library/constant_time.c

/**
 *  Constant-time functions
 *
 *  Copyright The Mbed TLS Contributors
 *  SPDX-License-Identifier: Apache-2.0
 *
 *  Licensed under the Apache License, Version 2.0 (the "License"); you may
 *  not use this file except in compliance with the License.
 *  You may obtain a copy of the License at
 *
 *  http://www.apache.org/licenses/LICENSE-2.0
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
 *  WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 */

 /*
 * The following functiona are implemented without using comparison operators, as those
 * might be translated to branches by some compilers on some platforms.
 */

#include "common.h"
#include "constant_time.h"
#include "mbedtls/error.h"
#include "mbedtls/platform_util.h"

#if defined(MBEDTLS_BIGNUM_C)
#include "mbedtls/bignum.h"
#endif

#if defined(MBEDTLS_SSL_TLS_C)
#include "ssl_misc.h"
#endif

#if defined(MBEDTLS_RSA_C)
#include "mbedtls/rsa.h"
#endif

#include <string.h>

int mbedtls_cf_memcmp( const void *a,
                       const void *b,
                       size_t n )
{
    size_t i;
    volatile const unsigned char *A = (volatile const unsigned char *) a;
    volatile const unsigned char *B = (volatile const unsigned char *) b;
    volatile unsigned char diff = 0;

    for( i = 0; i < n; i++ )
    {
        /* Read volatile data in order before computing diff.
         * This avoids IAR compiler warning:
         * 'the order of volatile accesses is undefined ..' */
        unsigned char x = A[i], y = B[i];
        diff |= x ^ y;
    }

    return( (int)diff );
}

unsigned mbedtls_cf_uint_mask( unsigned value )
{
    /* MSVC has a warning about unary minus on unsigned, but this is
     * well-defined and precisely what we want to do here */
#if defined(_MSC_VER)
#pragma warning( push )
#pragma warning( disable : 4146 )
#endif
    return( - ( ( value | - value ) >> ( sizeof( value ) * 8 - 1 ) ) );
#if defined(_MSC_VER)
#pragma warning( pop )
#endif
}

size_t mbedtls_cf_size_mask( size_t value )
{
    /* MSVC has a warning about unary minus on unsigned integer types,
     * but this is well-defined and precisely what we want to do here. */
#if defined(_MSC_VER)
#pragma warning( push )
#pragma warning( disable : 4146 )
#endif
    return( - ( ( value | - value ) >> ( sizeof( value ) * 8 - 1 ) ) );
#if defined(_MSC_VER)
#pragma warning( pop )
#endif
}

#if defined(MBEDTLS_BIGNUM_C)

mbedtls_mpi_uint mbedtls_cf_mpi_uint_mask( mbedtls_mpi_uint value )
{
    /* MSVC has a warning about unary minus on unsigned, but this is
     * well-defined and precisely what we want to do here */
#if defined(_MSC_VER)
#pragma warning( push )
#pragma warning( disable : 4146 )
#endif
    return( - ( ( value | - value ) >> ( sizeof( value ) * 8 - 1 ) ) );
#if defined(_MSC_VER)
#pragma warning( pop )
#endif
}

#endif /* MBEDTLS_BIGNUM_C */

/** Constant-flow mask generation for "less than" comparison:
 * - if \p x < \p y, return all-bits 1, that is (size_t) -1
 * - otherwise, return all bits 0, that is 0
 *
 * This function can be used to write constant-time code by replacing branches
 * with bit operations using masks.
 *
 * \param x     The first value to analyze.
 * \param y     The second value to analyze.
 *
 * \return      All-bits-one if \p x is less than \p y, otherwise zero.
 */
static size_t mbedtls_cf_size_mask_lt( size_t x,
                                       size_t y )
{
    /* This has the most significant bit set if and only if x < y */
    const size_t sub = x - y;

    /* sub1 = (x < y) ? 1 : 0 */
    const size_t sub1 = sub >> ( sizeof( sub ) * 8 - 1 );

    /* mask = (x < y) ? 0xff... : 0x00... */
    const size_t mask = mbedtls_cf_size_mask( sub1 );

    return( mask );
}

size_t mbedtls_cf_size_mask_ge( size_t x,
                                size_t y )
{
    return( ~mbedtls_cf_size_mask_lt( x, y ) );
}

unsigned mbedtls_cf_size_bool_eq( size_t x,
                                  size_t y )
{
    /* diff = 0 if x == y, non-zero otherwise */
    const size_t diff = x ^ y;

    /* MSVC has a warning about unary minus on unsigned integer types,
     * but this is well-defined and precisely what we want to do here. */
#if defined(_MSC_VER)
#pragma warning( push )
#pragma warning( disable : 4146 )
#endif

    /* diff_msb's most significant bit is equal to x != y */
    const size_t diff_msb = ( diff | (size_t) -diff );

#if defined(_MSC_VER)
#pragma warning( pop )
#endif

    /* diff1 = (x != y) ? 1 : 0 */
    const unsigned diff1 = diff_msb >> ( sizeof( diff_msb ) * 8 - 1 );

    return( 1 ^ diff1 );
}

unsigned mbedtls_cf_size_gt( size_t x,
                             size_t y )
{
    /* Return the sign bit (1 for negative) of (y - x). */
    return( ( y - x ) >> ( sizeof( size_t ) * 8 - 1 ) );
}

#if defined(MBEDTLS_BIGNUM_C)

unsigned mbedtls_cf_mpi_uint_lt( const mbedtls_mpi_uint x,
                                 const mbedtls_mpi_uint y )
{
    mbedtls_mpi_uint ret;
    mbedtls_mpi_uint cond;

    /*
     * Check if the most significant bits (MSB) of the operands are different.
     */
    cond = ( x ^ y );
    /*
     * If the MSB are the same then the difference x-y will be negative (and
     * have its MSB set to 1 during conversion to unsigned) if and only if x<y.
     */
    ret = ( x - y ) & ~cond;
    /*
     * If the MSB are different, then the operand with the MSB of 1 is the
     * bigger. (That is if y has MSB of 1, then x<y is true and it is false if
     * the MSB of y is 0.)
     */
    ret |= y & cond;


    ret = ret >> ( sizeof( mbedtls_mpi_uint ) * 8 - 1 );

    return (unsigned) ret;
}

#endif /* MBEDTLS_BIGNUM_C */

unsigned mbedtls_cf_uint_if( unsigned condition,
                             unsigned if1,
                             unsigned if0 )
{
    unsigned mask = mbedtls_cf_uint_mask( condition );
    return( ( mask & if1 ) | (~mask & if0 ) );
}

size_t mbedtls_cf_size_if( unsigned condition,
                           size_t if1,
                           size_t if0 )
{
    size_t mask = mbedtls_cf_size_mask( condition );
    return( ( mask & if1 ) | (~mask & if0 ) );
}

int mbedtls_cf_cond_select_sign( unsigned char condition,
                                 int if1,
                                 int if0 )
{
    /* In order to avoid questions about what we can reasonnably assume about
     * the representations of signed integers, move everything to unsigned
     * by taking advantage of the fact that a and b are either +1 or -1. */
    unsigned uif1 = if1 + 1;
    unsigned uif0 = if0 + 1;

    /* condition was 0 or 1, mask is 0 or 2 as are ua and ub */
    const unsigned mask = condition << 1;

    /* select ua or ub */
    unsigned ur = ( uif0 & ~mask ) | ( uif1 & mask );

    /* ur is now 0 or 2, convert back to -1 or +1 */
    return( (int) ur - 1 );
}

#if defined(MBEDTLS_BIGNUM_C)

void mbedtls_cf_mpi_uint_cond_assign( size_t n,
                                      mbedtls_mpi_uint *dest,
                                      const mbedtls_mpi_uint *src,
                                      unsigned char condition )
{
    size_t i;

    /* MSVC has a warning about unary minus on unsigned integer types,
     * but this is well-defined and precisely what we want to do here. */
#if defined(_MSC_VER)
#pragma warning( push )
#pragma warning( disable : 4146 )
#endif

    /* all-bits 1 if condition is 1, all-bits 0 if condition is 0 */
    const mbedtls_mpi_uint mask = -condition;

#if defined(_MSC_VER)
#pragma warning( pop )
#endif

    for( i = 0; i < n; i++ )
        dest[i] = ( src[i] & mask ) | ( dest[i] & ~mask );
}

#endif /* MBEDTLS_BIGNUM_C */

void mbedtls_cf_mem_move_to_left( void *start,
                                  size_t total,
                                  size_t offset )
{
    volatile unsigned char *buf = start;
    size_t i, n;
    if( total == 0 )
        return;
    for( i = 0; i < total; i++ )
    {
        unsigned no_op = mbedtls_cf_size_gt( total - offset, i );
        /* The first `total - offset` passes are a no-op. The last
         * `offset` passes shift the data one byte to the left and
         * zero out the last byte. */
        for( n = 0; n < total - 1; n++ )
        {
            unsigned char current = buf[n];
            unsigned char next = buf[n+1];
            buf[n] = mbedtls_cf_uint_if( no_op, current, next );
        }
        buf[total-1] = mbedtls_cf_uint_if( no_op, buf[total-1], 0 );
    }
}

void mbedtls_cf_memcpy_if_eq( unsigned char *dest,
                              const unsigned char *src,
                              size_t len,
                              size_t c1,
                              size_t c2 )
{
    /* mask = c1 == c2 ? 0xff : 0x00 */
    const size_t equal = mbedtls_cf_size_bool_eq( c1, c2 );
    const unsigned char mask = (unsigned char) mbedtls_cf_size_mask( equal );

    /* dest[i] = c1 == c2 ? src[i] : dest[i] */
    for( size_t i = 0; i < len; i++ )
        dest[i] = ( src[i] & mask ) | ( dest[i] & ~mask );
}

void mbedtls_cf_memcpy_offset( unsigned char *dest,
                               const unsigned char *src,
                               size_t offset,
                               size_t offset_min,
                               size_t offset_max,
                               size_t len )
{
    size_t offsetval;

    for( offsetval = offset_min; offsetval <= offset_max; offsetval++ )
    {
        mbedtls_cf_memcpy_if_eq( dest, src + offsetval, len,
                                 offsetval, offset );
    }
}

#if defined(MBEDTLS_SSL_SOME_SUITES_USE_TLS_CBC)

int mbedtls_cf_hmac( mbedtls_md_context_t *ctx,
                     const unsigned char *add_data,
                     size_t add_data_len,
                     const unsigned char *data,
                     size_t data_len_secret,
                     size_t min_data_len,
                     size_t max_data_len,
                     unsigned char *output )
{
    /*
     * This function breaks the HMAC abstraction and uses the md_clone()
     * extension to the MD API in order to get constant-flow behaviour.
     *
     * HMAC(msg) is defined as HASH(okey + HASH(ikey + msg)) where + means
     * concatenation, and okey/ikey are the XOR of the key with some fixed bit
     * patterns (see RFC 2104, sec. 2), which are stored in ctx->hmac_ctx.
     *
     * We'll first compute inner_hash = HASH(ikey + msg) by hashing up to
     * minlen, then cloning the context, and for each byte up to maxlen
     * finishing up the hash computation, keeping only the correct result.
     *
     * Then we only need to compute HASH(okey + inner_hash) and we're done.
     */
    const mbedtls_md_type_t md_alg = mbedtls_md_get_type( ctx->md_info );
    /* TLS 1.2 only supports SHA-384, SHA-256, SHA-1, MD-5,
     * all of which have the same block size except SHA-384. */
    const size_t block_size = md_alg == MBEDTLS_MD_SHA384 ? 128 : 64;
    const unsigned char * const ikey = ctx->hmac_ctx;
    const unsigned char * const okey = ikey + block_size;
    const size_t hash_size = mbedtls_md_get_size( ctx->md_info );

    unsigned char aux_out[MBEDTLS_MD_MAX_SIZE];
    mbedtls_md_context_t aux;
    size_t offset;
    int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;

    mbedtls_md_init( &aux );

#define MD_CHK( func_call ) \
    do {                    \
        ret = (func_call);  \
        if( ret != 0 )      \
            goto cleanup;   \
    } while( 0 )

    MD_CHK( mbedtls_md_setup( &aux, ctx->md_info, 0 ) );

    /* After hmac_start() of hmac_reset(), ikey has already been hashed,
     * so we can start directly with the message */
    MD_CHK( mbedtls_md_update( ctx, add_data, add_data_len ) );
    MD_CHK( mbedtls_md_update( ctx, data, min_data_len ) );

    /* For each possible length, compute the hash up to that point */
    for( offset = min_data_len; offset <= max_data_len; offset++ )
    {
        MD_CHK( mbedtls_md_clone( &aux, ctx ) );
        MD_CHK( mbedtls_md_finish( &aux, aux_out ) );
        /* Keep only the correct inner_hash in the output buffer */
        mbedtls_cf_memcpy_if_eq( output, aux_out, hash_size,
                                 offset, data_len_secret );

        if( offset < max_data_len )
            MD_CHK( mbedtls_md_update( ctx, data + offset, 1 ) );
    }

    /* The context needs to finish() before it starts() again */
    MD_CHK( mbedtls_md_finish( ctx, aux_out ) );

    /* Now compute HASH(okey + inner_hash) */
    MD_CHK( mbedtls_md_starts( ctx ) );
    MD_CHK( mbedtls_md_update( ctx, okey, block_size ) );
    MD_CHK( mbedtls_md_update( ctx, output, hash_size ) );
    MD_CHK( mbedtls_md_finish( ctx, output ) );

    /* Done, get ready for next time */
    MD_CHK( mbedtls_md_hmac_reset( ctx ) );

#undef MD_CHK

cleanup:
    mbedtls_md_free( &aux );
    return( ret );
}

#endif /* MBEDTLS_SSL_SOME_SUITES_USE_TLS_CBC */

#if defined(MBEDTLS_BIGNUM_C)

#define MPI_VALIDATE_RET( cond )                                       \
    MBEDTLS_INTERNAL_VALIDATE_RET( cond, MBEDTLS_ERR_MPI_BAD_INPUT_DATA )

/*
 * Conditionally assign X = Y, without leaking information
 * about whether the assignment was made or not.
 * (Leaking information about the respective sizes of X and Y is ok however.)
 */
int mbedtls_mpi_safe_cond_assign( mbedtls_mpi *X,
                                  const mbedtls_mpi *Y,
                                  unsigned char assign )
{
    int ret = 0;
    size_t i;
    mbedtls_mpi_uint limb_mask;
    MPI_VALIDATE_RET( X != NULL );
    MPI_VALIDATE_RET( Y != NULL );

    /* all-bits 1 if assign is 1, all-bits 0 if assign is 0 */
    limb_mask = mbedtls_cf_mpi_uint_mask( assign );;

    MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, Y->n ) );

    X->s = mbedtls_cf_cond_select_sign( assign, Y->s, X->s );

    mbedtls_cf_mpi_uint_cond_assign( Y->n, X->p, Y->p, assign );

    for( i = Y->n; i < X->n; i++ )
        X->p[i] &= ~limb_mask;

cleanup:
    return( ret );
}

/*
 * Conditionally swap X and Y, without leaking information
 * about whether the swap was made or not.
 * Here it is not ok to simply swap the pointers, which whould lead to
 * different memory access patterns when X and Y are used afterwards.
 */
int mbedtls_mpi_safe_cond_swap( mbedtls_mpi *X,
                                mbedtls_mpi *Y,
                                unsigned char swap )
{
    int ret, s;
    size_t i;
    mbedtls_mpi_uint limb_mask;
    mbedtls_mpi_uint tmp;
    MPI_VALIDATE_RET( X != NULL );
    MPI_VALIDATE_RET( Y != NULL );

    if( X == Y )
        return( 0 );

    /* all-bits 1 if swap is 1, all-bits 0 if swap is 0 */
    limb_mask = mbedtls_cf_mpi_uint_mask( swap );

    MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, Y->n ) );
    MBEDTLS_MPI_CHK( mbedtls_mpi_grow( Y, X->n ) );

    s = X->s;
    X->s = mbedtls_cf_cond_select_sign( swap, Y->s, X->s );
    Y->s = mbedtls_cf_cond_select_sign( swap, s, Y->s );


    for( i = 0; i < X->n; i++ )
    {
        tmp = X->p[i];
        X->p[i] = ( X->p[i] & ~limb_mask ) | ( Y->p[i] & limb_mask );
        Y->p[i] = ( Y->p[i] & ~limb_mask ) | (     tmp & limb_mask );
    }

cleanup:
    return( ret );
}

/*
 * Compare signed values in constant time
 */
int mbedtls_mpi_lt_mpi_ct( const mbedtls_mpi *X,
                           const mbedtls_mpi *Y,
                           unsigned *ret )
{
    size_t i;
    /* The value of any of these variables is either 0 or 1 at all times. */
    unsigned cond, done, X_is_negative, Y_is_negative;

    MPI_VALIDATE_RET( X != NULL );
    MPI_VALIDATE_RET( Y != NULL );
    MPI_VALIDATE_RET( ret != NULL );

    if( X->n != Y->n )
        return MBEDTLS_ERR_MPI_BAD_INPUT_DATA;

    /*
     * Set sign_N to 1 if N >= 0, 0 if N < 0.
     * We know that N->s == 1 if N >= 0 and N->s == -1 if N < 0.
     */
    X_is_negative = ( X->s & 2 ) >> 1;
    Y_is_negative = ( Y->s & 2 ) >> 1;

    /*
     * If the signs are different, then the positive operand is the bigger.
     * That is if X is negative (X_is_negative == 1), then X < Y is true and it
     * is false if X is positive (X_is_negative == 0).
     */
    cond = ( X_is_negative ^ Y_is_negative );
    *ret = cond & X_is_negative;

    /*
     * This is a constant-time function. We might have the result, but we still
     * need to go through the loop. Record if we have the result already.
     */
    done = cond;

    for( i = X->n; i > 0; i-- )
    {
        /*
         * If Y->p[i - 1] < X->p[i - 1] then X < Y is true if and only if both
         * X and Y are negative.
         *
         * Again even if we can make a decision, we just mark the result and
         * the fact that we are done and continue looping.
         */
        cond = mbedtls_cf_mpi_uint_lt( Y->p[i - 1], X->p[i - 1] );
        *ret |= cond & ( 1 - done ) & X_is_negative;
        done |= cond;

        /*
         * If X->p[i - 1] < Y->p[i - 1] then X < Y is true if and only if both
         * X and Y are positive.
         *
         * Again even if we can make a decision, we just mark the result and
         * the fact that we are done and continue looping.
         */
        cond = mbedtls_cf_mpi_uint_lt( X->p[i - 1], Y->p[i - 1] );
        *ret |= cond & ( 1 - done ) & ( 1 - X_is_negative );
        done |= cond;
    }

    return( 0 );
}

#endif /* MBEDTLS_BIGNUM_C */

#if defined(MBEDTLS_PKCS1_V15) && defined(MBEDTLS_RSA_C) && !defined(MBEDTLS_RSA_ALT)

int mbedtls_cf_rsaes_pkcs1_v15_unpadding( unsigned char *input,
                                          size_t ilen,
                                          unsigned char *output,
                                          size_t output_max_len,
                                          size_t *olen )
{
    int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED;
    size_t i, plaintext_max_size;

    /* The following variables take sensitive values: their value must
     * not leak into the observable behavior of the function other than
     * the designated outputs (output, olen, return value). Otherwise
     * this would open the execution of the function to
     * side-channel-based variants of the Bleichenbacher padding oracle
     * attack. Potential side channels include overall timing, memory
     * access patterns (especially visible to an adversary who has access
     * to a shared memory cache), and branches (especially visible to
     * an adversary who has access to a shared code cache or to a shared
     * branch predictor). */
    size_t pad_count = 0;
    unsigned bad = 0;
    unsigned char pad_done = 0;
    size_t plaintext_size = 0;
    unsigned output_too_large;

    plaintext_max_size = ( output_max_len > ilen - 11 ) ? ilen - 11
                                                        : output_max_len;

    /* Check and get padding length in constant time and constant
     * memory trace. The first byte must be 0. */
    bad |= input[0];


    /* Decode EME-PKCS1-v1_5 padding: 0x00 || 0x02 || PS || 0x00
     * where PS must be at least 8 nonzero bytes. */
    bad |= input[1] ^ MBEDTLS_RSA_CRYPT;

    /* Read the whole buffer. Set pad_done to nonzero if we find
     * the 0x00 byte and remember the padding length in pad_count. */
    for( i = 2; i < ilen; i++ )
    {
        pad_done  |= ((input[i] | (unsigned char)-input[i]) >> 7) ^ 1;
        pad_count += ((pad_done | (unsigned char)-pad_done) >> 7) ^ 1;
    }


    /* If pad_done is still zero, there's no data, only unfinished padding. */
    bad |= mbedtls_cf_uint_if( pad_done, 0, 1 );

    /* There must be at least 8 bytes of padding. */
    bad |= mbedtls_cf_size_gt( 8, pad_count );

    /* If the padding is valid, set plaintext_size to the number of
     * remaining bytes after stripping the padding. If the padding
     * is invalid, avoid leaking this fact through the size of the
     * output: use the maximum message size that fits in the output
     * buffer. Do it without branches to avoid leaking the padding
     * validity through timing. RSA keys are small enough that all the
     * size_t values involved fit in unsigned int. */
    plaintext_size = mbedtls_cf_uint_if(
                        bad, (unsigned) plaintext_max_size,
                        (unsigned) ( ilen - pad_count - 3 ) );

    /* Set output_too_large to 0 if the plaintext fits in the output
     * buffer and to 1 otherwise. */
    output_too_large = mbedtls_cf_size_gt( plaintext_size,
                                           plaintext_max_size );

    /* Set ret without branches to avoid timing attacks. Return:
     * - INVALID_PADDING if the padding is bad (bad != 0).
     * - OUTPUT_TOO_LARGE if the padding is good but the decrypted
     *   plaintext does not fit in the output buffer.
     * - 0 if the padding is correct. */
    ret = - (int) mbedtls_cf_uint_if(
                    bad, - MBEDTLS_ERR_RSA_INVALID_PADDING,
                    mbedtls_cf_uint_if( output_too_large,
                                        - MBEDTLS_ERR_RSA_OUTPUT_TOO_LARGE,
                                        0 ) );

    /* If the padding is bad or the plaintext is too large, zero the
     * data that we're about to copy to the output buffer.
     * We need to copy the same amount of data
     * from the same buffer whether the padding is good or not to
     * avoid leaking the padding validity through overall timing or
     * through memory or cache access patterns. */
    bad = mbedtls_cf_uint_mask( bad | output_too_large );
    for( i = 11; i < ilen; i++ )
        input[i] &= ~bad;

    /* If the plaintext is too large, truncate it to the buffer size.
     * Copy anyway to avoid revealing the length through timing, because
     * revealing the length is as bad as revealing the padding validity
     * for a Bleichenbacher attack. */
    plaintext_size = mbedtls_cf_uint_if( output_too_large,
                                         (unsigned) plaintext_max_size,
                                         (unsigned) plaintext_size );

    /* Move the plaintext to the leftmost position where it can start in
     * the working buffer, i.e. make it start plaintext_max_size from
     * the end of the buffer. Do this with a memory access trace that
     * does not depend on the plaintext size. After this move, the
     * starting location of the plaintext is no longer sensitive
     * information. */
    mbedtls_cf_mem_move_to_left( input + ilen - plaintext_max_size,
                                 plaintext_max_size,
                                 plaintext_max_size - plaintext_size );

    /* Finally copy the decrypted plaintext plus trailing zeros into the output
     * buffer. If output_max_len is 0, then output may be an invalid pointer
     * and the result of memcpy() would be undefined; prevent undefined
     * behavior making sure to depend only on output_max_len (the size of the
     * user-provided output buffer), which is independent from plaintext
     * length, validity of padding, success of the decryption, and other
     * secrets. */
    if( output_max_len != 0 )
        memcpy( output, input + ilen - plaintext_max_size, plaintext_max_size );

    /* Report the amount of data we copied to the output buffer. In case
     * of errors (bad padding or output too large), the value of *olen
     * when this function returns is not specified. Making it equivalent
     * to the good case limits the risks of leaking the padding validity. */
    *olen = plaintext_size;

    return( ret );
}

#endif /* MBEDTLS_PKCS1_V15 && MBEDTLS_RSA_C && ! MBEDTLS_RSA_ALT */