/* $OpenBSD: bn_exp.c,v 1.31 2017/05/02 03:59:44 deraadt 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"
#include "constant_time_locl.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() */
		BNerror(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);
}

static int
BN_mod_exp_internal(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
    BN_CTX *ctx, int ct)
{
	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.)
	 */

	if (BN_is_odd(m)) {
		if (a->top == 1 && !a->neg && !ct) {
			BN_ULONG A = a->d[0];
			ret = BN_mod_exp_mont_word(r, A,p, m,ctx, NULL);
		} else
			ret = BN_mod_exp_mont_ct(r, a,p, m,ctx, NULL);
	} else	{
		ret = BN_mod_exp_recp(r, a,p, m, 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)
{
	return BN_mod_exp_internal(r, a, p, m, ctx,
	    (BN_get_flags(p, BN_FLG_CONSTTIME) != 0));
}

int
BN_mod_exp_ct(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
    BN_CTX *ctx)
{
	return BN_mod_exp_internal(r, a, p, m, ctx, 1);
}


int
BN_mod_exp_nonct(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
    BN_CTX *ctx)
{
	return BN_mod_exp_internal(r, a, p, m, ctx, 0);
}


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() */
		BNerror(ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
		return -1;
	}

	bits = BN_num_bits(p);
	if (bits == 0) {
		/* x**0 mod 1 is still zero. */
		if (BN_is_one(m)) {
			ret = 1;
			BN_zero(r);
		} else
			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);
}

static int
BN_mod_exp_mont_internal(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
    BN_CTX *ctx, BN_MONT_CTX *in_mont, int ct)
{
	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 (ct) {
		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)) {
		BNerror(BN_R_CALLED_WITH_EVEN_MODULUS);
		return (0);
	}

	bits = BN_num_bits(p);
	if (bits == 0) {
		/* x**0 mod 1 is still zero. */
		if (BN_is_one(m)) {
			ret = 1;
			BN_zero(rr);
		} else
			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);
}

int
BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
    BN_CTX *ctx, BN_MONT_CTX *in_mont)
{
	return BN_mod_exp_mont_internal(rr, a, p, m, ctx, in_mont,
	    (BN_get_flags(p, BN_FLG_CONSTTIME) != 0));
}

int
BN_mod_exp_mont_ct(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
    BN_CTX *ctx, BN_MONT_CTX *in_mont)
{
	return BN_mod_exp_mont_internal(rr, a, p, m, ctx, in_mont, 1);
}

int
BN_mod_exp_mont_nonct(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
    BN_CTX *ctx, BN_MONT_CTX *in_mont)
{
	return BN_mod_exp_mont_internal(rr, a, p, m, ctx, in_mont, 0);
}

/* 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 window)
{
	int i, j;
	int width = 1 << window;
	BN_ULONG *table = (BN_ULONG *)buf;

	if (top > b->top)
		top = b->top; /* this works because 'buf' is explicitly zeroed */

	for (i = 0, j = idx; i < top; i++, j += width) {
		table[j] = b->d[i];
	}

	return 1;
}

static int
MOD_EXP_CTIME_COPY_FROM_PREBUF(BIGNUM *b, int top, unsigned char *buf, int idx,
    int window)
{
	int i, j;
	int width = 1 << window;
	volatile BN_ULONG *table = (volatile BN_ULONG *)buf;

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

	if (window <= 3) {
		for (i = 0; i < top; i++, table += width) {
		    BN_ULONG acc = 0;

		    for (j = 0; j < width; j++) {
			acc |= table[j] &
			       ((BN_ULONG)0 - (constant_time_eq_int(j,idx)&1));
		    }

		    b->d[i] = acc;
		}
	} else {
		int xstride = 1 << (window - 2);
		BN_ULONG y0, y1, y2, y3;

		i = idx >> (window - 2);        /* equivalent of idx / xstride */
		idx &= xstride - 1;             /* equivalent of idx % xstride */

		y0 = (BN_ULONG)0 - (constant_time_eq_int(i,0)&1);
		y1 = (BN_ULONG)0 - (constant_time_eq_int(i,1)&1);
		y2 = (BN_ULONG)0 - (constant_time_eq_int(i,2)&1);
		y3 = (BN_ULONG)0 - (constant_time_eq_int(i,3)&1);

		for (i = 0; i < top; i++, table += width) {
		    BN_ULONG acc = 0;

		    for (j = 0; j < xstride; j++) {
			acc |= ( (table[j + 0 * xstride] & y0) |
				 (table[j + 1 * xstride] & y1) |
				 (table[j + 2 * xstride] & y2) |
				 (table[j + 3 * xstride] & y3) )
			       & ((BN_ULONG)0 - (constant_time_eq_int(j,idx)&1));
		    }

		    b->d[i] = acc;
		}
	}
	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);

	if (!BN_is_odd(m)) {
		BNerror(BN_R_CALLED_WITH_EVEN_MODULUS);
		return (0);
	}

	top = m->top;

	bits = BN_num_bits(p);
	if (bits == 0) {
		/* x**0 mod 1 is still zero. */
		if (BN_is_one(m)) {
			ret = 1;
			BN_zero(rr);
		} else
			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 = calloc(powerbufLen +
	    MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH, 1)) == NULL)
		goto err;
	powerbuf = MOD_EXP_CTIME_ALIGN(powerbufFree);

	/* 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_ct(&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,
		    window))
			goto err;
		if (!MOD_EXP_CTIME_COPY_TO_PREBUF(&am,  top, powerbuf, 1,
		    window))
			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, window))
				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, window))
					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, window))
			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, window))
				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);
	freezero(powerbufFree, powerbufLen + MOD_EXP_CTIME_MIN_CACHE_LINE_WIDTH);
	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_ct(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() */
		BNerror(ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
		return -1;
	}

	bn_check_top(p);
	bn_check_top(m);

	if (!BN_is_odd(m)) {
		BNerror(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) {
		/* x**0 mod 1 is still zero. */
		if (BN_is_one(m)) {
			ret = 1;
			BN_zero(rr);
		} else
			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() */
		BNerror(ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
		return -1;
	}

	bits = BN_num_bits(p);
	if (bits == 0) {
		/* x**0 mod 1 is still zero. */
		if (BN_is_one(m)) {
			ret = 1;
			BN_zero(r);
		} else
			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);
}
