Import LibreSSL v2.4.2 to vendor branch

[dragonfly.git] / crypto / libressl / crypto / bn / bn_exp.c

/* $OpenBSD: bn_exp.c,v 1.22 2015/03/21 08:05:20 doug Exp $ */
/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com)
 * All rights reserved.
 *
 * This package is an SSL implementation written
 * by Eric Young (eay@cryptsoft.com).
 * 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 (tjh@cryptsoft.com).
 *
 * 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 (eay@cryptsoft.com)"
 *    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 (tjh@cryptsoft.com)"
 *
 * 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
 *    openssl-core@openssl.org.
 *
 * 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
 * (eay@cryptsoft.com).  This product includes software written by Tim
 * Hudson (tjh@cryptsoft.com).
 *
 */

#include <stdlib.h>
#include <string.h>

#include <openssl/err.h>

#include "bn_lcl.h"

/* 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 && rr != NULL)
                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);
        if ((aa = BN_CTX_get(ctx)) == NULL)
                goto err;
        if ((val[0] = BN_CTX_get(ctx)) == NULL)
                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);
        if ((d = BN_CTX_get(ctx)) == NULL)
                goto err;
        if ((r = BN_CTX_get(ctx)) == NULL)
                goto err;
        if ((val[0] = BN_CTX_get(ctx)) == NULL)
                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));
        if ((powerbufFree = malloc(powerbufLen +
            MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH)) == NULL)
                goto err;

        powerbuf = MOD_EXP_CTIME_ALIGN(powerbufFree);
        memset(powerbuf, 0, powerbufLen);

        /* 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 && top > 1) {
                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) {
                explicit_bzero(powerbuf, powerbufLen);
                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);
        if ((d = BN_CTX_get(ctx)) == NULL)
                goto err;
        if ((r = BN_CTX_get(ctx)) == NULL)
                goto err;
        if ((t = BN_CTX_get(ctx)) == 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);
        if ((d = BN_CTX_get(ctx)) == NULL)
                goto err;
        if ((val[0] = BN_CTX_get(ctx)) == NULL)
                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);
}