
|
|
p[0] = b;
|
|
|
|
return( esp_mpi_mul_mpi( X, A, &_B ) );
|
|
}
|
|
|
|
/*
|
|
* Unsigned integer divide - double esp_mpi_uint dividend, u1/u0, and
|
|
* esp_mpi_uint divisor, d
|
|
*/
|
|
static esp_mpi_uint int_div_int( esp_mpi_uint u1,
|
|
esp_mpi_uint u0, esp_mpi_uint d, esp_mpi_uint *r )
|
|
{
|
|
#if defined(HAVE_UDBL)
|
|
t_udbl dividend, quotient;
|
|
#else
|
|
const esp_mpi_uint radix = (esp_mpi_uint) 1 << biH;
|
|
const esp_mpi_uint uint_halfword_mask = ( (esp_mpi_uint) 1 << biH ) - 1;
|
|
esp_mpi_uint d0, d1, q0, q1, rAX, r0, quotient;
|
|
esp_mpi_uint u0_msw, u0_lsw;
|
|
size_t s;
|
|
#endif
|
|
|
|
/*
|
|
* Check for overflow
|
|
*/
|
|
if( 0 == d || u1 >= d )
|
|
{
|
|
if (r != NULL) *r = ~0;
|
|
|
|
return ( ~0 );
|
|
}
|
|
|
|
#if defined(HAVE_UDBL)
|
|
dividend = (t_udbl) u1 << biL;
|
|
dividend |= (t_udbl) u0;
|
|
quotient = dividend / d;
|
|
if( quotient > ( (t_udbl) 1 << biL ) - 1 )
|
|
quotient = ( (t_udbl) 1 << biL ) - 1;
|
|
|
|
if( r != NULL )
|
|
*r = (esp_mpi_uint)( dividend - (quotient * d ) );
|
|
|
|
return (esp_mpi_uint) quotient;
|
|
#else
|
|
|
|
/*
|
|
* Algorithm D, Section 4.3.1 - The Art of Computer Programming
|
|
* Vol. 2 - Seminumerical Algorithms, Knuth
|
|
*/
|
|
|
|
/*
|
|
* Normalize the divisor, d, and dividend, u0, u1
|
|
*/
|
|
s = clz( d );
|
|
d = d << s;
|
|
|
|
u1 = u1 << s;
|
|
u1 |= ( u0 >> ( biL - s ) ) & ( -(esp_mpi_sint)s >> ( biL - 1 ) );
|
|
u0 = u0 << s;
|
|
|
|
d1 = d >> biH;
|
|
d0 = d & uint_halfword_mask;
|
|
|
|
u0_msw = u0 >> biH;
|
|
u0_lsw = u0 & uint_halfword_mask;
|
|
|
|
/*
|
|
* Find the first quotient and remainder
|
|
*/
|
|
q1 = u1 / d1;
|
|
r0 = u1 - d1 * q1;
|
|
|
|
while( q1 >= radix || ( q1 * d0 > radix * r0 + u0_msw ) )
|
|
{
|
|
q1 -= 1;
|
|
r0 += d1;
|
|
|
|
if ( r0 >= radix ) break;
|
|
}
|
|
|
|
rAX = ( u1 * radix ) + ( u0_msw - q1 * d );
|
|
q0 = rAX / d1;
|
|
r0 = rAX - q0 * d1;
|
|
|
|
while( q0 >= radix || ( q0 * d0 > radix * r0 + u0_lsw ) )
|
|
{
|
|
q0 -= 1;
|
|
r0 += d1;
|
|
|
|
if ( r0 >= radix ) break;
|
|
}
|
|
|
|
if (r != NULL)
|
|
*r = ( rAX * radix + u0_lsw - q0 * d ) >> s;
|
|
|
|
quotient = q1 * radix + q0;
|
|
|
|
return quotient;
|
|
#endif
|
|
}
|
|
|
|
/*
|
|
* Division by mpi: A = Q * B + R (HAC 14.20)
|
|
*/
|
|
int esp_mpi_div_mpi( mpi *Q, mpi *R, const mpi *A, const mpi *B )
|
|
{
|
|
int ret;
|
|
size_t i, n, t, k;
|
|
mpi X, Y, Z, T1, T2;
|
|
|
|
if( esp_mpi_cmp_int( B, 0 ) == 0 )
|
|
return( ERR_MPI_DIVISION_BY_ZERO );
|
|
|
|
esp_mpi_init( &X ); esp_mpi_init( &Y ); esp_mpi_init( &Z );
|
|
esp_mpi_init( &T1 ); esp_mpi_init( &T2 );
|
|
|
|
if( esp_mpi_cmp_abs( A, B ) < 0 )
|
|
{
|
|
if( Q != NULL ) MPI_CHK( esp_mpi_lset( Q, 0 ) );
|
|
if( R != NULL ) MPI_CHK( esp_mpi_copy( R, A ) );
|
|
return( 0 );
|
|
}
|
|
|
|
MPI_CHK( esp_mpi_copy( &X, A ) );
|
|
MPI_CHK( esp_mpi_copy( &Y, B ) );
|
|
X.s = Y.s = 1;
|
|
|
|
MPI_CHK( esp_mpi_grow( &Z, A->n + 2 ) );
|
|
MPI_CHK( esp_mpi_lset( &Z, 0 ) );
|
|
MPI_CHK( esp_mpi_grow( &T1, 2 ) );
|
|
MPI_CHK( esp_mpi_grow( &T2, 3 ) );
|
|
|
|
k = esp_mpi_bitlen( &Y ) % biL;
|
|
if( k < biL - 1 )
|
|
{
|
|
k = biL - 1 - k;
|
|
MPI_CHK( esp_mpi_shift_l( &X, k ) );
|
|
MPI_CHK( esp_mpi_shift_l( &Y, k ) );
|
|
}
|
|
else k = 0;
|
|
|
|
n = X.n - 1;
|
|
t = Y.n - 1;
|
|
MPI_CHK( esp_mpi_shift_l( &Y, biL * ( n - t ) ) );
|
|
|
|
while( esp_mpi_cmp_mpi( &X, &Y ) >= 0 )
|
|
{
|
|
Z.p[n - t]++;
|
|
MPI_CHK( esp_mpi_sub_mpi( &X, &X, &Y ) );
|
|
}
|
|
MPI_CHK( esp_mpi_shift_r( &Y, biL * ( n - t ) ) );
|
|
|
|
for( i = n; i > t ; i-- )
|
|
{
|
|
if( X.p[i] >= Y.p[t] )
|
|
Z.p[i - t - 1] = ~0;
|
|
else
|
|
{
|
|
Z.p[i - t - 1] = int_div_int( X.p[i], X.p[i - 1],
|
|
Y.p[t], NULL);
|
|
}
|
|
|
|
Z.p[i - t - 1]++;
|
|
do
|
|
{
|
|
Z.p[i - t - 1]--;
|
|
|
|
MPI_CHK( esp_mpi_lset( &T1, 0 ) );
|
|
T1.p[0] = ( t < 1 ) ? 0 : Y.p[t - 1];
|
|
T1.p[1] = Y.p[t];
|
|
MPI_CHK( esp_mpi_mul_int( &T1, &T1, Z.p[i - t - 1] ) );
|
|
|
|
MPI_CHK( esp_mpi_lset( &T2, 0 ) );
|
|
T2.p[0] = ( i < 2 ) ? 0 : X.p[i - 2];
|
|
T2.p[1] = ( i < 1 ) ? 0 : X.p[i - 1];
|
|
T2.p[2] = X.p[i];
|
|
}
|
|
while( esp_mpi_cmp_mpi( &T1, &T2 ) > 0 );
|
|
|
|
MPI_CHK( esp_mpi_mul_int( &T1, &Y, Z.p[i - t - 1] ) );
|
|
MPI_CHK( esp_mpi_shift_l( &T1, biL * ( i - t - 1 ) ) );
|
|
MPI_CHK( esp_mpi_sub_mpi( &X, &X, &T1 ) );
|
|
|
|
if( esp_mpi_cmp_int( &X, 0 ) < 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_copy( &T1, &Y ) );
|
|
MPI_CHK( esp_mpi_shift_l( &T1, biL * ( i - t - 1 ) ) );
|
|
MPI_CHK( esp_mpi_add_mpi( &X, &X, &T1 ) );
|
|
Z.p[i - t - 1]--;
|
|
}
|
|
}
|
|
|
|
if( Q != NULL )
|
|
{
|
|
MPI_CHK( esp_mpi_copy( Q, &Z ) );
|
|
Q->s = A->s * B->s;
|
|
}
|
|
|
|
if( R != NULL )
|
|
{
|
|
MPI_CHK( esp_mpi_shift_r( &X, k ) );
|
|
X.s = A->s;
|
|
MPI_CHK( esp_mpi_copy( R, &X ) );
|
|
|
|
if( esp_mpi_cmp_int( R, 0 ) == 0 )
|
|
R->s = 1;
|
|
}
|
|
|
|
cleanup:
|
|
|
|
esp_mpi_free( &X ); esp_mpi_free( &Y ); esp_mpi_free( &Z );
|
|
esp_mpi_free( &T1 ); esp_mpi_free( &T2 );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Division by int: A = Q * b + R
|
|
*/
|
|
int esp_mpi_div_int( mpi *Q, mpi *R, const mpi *A, esp_mpi_sint b )
|
|
{
|
|
mpi _B;
|
|
esp_mpi_uint p[1];
|
|
|
|
p[0] = ( b < 0 ) ? -b : b;
|
|
_B.s = ( b < 0 ) ? -1 : 1;
|
|
_B.n = 1;
|
|
_B.p = p;
|
|
|
|
return( esp_mpi_div_mpi( Q, R, A, &_B ) );
|
|
}
|
|
|
|
/*
|
|
* Modulo: R = A mod B
|
|
*/
|
|
int esp_mpi_mod_mpi( mpi *R, const mpi *A, const mpi *B )
|
|
{
|
|
int ret;
|
|
|
|
if( esp_mpi_cmp_int( B, 0 ) < 0 )
|
|
return( ERR_MPI_NEGATIVE_VALUE );
|
|
|
|
MPI_CHK( esp_mpi_div_mpi( NULL, R, A, B ) );
|
|
|
|
while( esp_mpi_cmp_int( R, 0 ) < 0 )
|
|
MPI_CHK( esp_mpi_add_mpi( R, R, B ) );
|
|
|
|
while( esp_mpi_cmp_mpi( R, B ) >= 0 )
|
|
MPI_CHK( esp_mpi_sub_mpi( R, R, B ) );
|
|
|
|
cleanup:
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Modulo: r = A mod b
|
|
*/
|
|
int esp_mpi_mod_int( esp_mpi_uint *r, const mpi *A, esp_mpi_sint b )
|
|
{
|
|
size_t i;
|
|
esp_mpi_uint x, y, z;
|
|
|
|
if( b == 0 )
|
|
return( ERR_MPI_DIVISION_BY_ZERO );
|
|
|
|
if( b < 0 )
|
|
return( ERR_MPI_NEGATIVE_VALUE );
|
|
|
|
/*
|
|
* handle trivial cases
|
|
*/
|
|
if( b == 1 )
|
|
{
|
|
*r = 0;
|
|
return( 0 );
|
|
}
|
|
|
|
if( b == 2 )
|
|
{
|
|
*r = A->p[0] & 1;
|
|
return( 0 );
|
|
}
|
|
|
|
/*
|
|
* general case
|
|
*/
|
|
for( i = A->n, y = 0; i > 0; i-- )
|
|
{
|
|
x = A->p[i - 1];
|
|
y = ( y << biH ) | ( x >> biH );
|
|
z = y / b;
|
|
y -= z * b;
|
|
|
|
x <<= biH;
|
|
y = ( y << biH ) | ( x >> biH );
|
|
z = y / b;
|
|
y -= z * b;
|
|
}
|
|
|
|
/*
|
|
* If A is negative, then the current y represents a negative value.
|
|
* Flipping it to the positive side.
|
|
*/
|
|
if( A->s < 0 && y != 0 )
|
|
y = b - y;
|
|
|
|
*r = y;
|
|
|
|
return( 0 );
|
|
}
|
|
|
|
/*
|
|
* Fast Montgomery initialization (thanks to Tom St Denis)
|
|
*/
|
|
static void esp_mpi_montg_init( esp_mpi_uint *mm, const mpi *N )
|
|
{
|
|
esp_mpi_uint x, m0 = N->p[0];
|
|
unsigned int i;
|
|
|
|
x = m0;
|
|
x += ( ( m0 + 2 ) & 4 ) << 1;
|
|
|
|
for( i = biL; i >= 8; i /= 2 )
|
|
x *= ( 2 - ( m0 * x ) );
|
|
|
|
*mm = ~x + 1;
|
|
}
|
|
|
|
/*
|
|
* Montgomery multiplication: A = A * B * R^-1 mod N (HAC 14.36)
|
|
*/
|
|
static void esp_mpi_montmul( mpi *A, const mpi *B, const mpi *N, esp_mpi_uint mm,
|
|
const mpi *T )
|
|
{
|
|
size_t n, m;
|
|
esp_mpi_uint *d = NULL;
|
|
|
|
memset( T->p, 0, T->n * ciL );
|
|
|
|
d = T->p;
|
|
n = N->n;
|
|
m = ( B->n < n ) ? B->n : n;
|
|
|
|
BIGNUM_LOCK();
|
|
if (ets_bigint_montgomery_mult_prepare(N->p, B->p, d, m, n, false)) {
|
|
ets_bigint_wait_finish();
|
|
|
|
ets_bigint_montgomery_mult_getz(A->p, n);
|
|
} else{
|
|
esp_mpi_printf("Montgomery multiplication failed\n");
|
|
}
|
|
BIGNUM_UNLOCK();
|
|
|
|
}
|
|
|
|
/*
|
|
* Montgomery reduction: A = A * R^-1 mod N
|
|
*/
|
|
static void esp_mpi_montred( mpi *A, const mpi *N, esp_mpi_uint mm, const mpi *T )
|
|
{
|
|
esp_mpi_uint z = 1;
|
|
mpi U;
|
|
|
|
U.n = U.s = (int) z;
|
|
U.p = &z;
|
|
|
|
esp_mpi_montmul( A, &U, N, mm, T );
|
|
}
|
|
|
|
/*
|
|
* Sliding-window exponentiation: X = A^E mod N (HAC 14.85)
|
|
*/
|
|
int esp_mpi_exp_mod( mpi *X, const mpi *A, const mpi *E, const mpi *N, mpi *_RR )
|
|
{
|
|
int ret;
|
|
size_t wbits, wsize, one = 1;
|
|
size_t i, j, nblimbs;
|
|
size_t bufsize, nbits;
|
|
esp_mpi_uint ei, mm, state;
|
|
mpi RR, T, W[ 2 << MPI_WINDOW_SIZE ], Apos;
|
|
int neg;
|
|
|
|
if( esp_mpi_cmp_int( N, 0 ) < 0 || ( N->p[0] & 1 ) == 0 )
|
|
return( ERR_MPI_BAD_INPUT_DATA );
|
|
|
|
if( esp_mpi_cmp_int( E, 0 ) < 0 )
|
|
return( ERR_MPI_BAD_INPUT_DATA );
|
|
|
|
/*
|
|
* Init temps and window size
|
|
*/
|
|
esp_mpi_montg_init( &mm, N );
|
|
esp_mpi_init( &RR ); esp_mpi_init( &T );
|
|
esp_mpi_init( &Apos );
|
|
memset( W, 0, sizeof( W ) );
|
|
|
|
i = esp_mpi_bitlen( E );
|
|
|
|
wsize = ( i > 671 ) ? 6 : ( i > 239 ) ? 5 :
|
|
( i > 79 ) ? 4 : ( i > 23 ) ? 3 : 1;
|
|
|
|
if( wsize > MPI_WINDOW_SIZE )
|
|
wsize = MPI_WINDOW_SIZE;
|
|
|
|
j = N->n + 1;
|
|
MPI_CHK( esp_mpi_grow( X, j ) );
|
|
MPI_CHK( esp_mpi_grow( &W[1], j ) );
|
|
MPI_CHK( esp_mpi_grow( &T, j * 2 ) );
|
|
|
|
/*
|
|
* Compensate for negative A (and correct at the end)
|
|
*/
|
|
neg = ( A->s == -1 );
|
|
if( neg )
|
|
{
|
|
MPI_CHK( esp_mpi_copy( &Apos, A ) );
|
|
Apos.s = 1;
|
|
A = &Apos;
|
|
}
|
|
|
|
/*
|
|
* If 1st call, pre-compute R^2 mod N
|
|
*/
|
|
if( _RR == NULL || _RR->p == NULL )
|
|
{
|
|
MPI_CHK( esp_mpi_lset( &RR, 1 ) );
|
|
MPI_CHK( esp_mpi_shift_l( &RR, N->n * 2 * biL ) );
|
|
MPI_CHK( esp_mpi_mod_mpi( &RR, &RR, N ) );
|
|
|
|
if( _RR != NULL )
|
|
memcpy( _RR, &RR, sizeof( mpi ) );
|
|
}
|
|
else
|
|
memcpy( &RR, _RR, sizeof( mpi ) );
|
|
|
|
/*
|
|
* W[1] = A * R^2 * R^-1 mod N = A * R mod N
|
|
*/
|
|
if( esp_mpi_cmp_mpi( A, N ) >= 0 )
|
|
MPI_CHK( esp_mpi_mod_mpi( &W[1], A, N ) );
|
|
else
|
|
MPI_CHK( esp_mpi_copy( &W[1], A ) );
|
|
|
|
esp_mpi_montmul( &W[1], &RR, N, mm, &T );
|
|
|
|
/*
|
|
* X = R^2 * R^-1 mod N = R mod N
|
|
*/
|
|
MPI_CHK( esp_mpi_copy( X, &RR ) );
|
|
esp_mpi_montred( X, N, mm, &T );
|
|
|
|
if( wsize > 1 )
|
|
{
|
|
/*
|
|
* W[1 << (wsize - 1)] = W[1] ^ (wsize - 1)
|
|
*/
|
|
j = one << ( wsize - 1 );
|
|
|
|
MPI_CHK( esp_mpi_grow( &W[j], N->n + 1 ) );
|
|
MPI_CHK( esp_mpi_copy( &W[j], &W[1] ) );
|
|
|
|
for( i = 0; i < wsize - 1; i++ )
|
|
esp_mpi_montmul( &W[j], &W[j], N, mm, &T );
|
|
|
|
/*
|
|
* W[i] = W[i - 1] * W[1]
|
|
*/
|
|
for( i = j + 1; i < ( one << wsize ); i++ )
|
|
{
|
|
MPI_CHK( esp_mpi_grow( &W[i], N->n + 1 ) );
|
|
MPI_CHK( esp_mpi_copy( &W[i], &W[i - 1] ) );
|
|
|
|
esp_mpi_montmul( &W[i], &W[1], N, mm, &T );
|
|
}
|
|
}
|
|
|
|
nblimbs = E->n;
|
|
bufsize = 0;
|
|
nbits = 0;
|
|
wbits = 0;
|
|
state = 0;
|
|
|
|
while( 1 )
|
|
{
|
|
if( bufsize == 0 )
|
|
{
|
|
if( nblimbs == 0 )
|
|
break;
|
|
|
|
nblimbs--;
|
|
|
|
bufsize = sizeof( esp_mpi_uint ) << 3;
|
|
}
|
|
|
|
bufsize--;
|
|
|
|
ei = (E->p[nblimbs] >> bufsize) & 1;
|
|
|
|
/*
|
|
* skip leading 0s
|
|
*/
|
|
if( ei == 0 && state == 0 )
|
|
continue;
|
|
|
|
if( ei == 0 && state == 1 )
|
|
{
|
|
/*
|
|
* out of window, square X
|
|
*/
|
|
esp_mpi_montmul( X, X, N, mm, &T );
|
|
continue;
|
|
}
|
|
|
|
/*
|
|
* add ei to current window
|
|
*/
|
|
state = 2;
|
|
|
|
nbits++;
|
|
wbits |= ( ei << ( wsize - nbits ) );
|
|
|
|
if( nbits == wsize )
|
|
{
|
|
/*
|
|
* X = X^wsize R^-1 mod N
|
|
*/
|
|
for( i = 0; i < wsize; i++ )
|
|
esp_mpi_montmul( X, X, N, mm, &T );
|
|
|
|
/*
|
|
* X = X * W[wbits] R^-1 mod N
|
|
*/
|
|
esp_mpi_montmul( X, &W[wbits], N, mm, &T );
|
|
|
|
state--;
|
|
nbits = 0;
|
|
wbits = 0;
|
|
}
|
|
}
|
|
|
|
/*
|
|
* process the remaining bits
|
|
*/
|
|
for( i = 0; i < nbits; i++ )
|
|
{
|
|
esp_mpi_montmul( X, X, N, mm, &T );
|
|
|
|
wbits <<= 1;
|
|
|
|
if( ( wbits & ( one << wsize ) ) != 0 )
|
|
esp_mpi_montmul( X, &W[1], N, mm, &T );
|
|
}
|
|
|
|
/*
|
|
* X = A^E * R * R^-1 mod N = A^E mod N
|
|
*/
|
|
esp_mpi_montred( X, N, mm, &T );
|
|
|
|
if( neg )
|
|
{
|
|
X->s = -1;
|
|
MPI_CHK( esp_mpi_add_mpi( X, N, X ) );
|
|
}
|
|
|
|
cleanup:
|
|
|
|
for( i = ( one << ( wsize - 1 ) ); i < ( one << wsize ); i++ )
|
|
esp_mpi_free( &W[i] );
|
|
|
|
esp_mpi_free( &W[1] ); esp_mpi_free( &T ); esp_mpi_free( &Apos );
|
|
|
|
if( _RR == NULL || _RR->p == NULL )
|
|
esp_mpi_free( &RR );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Greatest common divisor: G = gcd(A, B) (HAC 14.54)
|
|
*/
|
|
int esp_mpi_gcd( mpi *G, const mpi *A, const mpi *B )
|
|
{
|
|
int ret;
|
|
size_t lz, lzt;
|
|
mpi TG, TA, TB;
|
|
|
|
esp_mpi_init( &TG ); esp_mpi_init( &TA ); esp_mpi_init( &TB );
|
|
|
|
MPI_CHK( esp_mpi_copy( &TA, A ) );
|
|
MPI_CHK( esp_mpi_copy( &TB, B ) );
|
|
|
|
lz = esp_mpi_lsb( &TA );
|
|
lzt = esp_mpi_lsb( &TB );
|
|
|
|
if( lzt < lz )
|
|
lz = lzt;
|
|
|
|
MPI_CHK( esp_mpi_shift_r( &TA, lz ) );
|
|
MPI_CHK( esp_mpi_shift_r( &TB, lz ) );
|
|
|
|
TA.s = TB.s = 1;
|
|
|
|
while( esp_mpi_cmp_int( &TA, 0 ) != 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_shift_r( &TA, esp_mpi_lsb( &TA ) ) );
|
|
MPI_CHK( esp_mpi_shift_r( &TB, esp_mpi_lsb( &TB ) ) );
|
|
|
|
if( esp_mpi_cmp_mpi( &TA, &TB ) >= 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_sub_abs( &TA, &TA, &TB ) );
|
|
MPI_CHK( esp_mpi_shift_r( &TA, 1 ) );
|
|
}
|
|
else
|
|
{
|
|
MPI_CHK( esp_mpi_sub_abs( &TB, &TB, &TA ) );
|
|
MPI_CHK( esp_mpi_shift_r( &TB, 1 ) );
|
|
}
|
|
}
|
|
|
|
MPI_CHK( esp_mpi_shift_l( &TB, lz ) );
|
|
MPI_CHK( esp_mpi_copy( G, &TB ) );
|
|
|
|
cleanup:
|
|
|
|
esp_mpi_free( &TG ); esp_mpi_free( &TA ); esp_mpi_free( &TB );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Fill X with size bytes of random.
|
|
*
|
|
* Use a temporary bytes representation to make sure the result is the same
|
|
* regardless of the platform endianness (useful when f_rng is actually
|
|
* deterministic, eg for tests).
|
|
*/
|
|
int esp_mpi_fill_random( mpi *X, size_t size,
|
|
int (*f_rng)(void *, unsigned char *, size_t),
|
|
void *p_rng )
|
|
{
|
|
int ret;
|
|
unsigned char buf[MPI_MAX_SIZE];
|
|
|
|
if( size > MPI_MAX_SIZE )
|
|
return( ERR_MPI_BAD_INPUT_DATA );
|
|
|
|
MPI_CHK( f_rng( p_rng, buf, size ) );
|
|
MPI_CHK( esp_mpi_read_binary( X, buf, size ) );
|
|
|
|
cleanup:
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Modular inverse: X = A^-1 mod N (HAC 14.61 / 14.64)
|
|
*/
|
|
int esp_mpi_inv_mod( mpi *X, const mpi *A, const mpi *N )
|
|
{
|
|
int ret;
|
|
mpi G, TA, TU, U1, U2, TB, TV, V1, V2;
|
|
|
|
if( esp_mpi_cmp_int( N, 0 ) <= 0 )
|
|
return( ERR_MPI_BAD_INPUT_DATA );
|
|
|
|
esp_mpi_init( &TA ); esp_mpi_init( &TU ); esp_mpi_init( &U1 ); esp_mpi_init( &U2 );
|
|
esp_mpi_init( &G ); esp_mpi_init( &TB ); esp_mpi_init( &TV );
|
|
esp_mpi_init( &V1 ); esp_mpi_init( &V2 );
|
|
|
|
MPI_CHK( esp_mpi_gcd( &G, A, N ) );
|
|
|
|
if( esp_mpi_cmp_int( &G, 1 ) != 0 )
|
|
{
|
|
ret = ERR_MPI_NOT_ACCEPTABLE;
|
|
goto cleanup;
|
|
}
|
|
|
|
MPI_CHK( esp_mpi_mod_mpi( &TA, A, N ) );
|
|
MPI_CHK( esp_mpi_copy( &TU, &TA ) );
|
|
MPI_CHK( esp_mpi_copy( &TB, N ) );
|
|
MPI_CHK( esp_mpi_copy( &TV, N ) );
|
|
|
|
MPI_CHK( esp_mpi_lset( &U1, 1 ) );
|
|
MPI_CHK( esp_mpi_lset( &U2, 0 ) );
|
|
MPI_CHK( esp_mpi_lset( &V1, 0 ) );
|
|
MPI_CHK( esp_mpi_lset( &V2, 1 ) );
|
|
|
|
do
|
|
{
|
|
while( ( TU.p[0] & 1 ) == 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_shift_r( &TU, 1 ) );
|
|
|
|
if( ( U1.p[0] & 1 ) != 0 || ( U2.p[0] & 1 ) != 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_add_mpi( &U1, &U1, &TB ) );
|
|
MPI_CHK( esp_mpi_sub_mpi( &U2, &U2, &TA ) );
|
|
}
|
|
|
|
MPI_CHK( esp_mpi_shift_r( &U1, 1 ) );
|
|
MPI_CHK( esp_mpi_shift_r( &U2, 1 ) );
|
|
}
|
|
|
|
while( ( TV.p[0] & 1 ) == 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_shift_r( &TV, 1 ) );
|
|
|
|
if( ( V1.p[0] & 1 ) != 0 || ( V2.p[0] & 1 ) != 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_add_mpi( &V1, &V1, &TB ) );
|
|
MPI_CHK( esp_mpi_sub_mpi( &V2, &V2, &TA ) );
|
|
}
|
|
|
|
MPI_CHK( esp_mpi_shift_r( &V1, 1 ) );
|
|
MPI_CHK( esp_mpi_shift_r( &V2, 1 ) );
|
|
}
|
|
|
|
if( esp_mpi_cmp_mpi( &TU, &TV ) >= 0 )
|
|
{
|
|
MPI_CHK( esp_mpi_sub_mpi( &TU, &TU, &TV ) );
|
|
MPI_CHK( esp_mpi_sub_mpi( &U1, &U1, &V1 ) );
|
|
MPI_CHK( esp_mpi_sub_mpi( &U2, &U2, &V2 ) );
|
|
}
|
|
else
|
|
{
|
|
MPI_CHK( esp_mpi_sub_mpi( &TV, &TV, &TU ) );
|
|
MPI_CHK( esp_mpi_sub_mpi( &V1, &V1, &U1 ) );
|
|
MPI_CHK( esp_mpi_sub_mpi( &V2, &V2, &U2 ) );
|
|
}
|
|
}
|
|
while( esp_mpi_cmp_int( &TU, 0 ) != 0 );
|
|
|
|
while( esp_mpi_cmp_int( &V1, 0 ) < 0 )
|
|
MPI_CHK( esp_mpi_add_mpi( &V1, &V1, N ) );
|
|
|
|
while( esp_mpi_cmp_mpi( &V1, N ) >= 0 )
|
|
MPI_CHK( esp_mpi_sub_mpi( &V1, &V1, N ) );
|
|
|
|
MPI_CHK( esp_mpi_copy( X, &V1 ) );
|
|
|
|
cleanup:
|
|
|
|
esp_mpi_free( &TA ); esp_mpi_free( &TU ); esp_mpi_free( &U1 ); esp_mpi_free( &U2 );
|
|
esp_mpi_free( &G ); esp_mpi_free( &TB ); esp_mpi_free( &TV );
|
|
esp_mpi_free( &V1 ); esp_mpi_free( &V2 );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
static const int small_prime[] =
|
|
{
|
|
3, 5, 7, 11, 13, 17, 19, 23,
|
|
29, 31, 37, 41, 43, 47, 53, 59,
|
|
61, 67, 71, 73, 79, 83, 89, 97,
|
|
101, 103, 107, 109, 113, 127, 131, 137,
|
|
139, 149, 151, 157, 163, 167, 173, 179,
|
|
181, 191, 193, 197, 199, 211, 223, 227,
|
|
229, 233, 239, 241, 251, 257, 263, 269,
|
|
271, 277, 281, 283, 293, 307, 311, 313,
|
|
317, 331, 337, 347, 349, 353, 359, 367,
|
|
373, 379, 383, 389, 397, 401, 409, 419,
|
|
421, 431, 433, 439, 443, 449, 457, 461,
|
|
463, 467, 479, 487, 491, 499, 503, 509,
|
|
521, 523, 541, 547, 557, 563, 569, 571,
|
|
577, 587, 593, 599, 601, 607, 613, 617,
|
|
619, 631, 641, 643, 647, 653, 659, 661,
|
|
673, 677, 683, 691, 701, 709, 719, 727,
|
|
733, 739, 743, 751, 757, 761, 769, 773,
|
|
787, 797, 809, 811, 821, 823, 827, 829,
|
|
839, 853, 857, 859, 863, 877, 881, 883,
|
|
887, 907, 911, 919, 929, 937, 941, 947,
|
|
953, 967, 971, 977, 983, 991, 997, -103
|
|
};
|
|
|
|
/*
|
|
* Small divisors test (X must be positive)
|
|
*
|
|
* Return values:
|
|
* 0: no small factor (possible prime, more tests needed)
|
|
* 1: certain prime
|
|
* ERR_MPI_NOT_ACCEPTABLE: certain non-prime
|
|
* other negative: error
|
|
*/
|
|
static int esp_mpi_check_small_factors( const mpi *X )
|
|
{
|
|
int ret = 0;
|
|
size_t i;
|
|
esp_mpi_uint r;
|
|
|
|
if( ( X->p[0] & 1 ) == 0 )
|
|
return( ERR_MPI_NOT_ACCEPTABLE );
|
|
|
|
for( i = 0; small_prime[i] > 0; i++ )
|
|
{
|
|
if( esp_mpi_cmp_int( X, small_prime[i] ) <= 0 )
|
|
return( 1 );
|
|
|
|
MPI_CHK( esp_mpi_mod_int( &r, X, small_prime[i] ) );
|
|
|
|
if( r == 0 )
|
|
return( ERR_MPI_NOT_ACCEPTABLE );
|
|
}
|
|
|
|
cleanup:
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Miller-Rabin pseudo-primality test (HAC 4.24)
|
|
*/
|
|
static int esp_mpi_miller_rabin( const mpi *X,
|
|
int (*f_rng)(void *, unsigned char *, size_t),
|
|
void *p_rng )
|
|
{
|
|
int ret, count;
|
|
size_t i, j, k, n, s;
|
|
mpi W, R, T, A, RR;
|
|
|
|
esp_mpi_init( &W ); esp_mpi_init( &R ); esp_mpi_init( &T ); esp_mpi_init( &A );
|
|
esp_mpi_init( &RR );
|
|
|
|
/*
|
|
* W = |X| - 1
|
|
* R = W >> lsb( W )
|
|
*/
|
|
MPI_CHK( esp_mpi_sub_int( &W, X, 1 ) );
|
|
s = esp_mpi_lsb( &W );
|
|
MPI_CHK( esp_mpi_copy( &R, &W ) );
|
|
MPI_CHK( esp_mpi_shift_r( &R, s ) );
|
|
|
|
i = esp_mpi_bitlen( X );
|
|
/*
|
|
* HAC, table 4.4
|
|
*/
|
|
n = ( ( i >= 1300 ) ? 2 : ( i >= 850 ) ? 3 :
|
|
( i >= 650 ) ? 4 : ( i >= 350 ) ? 8 :
|
|
( i >= 250 ) ? 12 : ( i >= 150 ) ? 18 : 27 );
|
|
|
|
for( i = 0; i < n; i++ )
|
|
{
|
|
/*
|
|
* pick a random A, 1 < A < |X| - 1
|
|
*/
|
|
MPI_CHK( esp_mpi_fill_random( &A, X->n * ciL, f_rng, p_rng ) );
|
|
|
|
if( esp_mpi_cmp_mpi( &A, &W ) >= 0 )
|
|
{
|
|
j = esp_mpi_bitlen( &A ) - esp_mpi_bitlen( &W );
|
|
MPI_CHK( esp_mpi_shift_r( &A, j + 1 ) );
|
|
}
|
|
A.p[0] |= 3;
|
|
|
|
count = 0;
|
|
do {
|
|
MPI_CHK( esp_mpi_fill_random( &A, X->n * ciL, f_rng, p_rng ) );
|
|
|
|
j = esp_mpi_bitlen( &A );
|
|
k = esp_mpi_bitlen( &W );
|
|
if (j > k) {
|
|
MPI_CHK( esp_mpi_shift_r( &A, j - k ) );
|
|
}
|
|
|
|
if (count++ > 30) {
|
|
return ERR_MPI_NOT_ACCEPTABLE;
|
|
}
|
|
|
|
} while ( esp_mpi_cmp_mpi( &A, &W ) >= 0 ||
|
|
esp_mpi_cmp_int( &A, 1 ) <= 0 );
|
|
|
|
/*
|
|
* A = A^R mod |X|
|
|
*/
|
|
MPI_CHK( esp_mpi_exp_mod( &A, &A, &R, X, &RR ) );
|
|
|
|
if( esp_mpi_cmp_mpi( &A, &W ) == 0 ||
|
|
esp_mpi_cmp_int( &A, 1 ) == 0 )
|
|
continue;
|
|
|
|
j = 1;
|
|
while( j < s && esp_mpi_cmp_mpi( &A, &W ) != 0 )
|
|
{
|
|
/*
|
|
* A = A * A mod |X|
|
|
*/
|
|
MPI_CHK( esp_mpi_mul_mpi( &T, &A, &A ) );
|
|
MPI_CHK( esp_mpi_mod_mpi( &A, &T, X ) );
|
|
|
|
if( esp_mpi_cmp_int( &A, 1 ) == 0 )
|
|
break;
|
|
|
|
j++;
|
|
}
|
|
|
|
/*
|
|
* not prime if A != |X| - 1 or A == 1
|
|
*/
|
|
if( esp_mpi_cmp_mpi( &A, &W ) != 0 ||
|
|
esp_mpi_cmp_int( &A, 1 ) == 0 )
|
|
{
|
|
ret = ERR_MPI_NOT_ACCEPTABLE;
|
|
break;
|
|
}
|
|
}
|
|
|
|
cleanup:
|
|
esp_mpi_free( &W ); esp_mpi_free( &R ); esp_mpi_free( &T ); esp_mpi_free( &A );
|
|
esp_mpi_free( &RR );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
/*
|
|
* Pseudo-primality test: small factors, then Miller-Rabin
|
|
*/
|
|
int esp_mpi_is_prime( const mpi *X,
|
|
int (*f_rng)(void *, unsigned char *, size_t),
|
|
void *p_rng )
|
|
{
|
|
int ret;
|
|
mpi XX;
|
|
|
|
XX.s = 1;
|
|
XX.n = X->n;
|
|
XX.p = X->p;
|
|
|
|
if( esp_mpi_cmp_int( &XX, 0 ) == 0 ||
|
|
esp_mpi_cmp_int( &XX, 1 ) == 0 )
|
|
return( ERR_MPI_NOT_ACCEPTABLE );
|
|
|
|
if( esp_mpi_cmp_int( &XX, 2 ) == 0 )
|
|
return( 0 );
|
|
|
|
if( ( ret = esp_mpi_check_small_factors( &XX ) ) != 0 )
|
|
{
|
|
if( ret == 1 )
|
|
return( 0 );
|
|
|
|
return( ret );
|
|
}
|
|
|
|
return( esp_mpi_miller_rabin( &XX, f_rng, p_rng ) );
|
|
}
|
|
|
|
/*
|
|
* Prime number generation
|
|
*/
|
|
int esp_mpi_gen_prime( mpi *X, size_t nbits, int dh_flag,
|
|
int (*f_rng)(void *, unsigned char *, size_t),
|
|
void *p_rng )
|
|
{
|
|
int ret;
|
|
size_t k, n;
|
|
esp_mpi_uint r;
|
|
mpi Y;
|
|
|
|
if( nbits < 3 || nbits > MPI_MAX_BITS )
|
|
return( ERR_MPI_BAD_INPUT_DATA );
|
|
|
|
esp_mpi_init( &Y );
|
|
|
|
n = BITS_TO_LIMBS( nbits );
|
|
|
|
MPI_CHK( esp_mpi_fill_random( X, n * ciL, f_rng, p_rng ) );
|
|
|
|
k = esp_mpi_bitlen( X );
|
|
if( k > nbits ) MPI_CHK( esp_mpi_shift_r( X, k - nbits + 1 ) );
|
|
|
|
esp_mpi_set_bit( X, nbits-1, 1 );
|
|
|
|
X->p[0] |= 1;
|
|
|
|
if( dh_flag == 0 )
|
|
{
|
|
while( ( ret = esp_mpi_is_prime( X, f_rng, p_rng ) ) != 0 )
|
|
{
|
|
if( ret != ERR_MPI_NOT_ACCEPTABLE )
|
|
goto cleanup;
|
|
|
|
MPI_CHK( esp_mpi_add_int( X, X, 2 ) );
|
|
}
|
|
}
|
|
else
|
|
{
|
|
/*
|
|
* An necessary condition for Y and X = 2Y + 1 to be prime
|
|
* is X = 2 mod 3 (which is equivalent to Y = 2 mod 3).
|
|
* Make sure it is satisfied, while keeping X = 3 mod 4
|
|
*/
|
|
|
|
X->p[0] |= 2;
|
|
|
|
MPI_CHK( esp_mpi_mod_int( &r, X, 3 ) );
|
|
if( r == 0 )
|
|
MPI_CHK( esp_mpi_add_int( X, X, 8 ) );
|
|
else if( r == 1 )
|
|
MPI_CHK( esp_mpi_add_int( X, X, 4 ) );
|
|
|
|
/* Set Y = (X-1) / 2, which is X / 2 because X is odd */
|
|
MPI_CHK( esp_mpi_copy( &Y, X ) );
|
|
MPI_CHK( esp_mpi_shift_r( &Y, 1 ) );
|
|
|
|
while( 1 )
|
|
{
|
|
/*
|
|
* First, check small factors for X and Y
|
|
* before doing Miller-Rabin on any of them
|
|
*/
|
|
if( ( ret = esp_mpi_check_small_factors( X ) ) == 0 &&
|
|
( ret = esp_mpi_check_small_factors( &Y ) ) == 0 &&
|
|
( ret = esp_mpi_miller_rabin( X, f_rng, p_rng ) ) == 0 &&
|
|
( ret = esp_mpi_miller_rabin( &Y, f_rng, p_rng ) ) == 0 )
|
|
{
|
|
break;
|
|
}
|
|
|
|
if( ret != ERR_MPI_NOT_ACCEPTABLE )
|
|
goto cleanup;
|
|
|
|
/*
|
|
* Next candidates. We want to preserve Y = (X-1) / 2 and
|
|
* Y = 1 mod 2 and Y = 2 mod 3 (eq X = 3 mod 4 and X = 2 mod 3)
|
|
* so up Y by 6 and X by 12.
|
|
*/
|
|
MPI_CHK( esp_mpi_add_int( X, X, 12 ) );
|
|
MPI_CHK( esp_mpi_add_int( &Y, &Y, 6 ) );
|
|
}
|
|
}
|
|
|
|
cleanup:
|
|
|
|
esp_mpi_free( &Y );
|
|
|
|
return( ret );
|
|
}
|
|
|