bn_exp.c

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/* crypto/bn/bn_exp.c */
/* Copyright (C) 1995-1998 Eric Young ([email protected])
 * All rights reserved.
 *
 * This package is an SSL implementation written
 * by Eric Young ([email protected]).
 * The implementation was written so as to conform with Netscapes SSL.
 * 
 * This library is free for commercial and non-commercial use as long as
 * the following conditions are aheared to.  The following conditions
 * apply to all code found in this distribution, be it the RC4, RSA,
 * lhash, DES, etc., code; not just the SSL code.  The SSL documentation
 * included with this distribution is covered by the same copyright terms
 * except that the holder is Tim Hudson ([email protected]).
 * 
 * Copyright remains Eric Young's, and as such any Copyright notices in
 * the code are not to be removed.
 * If this package is used in a product, Eric Young should be given attribution
 * as the author of the parts of the library used.
 * This can be in the form of a textual message at program startup or
 * in documentation (online or textual) provided with the package.
 * 
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. All advertising materials mentioning features or use of this software
 *    must display the following acknowledgement:
 *    "This product includes cryptographic software written by
 *     Eric Young ([email protected])"
 *    The word 'cryptographic' can be left out if the rouines from the library
 *    being used are not cryptographic related :-).
 * 4. If you include any Windows specific code (or a derivative thereof) from 
 *    the apps directory (application code) you must include an acknowledgement:
 *    "This product includes software written by Tim Hudson ([email protected])"
 * 
 * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 * 
 * The licence and distribution terms for any publically available version or
 * derivative of this code cannot be changed.  i.e. this code cannot simply be
 * copied and put under another distribution licence
 * [including the GNU Public Licence.]
 */
/* ====================================================================
 * Copyright (c) 1998-2005 The OpenSSL Project.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer. 
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 *
 * 3. All advertising materials mentioning features or use of this
 *    software must display the following acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
 *
 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
 *    endorse or promote products derived from this software without
 *    prior written permission. For written permission, please contact
 *    [email protected].
 *
 * 5. Products derived from this software may not be called "OpenSSL"
 *    nor may "OpenSSL" appear in their names without prior written
 *    permission of the OpenSSL Project.
 *
 * 6. Redistributions of any form whatsoever must retain the following
 *    acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
 *
 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
 * OF THE POSSIBILITY OF SUCH DAMAGE.
 * ====================================================================
 *
 * This product includes cryptographic software written by Eric Young
 * ([email protected]).  This product includes software written by Tim
 * Hudson ([email protected]).
 *
 */


#include "cryptlib.h"
#include "bn_lcl.h"

#include <stdlib.h>
#ifdef _WIN32
# include <malloc.h>
# ifndef alloca
#  define alloca _alloca
# endif
#elif defined(__GNUC__)
# ifndef alloca
#  define alloca(s) __builtin_alloca((s))
# endif
#endif

/* maximum precomputation table size for *variable* sliding windows */
#define TABLE_SIZE  32

/* this one works - simple but works */
int BN_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
    {
    int i,bits,ret=0;
    BIGNUM *v,*rr;

    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0)
        {
        /* BN_FLG_CONSTTIME only supported by BN_mod_exp_mont() */
        BNerr(BN_F_BN_EXP,ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
        return -1;
        }

    BN_CTX_start(ctx);
    if ((r == a) || (r == p))
        rr = BN_CTX_get(ctx);
    else
        rr = r;
    v = BN_CTX_get(ctx);
    if (rr == NULL || v == NULL) goto err;

    if (BN_copy(v,a) == NULL) goto err;
    bits=BN_num_bits(p);

    if (BN_is_odd(p))
        { if (BN_copy(rr,a) == NULL) goto err; }
    else    { if (!BN_one(rr)) goto err; }

    for (i=1; i<bits; i++)
        {
        if (!BN_sqr(v,v,ctx)) goto err;
        if (BN_is_bit_set(p,i))
            {
            if (!BN_mul(rr,rr,v,ctx)) goto err;
            }
        }
    ret=1;
err:
    if (r != rr) BN_copy(r,rr);
    BN_CTX_end(ctx);
    bn_check_top(r);
    return(ret);
    }


int BN_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
           BN_CTX *ctx)
    {
    int ret;

    bn_check_top(a);
    bn_check_top(p);
    bn_check_top(m);

    /* For even modulus  m = 2^k*m_odd,  it might make sense to compute
     * a^p mod m_odd  and  a^p mod 2^k  separately (with Montgomery
     * exponentiation for the odd part), using appropriate exponent
     * reductions, and combine the results using the CRT.
     *
     * For now, we use Montgomery only if the modulus is odd; otherwise,
     * exponentiation using the reciprocal-based quick remaindering
     * algorithm is used.
     *
     * (Timing obtained with expspeed.c [computations  a^p mod m
     * where  a, p, m  are of the same length: 256, 512, 1024, 2048,
     * 4096, 8192 bits], compared to the running time of the
     * standard algorithm:
     *
     *   BN_mod_exp_mont   33 .. 40 %  [AMD K6-2, Linux, debug configuration]
         *                     55 .. 77 %  [UltraSparc processor, but
     *                                  debug-solaris-sparcv8-gcc conf.]
     * 
     *   BN_mod_exp_recp   50 .. 70 %  [AMD K6-2, Linux, debug configuration]
     *                     62 .. 118 % [UltraSparc, debug-solaris-sparcv8-gcc]
     *
     * On the Sparc, BN_mod_exp_recp was faster than BN_mod_exp_mont
     * at 2048 and more bits, but at 512 and 1024 bits, it was
     * slower even than the standard algorithm!
     *
     * "Real" timings [linux-elf, solaris-sparcv9-gcc configurations]
     * should be obtained when the new Montgomery reduction code
     * has been integrated into OpenSSL.)
     */

#define MONT_MUL_MOD
#define MONT_EXP_WORD
#define RECP_MUL_MOD

#ifdef MONT_MUL_MOD
    /* I have finally been able to take out this pre-condition of
     * the top bit being set.  It was caused by an error in BN_div
     * with negatives.  There was also another problem when for a^b%m
     * a >= m.  eay 07-May-97 */
/*  if ((m->d[m->top-1]&BN_TBIT) && BN_is_odd(m)) */

    if (BN_is_odd(m))
        {
#  ifdef MONT_EXP_WORD
        if (a->top == 1 && !a->neg && (BN_get_flags(p, BN_FLG_CONSTTIME) == 0))
            {
            BN_ULONG A = a->d[0];
            ret=BN_mod_exp_mont_word(r,A,p,m,ctx,NULL);
            }
        else
#  endif
            ret=BN_mod_exp_mont(r,a,p,m,ctx,NULL);
        }
    else
#endif
#ifdef RECP_MUL_MOD
        { ret=BN_mod_exp_recp(r,a,p,m,ctx); }
#else
        { ret=BN_mod_exp_simple(r,a,p,m,ctx); }
#endif

    bn_check_top(r);
    return(ret);
    }


int BN_mod_exp_recp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
            const BIGNUM *m, BN_CTX *ctx)
    {
    int i,j,bits,ret=0,wstart,wend,window,wvalue;
    int start=1;
    BIGNUM *aa;
    /* Table of variables obtained from 'ctx' */
    BIGNUM *val[TABLE_SIZE];
    BN_RECP_CTX recp;

    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0)
        {
        /* BN_FLG_CONSTTIME only supported by BN_mod_exp_mont() */
        BNerr(BN_F_BN_MOD_EXP_RECP,ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
        return -1;
        }

    bits=BN_num_bits(p);

    if (bits == 0)
        {
        ret = BN_one(r);
        return ret;
        }

    BN_CTX_start(ctx);
    aa = BN_CTX_get(ctx);
    val[0] = BN_CTX_get(ctx);
    if(!aa || !val[0]) goto err;

    BN_RECP_CTX_init(&recp);
    if (m->neg)
        {
        /* ignore sign of 'm' */
        if (!BN_copy(aa, m)) goto err;
        aa->neg = 0;
        if (BN_RECP_CTX_set(&recp,aa,ctx) <= 0) goto err;
        }
    else
        {
        if (BN_RECP_CTX_set(&recp,m,ctx) <= 0) goto err;
        }

    if (!BN_nnmod(val[0],a,m,ctx)) goto err;        /* 1 */
    if (BN_is_zero(val[0]))
        {
        BN_zero(r);
        ret = 1;
        goto err;
        }

    window = BN_window_bits_for_exponent_size(bits);
    if (window > 1)
        {
        if (!BN_mod_mul_reciprocal(aa,val[0],val[0],&recp,ctx))
            goto err;               /* 2 */
        j=1<<(window-1);
        for (i=1; i<j; i++)
            {
            if(((val[i] = BN_CTX_get(ctx)) == NULL) ||
                    !BN_mod_mul_reciprocal(val[i],val[i-1],
                        aa,&recp,ctx))
                goto err;
            }
        }
        
    start=1;    /* This is used to avoid multiplication etc
             * when there is only the value '1' in the
             * buffer. */
    wvalue=0;   /* The 'value' of the window */
    wstart=bits-1;  /* The top bit of the window */
    wend=0;     /* The bottom bit of the window */

    if (!BN_one(r)) goto err;

    for (;;)
        {
        if (BN_is_bit_set(p,wstart) == 0)
            {
            if (!start)
                if (!BN_mod_mul_reciprocal(r,r,r,&recp,ctx))
                goto err;
            if (wstart == 0) break;
            wstart--;
            continue;
            }
        /* We now have wstart on a 'set' bit, we now need to work out
         * how bit a window to do.  To do this we need to scan
         * forward until the last set bit before the end of the
         * window */
        j=wstart;
        wvalue=1;
        wend=0;
        for (i=1; i<window; i++)
            {
            if (wstart-i < 0) break;
            if (BN_is_bit_set(p,wstart-i))
                {
                wvalue<<=(i-wend);
                wvalue|=1;
                wend=i;
                }
            }

        /* wend is the size of the current window */
        j=wend+1;
        /* add the 'bytes above' */
        if (!start)
            for (i=0; i<j; i++)
                {
                if (!BN_mod_mul_reciprocal(r,r,r,&recp,ctx))
                    goto err;
                }
        
        /* wvalue will be an odd number < 2^window */
        if (!BN_mod_mul_reciprocal(r,r,val[wvalue>>1],&recp,ctx))
            goto err;

        /* move the 'window' down further */
        wstart-=wend+1;
        wvalue=0;
        start=0;
        if (wstart < 0) break;
        }
    ret=1;
err:
    BN_CTX_end(ctx);
    BN_RECP_CTX_free(&recp);
    bn_check_top(r);
    return(ret);
    }


int BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
            const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont)
    {
    int i,j,bits,ret=0,wstart,wend,window,wvalue;
    int start=1;
    BIGNUM *d,*r;
    const BIGNUM *aa;
    /* Table of variables obtained from 'ctx' */
    BIGNUM *val[TABLE_SIZE];
    BN_MONT_CTX *mont=NULL;

    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0)
        {
        return BN_mod_exp_mont_consttime(rr, a, p, m, ctx, in_mont);
        }

    bn_check_top(a);
    bn_check_top(p);
    bn_check_top(m);

    if (!BN_is_odd(m))
        {
        BNerr(BN_F_BN_MOD_EXP_MONT,BN_R_CALLED_WITH_EVEN_MODULUS);
        return(0);
        }
    bits=BN_num_bits(p);
    if (bits == 0)
        {
        ret = BN_one(rr);
        return ret;
        }

    BN_CTX_start(ctx);
    d = BN_CTX_get(ctx);
    r = BN_CTX_get(ctx);
    val[0] = BN_CTX_get(ctx);
    if (!d || !r || !val[0]) goto err;

    /* If this is not done, things will break in the montgomery
     * part */

    if (in_mont != NULL)
        mont=in_mont;
    else
        {
        if ((mont=BN_MONT_CTX_new()) == NULL) goto err;
        if (!BN_MONT_CTX_set(mont,m,ctx)) goto err;
        }

    if (a->neg || BN_ucmp(a,m) >= 0)
        {
        if (!BN_nnmod(val[0],a,m,ctx))
            goto err;
        aa= val[0];
        }
    else
        aa=a;
    if (BN_is_zero(aa))
        {
        BN_zero(rr);
        ret = 1;
        goto err;
        }
    if (!BN_to_montgomery(val[0],aa,mont,ctx)) goto err; /* 1 */

    window = BN_window_bits_for_exponent_size(bits);
    if (window > 1)
        {
        if (!BN_mod_mul_montgomery(d,val[0],val[0],mont,ctx)) goto err; /* 2 */
        j=1<<(window-1);
        for (i=1; i<j; i++)
            {
            if(((val[i] = BN_CTX_get(ctx)) == NULL) ||
                    !BN_mod_mul_montgomery(val[i],val[i-1],
                        d,mont,ctx))
                goto err;
            }
        }

    start=1;    /* This is used to avoid multiplication etc
             * when there is only the value '1' in the
             * buffer. */
    wvalue=0;   /* The 'value' of the window */
    wstart=bits-1;  /* The top bit of the window */
    wend=0;     /* The bottom bit of the window */

    if (!BN_to_montgomery(r,BN_value_one(),mont,ctx)) goto err;
    for (;;)
        {
        if (BN_is_bit_set(p,wstart) == 0)
            {
            if (!start)
                {
                if (!BN_mod_mul_montgomery(r,r,r,mont,ctx))
                goto err;
                }
            if (wstart == 0) break;
            wstart--;
            continue;
            }
        /* We now have wstart on a 'set' bit, we now need to work out
         * how bit a window to do.  To do this we need to scan
         * forward until the last set bit before the end of the
         * window */
        j=wstart;
        wvalue=1;
        wend=0;
        for (i=1; i<window; i++)
            {
            if (wstart-i < 0) break;
            if (BN_is_bit_set(p,wstart-i))
                {
                wvalue<<=(i-wend);
                wvalue|=1;
                wend=i;
                }
            }

        /* wend is the size of the current window */
        j=wend+1;
        /* add the 'bytes above' */
        if (!start)
            for (i=0; i<j; i++)
                {
                if (!BN_mod_mul_montgomery(r,r,r,mont,ctx))
                    goto err;
                }
        
        /* wvalue will be an odd number < 2^window */
        if (!BN_mod_mul_montgomery(r,r,val[wvalue>>1],mont,ctx))
            goto err;

        /* move the 'window' down further */
        wstart-=wend+1;
        wvalue=0;
        start=0;
        if (wstart < 0) break;
        }
    if (!BN_from_montgomery(rr,r,mont,ctx)) goto err;
    ret=1;
err:
    if ((in_mont == NULL) && (mont != NULL)) BN_MONT_CTX_free(mont);
    BN_CTX_end(ctx);
    bn_check_top(rr);
    return(ret);
    }


/* BN_mod_exp_mont_consttime() stores the precomputed powers in a specific layout
 * so that accessing any of these table values shows the same access pattern as far
 * as cache lines are concerned.  The following functions are used to transfer a BIGNUM
 * from/to that table. */

static int MOD_EXP_CTIME_COPY_TO_PREBUF(const BIGNUM *b, int top, unsigned char *buf, int idx, int width)
    {
    size_t i, j;

    if (top > b->top)
        top = b->top; /* this works because 'buf' is explicitly zeroed */
    for (i = 0, j=idx; i < top * sizeof b->d[0]; i++, j+=width)
        {
        buf[j] = ((unsigned char*)b->d)[i];
        }

    return 1;
    }

static int MOD_EXP_CTIME_COPY_FROM_PREBUF(BIGNUM *b, int top, unsigned char *buf, int idx, int width)
    {
    size_t i, j;

    if (bn_wexpand(b, top) == NULL)
        return 0;

    for (i=0, j=idx; i < top * sizeof b->d[0]; i++, j+=width)
        {
        ((unsigned char*)b->d)[i] = buf[j];
        }

    b->top = top;
    bn_correct_top(b);
    return 1;
    }   

/* Given a pointer value, compute the next address that is a cache line multiple. */
#define MOD_EXP_CTIME_ALIGN(x_) \
    ((unsigned char*)(x_) + (MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH - (((size_t)(x_)) & (MOD_EXP_CTIME_MIN_CACHE_LINE_MASK))))

/* This variant of BN_mod_exp_mont() uses fixed windows and the special
 * precomputation memory layout to limit data-dependency to a minimum
 * to protect secret exponents (cf. the hyper-threading timing attacks
 * pointed out by Colin Percival,
 * http://www.daemonology.net/hyperthreading-considered-harmful/)
 */
int BN_mod_exp_mont_consttime(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
            const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont)
    {
    int i,bits,ret=0,window,wvalue;
    int top;
    BN_MONT_CTX *mont=NULL;

    int numPowers;
    unsigned char *powerbufFree=NULL;
    int powerbufLen = 0;
    unsigned char *powerbuf=NULL;
    BIGNUM tmp, am;

    bn_check_top(a);
    bn_check_top(p);
    bn_check_top(m);

    top = m->top;

    if (!(m->d[0] & 1))
        {
        BNerr(BN_F_BN_MOD_EXP_MONT_CONSTTIME,BN_R_CALLED_WITH_EVEN_MODULUS);
        return(0);
        }
    bits=BN_num_bits(p);
    if (bits == 0)
        {
        ret = BN_one(rr);
        return ret;
        }

    BN_CTX_start(ctx);

    /* Allocate a montgomery context if it was not supplied by the caller.
     * If this is not done, things will break in the montgomery part.
     */
    if (in_mont != NULL)
        mont=in_mont;
    else
        {
        if ((mont=BN_MONT_CTX_new()) == NULL) goto err;
        if (!BN_MONT_CTX_set(mont,m,ctx)) goto err;
        }

    /* Get the window size to use with size of p. */
    window = BN_window_bits_for_ctime_exponent_size(bits);
#if defined(OPENSSL_BN_ASM_MONT5)
    if (window==6 && bits<=1024) window=5;  /* ~5% improvement of 2048-bit RSA sign */
#endif

    /* Allocate a buffer large enough to hold all of the pre-computed
     * powers of am, am itself and tmp.
     */
    numPowers = 1 << window;
    powerbufLen = sizeof(m->d[0])*(top*numPowers +
                ((2*top)>numPowers?(2*top):numPowers));
#ifdef alloca
    if (powerbufLen < 3072)
        powerbufFree = alloca(powerbufLen+MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH);
    else
#endif
    if ((powerbufFree=(unsigned char*)OPENSSL_malloc(powerbufLen+MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH)) == NULL)
        goto err;
        
    powerbuf = MOD_EXP_CTIME_ALIGN(powerbufFree);
    memset(powerbuf, 0, powerbufLen);

#ifdef alloca
    if (powerbufLen < 3072)
        powerbufFree = NULL;
#endif

    /* lay down tmp and am right after powers table */
    tmp.d     = (BN_ULONG *)(powerbuf + sizeof(m->d[0])*top*numPowers);
    am.d      = tmp.d + top;
    tmp.top   = am.top  = 0;
    tmp.dmax  = am.dmax = top;
    tmp.neg   = am.neg  = 0;
    tmp.flags = am.flags = BN_FLG_STATIC_DATA;

    /* prepare a^0 in Montgomery domain */
#if 1
    if (!BN_to_montgomery(&tmp,BN_value_one(),mont,ctx))    goto err;
#else
    tmp.d[0] = (0-m->d[0])&BN_MASK2;    /* 2^(top*BN_BITS2) - m */
    for (i=1;i<top;i++)
        tmp.d[i] = (~m->d[i])&BN_MASK2;
    tmp.top = top;
#endif

    /* prepare a^1 in Montgomery domain */
    if (a->neg || BN_ucmp(a,m) >= 0)
        {
        if (!BN_mod(&am,a,m,ctx))           goto err;
        if (!BN_to_montgomery(&am,&am,mont,ctx))    goto err;
        }
    else    if (!BN_to_montgomery(&am,a,mont,ctx))      goto err;

#if defined(OPENSSL_BN_ASM_MONT5)
    /* This optimization uses ideas from http://eprint.iacr.org/2011/239,
     * specifically optimization of cache-timing attack countermeasures
     * and pre-computation optimization. */

    /* Dedicated window==4 case improves 512-bit RSA sign by ~15%, but as
     * 512-bit RSA is hardly relevant, we omit it to spare size... */ 
    if (window==5)
    {
    void bn_mul_mont_gather5(BN_ULONG *rp,const BN_ULONG *ap,
            const void *table,const BN_ULONG *np,
            const BN_ULONG *n0,int num,int power);
    void bn_scatter5(const BN_ULONG *inp,size_t num,
            void *table,size_t power);
    void bn_gather5(BN_ULONG *out,size_t num,
            void *table,size_t power);

    BN_ULONG *np=mont->N.d, *n0=mont->n0;

    /* BN_to_montgomery can contaminate words above .top
     * [in BN_DEBUG[_DEBUG] build]... */
    for (i=am.top; i<top; i++)  am.d[i]=0;
    for (i=tmp.top; i<top; i++) tmp.d[i]=0;

    bn_scatter5(tmp.d,top,powerbuf,0);
    bn_scatter5(am.d,am.top,powerbuf,1);
    bn_mul_mont(tmp.d,am.d,am.d,np,n0,top);
    bn_scatter5(tmp.d,top,powerbuf,2);

#if 0
    for (i=3; i<32; i++)
        {
        /* Calculate a^i = a^(i-1) * a */
        bn_mul_mont_gather5(tmp.d,am.d,powerbuf,np,n0,top,i-1);
        bn_scatter5(tmp.d,top,powerbuf,i);
        }
#else
    /* same as above, but uses squaring for 1/2 of operations */
    for (i=4; i<32; i*=2)
        {
        bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
        bn_scatter5(tmp.d,top,powerbuf,i);
        }
    for (i=3; i<8; i+=2)
        {
        int j;
        bn_mul_mont_gather5(tmp.d,am.d,powerbuf,np,n0,top,i-1);
        bn_scatter5(tmp.d,top,powerbuf,i);
        for (j=2*i; j<32; j*=2)
            {
            bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
            bn_scatter5(tmp.d,top,powerbuf,j);
            }
        }
    for (; i<16; i+=2)
        {
        bn_mul_mont_gather5(tmp.d,am.d,powerbuf,np,n0,top,i-1);
        bn_scatter5(tmp.d,top,powerbuf,i);
        bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
        bn_scatter5(tmp.d,top,powerbuf,2*i);
        }
    for (; i<32; i+=2)
        {
        bn_mul_mont_gather5(tmp.d,am.d,powerbuf,np,n0,top,i-1);
        bn_scatter5(tmp.d,top,powerbuf,i);
        }
#endif
    bits--;
    for (wvalue=0, i=bits%5; i>=0; i--,bits--)
        wvalue = (wvalue<<1)+BN_is_bit_set(p,bits);
    bn_gather5(tmp.d,top,powerbuf,wvalue);

    /* Scan the exponent one window at a time starting from the most
     * significant bits.
     */
    while (bits >= 0)
        {
        for (wvalue=0, i=0; i<5; i++,bits--)
            wvalue = (wvalue<<1)+BN_is_bit_set(p,bits);

        bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
        bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
        bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
        bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
        bn_mul_mont(tmp.d,tmp.d,tmp.d,np,n0,top);
        bn_mul_mont_gather5(tmp.d,tmp.d,powerbuf,np,n0,top,wvalue);
        }

    tmp.top=top;
    bn_correct_top(&tmp);
    }
    else
#endif
    {
    if (!MOD_EXP_CTIME_COPY_TO_PREBUF(&tmp, top, powerbuf, 0, numPowers)) goto err;
    if (!MOD_EXP_CTIME_COPY_TO_PREBUF(&am,  top, powerbuf, 1, numPowers)) goto err;

    /* If the window size is greater than 1, then calculate
     * val[i=2..2^winsize-1]. Powers are computed as a*a^(i-1)
     * (even powers could instead be computed as (a^(i/2))^2
     * to use the slight performance advantage of sqr over mul).
     */
    if (window > 1)
        {
        if (!BN_mod_mul_montgomery(&tmp,&am,&am,mont,ctx))  goto err;
        if (!MOD_EXP_CTIME_COPY_TO_PREBUF(&tmp, top, powerbuf, 2, numPowers)) goto err;
        for (i=3; i<numPowers; i++)
            {
            /* Calculate a^i = a^(i-1) * a */
            if (!BN_mod_mul_montgomery(&tmp,&am,&tmp,mont,ctx))
                goto err;
            if (!MOD_EXP_CTIME_COPY_TO_PREBUF(&tmp, top, powerbuf, i, numPowers)) goto err;
            }
        }

    bits--;
    for (wvalue=0, i=bits%window; i>=0; i--,bits--)
        wvalue = (wvalue<<1)+BN_is_bit_set(p,bits);
    if (!MOD_EXP_CTIME_COPY_FROM_PREBUF(&tmp,top,powerbuf,wvalue,numPowers)) goto err;
 
    /* Scan the exponent one window at a time starting from the most
     * significant bits.
     */
    while (bits >= 0)
        {
        wvalue=0; /* The 'value' of the window */
        
        /* Scan the window, squaring the result as we go */
        for (i=0; i<window; i++,bits--)
            {
            if (!BN_mod_mul_montgomery(&tmp,&tmp,&tmp,mont,ctx))    goto err;
            wvalue = (wvalue<<1)+BN_is_bit_set(p,bits);
            }
        
        /* Fetch the appropriate pre-computed value from the pre-buf */
        if (!MOD_EXP_CTIME_COPY_FROM_PREBUF(&am, top, powerbuf, wvalue, numPowers)) goto err;

        /* Multiply the result into the intermediate result */
        if (!BN_mod_mul_montgomery(&tmp,&tmp,&am,mont,ctx)) goto err;
        }
    }

    /* Convert the final result from montgomery to standard format */
    if (!BN_from_montgomery(rr,&tmp,mont,ctx)) goto err;
    ret=1;
err:
    if ((in_mont == NULL) && (mont != NULL)) BN_MONT_CTX_free(mont);
    if (powerbuf!=NULL)
        {
        OPENSSL_cleanse(powerbuf,powerbufLen);
        if (powerbufFree) OPENSSL_free(powerbufFree);
        }
    BN_CTX_end(ctx);
    return(ret);
    }

int BN_mod_exp_mont_word(BIGNUM *rr, BN_ULONG a, const BIGNUM *p,
                         const BIGNUM *m, BN_CTX *ctx, BN_MONT_CTX *in_mont)
    {
    BN_MONT_CTX *mont = NULL;
    int b, bits, ret=0;
    int r_is_one;
    BN_ULONG w, next_w;
    BIGNUM *d, *r, *t;
    BIGNUM *swap_tmp;
#define BN_MOD_MUL_WORD(r, w, m) \
        (BN_mul_word(r, (w)) && \
        (/* BN_ucmp(r, (m)) < 0 ? 1 :*/  \
            (BN_mod(t, r, m, ctx) && (swap_tmp = r, r = t, t = swap_tmp, 1))))
        /* BN_MOD_MUL_WORD is only used with 'w' large,
         * so the BN_ucmp test is probably more overhead
         * than always using BN_mod (which uses BN_copy if
         * a similar test returns true). */
        /* We can use BN_mod and do not need BN_nnmod because our
         * accumulator is never negative (the result of BN_mod does
         * not depend on the sign of the modulus).
         */
#define BN_TO_MONTGOMERY_WORD(r, w, mont) \
        (BN_set_word(r, (w)) && BN_to_montgomery(r, r, (mont), ctx))

    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0)
        {
        /* BN_FLG_CONSTTIME only supported by BN_mod_exp_mont() */
        BNerr(BN_F_BN_MOD_EXP_MONT_WORD,ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
        return -1;
        }

    bn_check_top(p);
    bn_check_top(m);

    if (!BN_is_odd(m))
        {
        BNerr(BN_F_BN_MOD_EXP_MONT_WORD,BN_R_CALLED_WITH_EVEN_MODULUS);
        return(0);
        }
    if (m->top == 1)
        a %= m->d[0]; /* make sure that 'a' is reduced */

    bits = BN_num_bits(p);
    if (bits == 0)
        {
        ret = BN_one(rr);
        return ret;
        }
    if (a == 0)
        {
        BN_zero(rr);
        ret = 1;
        return ret;
        }

    BN_CTX_start(ctx);
    d = BN_CTX_get(ctx);
    r = BN_CTX_get(ctx);
    t = BN_CTX_get(ctx);
    if (d == NULL || r == NULL || t == NULL) goto err;

    if (in_mont != NULL)
        mont=in_mont;
    else
        {
        if ((mont = BN_MONT_CTX_new()) == NULL) goto err;
        if (!BN_MONT_CTX_set(mont, m, ctx)) goto err;
        }

    r_is_one = 1; /* except for Montgomery factor */

    /* bits-1 >= 0 */

    /* The result is accumulated in the product r*w. */
    w = a; /* bit 'bits-1' of 'p' is always set */
    for (b = bits-2; b >= 0; b--)
        {
        /* First, square r*w. */
        next_w = w*w;
        if ((next_w/w) != w) /* overflow */
            {
            if (r_is_one)
                {
                if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err;
                r_is_one = 0;
                }
            else
                {
                if (!BN_MOD_MUL_WORD(r, w, m)) goto err;
                }
            next_w = 1;
            }
        w = next_w;
        if (!r_is_one)
            {
            if (!BN_mod_mul_montgomery(r, r, r, mont, ctx)) goto err;
            }

        /* Second, multiply r*w by 'a' if exponent bit is set. */
        if (BN_is_bit_set(p, b))
            {
            next_w = w*a;
            if ((next_w/a) != w) /* overflow */
                {
                if (r_is_one)
                    {
                    if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err;
                    r_is_one = 0;
                    }
                else
                    {
                    if (!BN_MOD_MUL_WORD(r, w, m)) goto err;
                    }
                next_w = a;
                }
            w = next_w;
            }
        }

    /* Finally, set r:=r*w. */
    if (w != 1)
        {
        if (r_is_one)
            {
            if (!BN_TO_MONTGOMERY_WORD(r, w, mont)) goto err;
            r_is_one = 0;
            }
        else
            {
            if (!BN_MOD_MUL_WORD(r, w, m)) goto err;
            }
        }

    if (r_is_one) /* can happen only if a == 1*/
        {
        if (!BN_one(rr)) goto err;
        }
    else
        {
        if (!BN_from_montgomery(rr, r, mont, ctx)) goto err;
        }
    ret = 1;
err:
    if ((in_mont == NULL) && (mont != NULL)) BN_MONT_CTX_free(mont);
    BN_CTX_end(ctx);
    bn_check_top(rr);
    return(ret);
    }


/* The old fallback, simple version :-) */
int BN_mod_exp_simple(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
        const BIGNUM *m, BN_CTX *ctx)
    {
    int i,j,bits,ret=0,wstart,wend,window,wvalue;
    int start=1;
    BIGNUM *d;
    /* Table of variables obtained from 'ctx' */
    BIGNUM *val[TABLE_SIZE];

    if (BN_get_flags(p, BN_FLG_CONSTTIME) != 0)
        {
        /* BN_FLG_CONSTTIME only supported by BN_mod_exp_mont() */
        BNerr(BN_F_BN_MOD_EXP_SIMPLE,ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
        return -1;
        }

    bits=BN_num_bits(p);

    if (bits == 0)
        {
        ret = BN_one(r);
        return ret;
        }

    BN_CTX_start(ctx);
    d = BN_CTX_get(ctx);
    val[0] = BN_CTX_get(ctx);
    if(!d || !val[0]) goto err;

    if (!BN_nnmod(val[0],a,m,ctx)) goto err;        /* 1 */
    if (BN_is_zero(val[0]))
        {
        BN_zero(r);
        ret = 1;
        goto err;
        }

    window = BN_window_bits_for_exponent_size(bits);
    if (window > 1)
        {
        if (!BN_mod_mul(d,val[0],val[0],m,ctx))
            goto err;               /* 2 */
        j=1<<(window-1);
        for (i=1; i<j; i++)
            {
            if(((val[i] = BN_CTX_get(ctx)) == NULL) ||
                    !BN_mod_mul(val[i],val[i-1],d,m,ctx))
                goto err;
            }
        }

    start=1;    /* This is used to avoid multiplication etc
             * when there is only the value '1' in the
             * buffer. */
    wvalue=0;   /* The 'value' of the window */
    wstart=bits-1;  /* The top bit of the window */
    wend=0;     /* The bottom bit of the window */

    if (!BN_one(r)) goto err;

    for (;;)
        {
        if (BN_is_bit_set(p,wstart) == 0)
            {
            if (!start)
                if (!BN_mod_mul(r,r,r,m,ctx))
                goto err;
            if (wstart == 0) break;
            wstart--;
            continue;
            }
        /* We now have wstart on a 'set' bit, we now need to work out
         * how bit a window to do.  To do this we need to scan
         * forward until the last set bit before the end of the
         * window */
        j=wstart;
        wvalue=1;
        wend=0;
        for (i=1; i<window; i++)
            {
            if (wstart-i < 0) break;
            if (BN_is_bit_set(p,wstart-i))
                {
                wvalue<<=(i-wend);
                wvalue|=1;
                wend=i;
                }
            }

        /* wend is the size of the current window */
        j=wend+1;
        /* add the 'bytes above' */
        if (!start)
            for (i=0; i<j; i++)
                {
                if (!BN_mod_mul(r,r,r,m,ctx))
                    goto err;
                }
        
        /* wvalue will be an odd number < 2^window */
        if (!BN_mod_mul(r,r,val[wvalue>>1],m,ctx))
            goto err;

        /* move the 'window' down further */
        wstart-=wend+1;
        wvalue=0;
        start=0;
        if (wstart < 0) break;
        }
    ret=1;
err:
    BN_CTX_end(ctx);
    bn_check_top(r);
    return(ret);
    }