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openssl/crypto/ec/ec_mult.c
Bodo Möller c78515f55e comments etc.
2001-11-15 22:35:41 +00:00

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/* crypto/ec/ec_mult.c */
/* ====================================================================
* Copyright (c) 1998-2001 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 <openssl/err.h>
#include "ec_lcl.h"
/* TODO: optional precomputation of multiples of the generator */
#if 1
/*
* wNAF-based interleaving multi-exponentation method
*/
/* Determine the width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'.
* This is an array r[] of values that are either zero or odd with an
* absolute value less than 2^w satisfying
* scalar = \sum_j r[j]*2^j
* where at most one of any w+1 consecutive digits is non-zero.
*/
static signed char *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len, BN_CTX *ctx)
{
BIGNUM *c;
int ok = 0;
signed char *r = NULL;
int sign = 1;
int bit, next_bit, mask;
size_t len, j;
BN_CTX_start(ctx);
c = BN_CTX_get(ctx);
if (c == NULL) goto err;
if (w <= 0 || w > 7) /* 'signed char' can represent integers with absolute values less than 2^7 */
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
}
bit = 1 << w; /* at most 128 */
next_bit = bit << 1; /* at most 256 */
mask = next_bit - 1; /* at most 255 */
if (!BN_copy(c, scalar)) goto err;
if (c->neg)
{
sign = -1;
c->neg = 0;
}
len = BN_num_bits(c) + 1; /* wNAF may be one digit longer than binary representation */
r = OPENSSL_malloc(len);
if (r == NULL) goto err;
j = 0;
while (!BN_is_zero(c))
{
int u = 0;
if (BN_is_odd(c))
{
if (c->d == NULL || c->top == 0)
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
}
u = c->d[0] & mask;
if (u & bit)
{
u -= next_bit;
/* u < 0 */
if (!BN_add_word(c, -u)) goto err;
}
else
{
/* u > 0 */
if (!BN_sub_word(c, u)) goto err;
}
if (u <= -bit || u >= bit || !(u & 1) || c->neg)
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
}
}
r[j++] = sign * u;
if (BN_is_odd(c))
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
}
if (!BN_rshift1(c, c)) goto err;
}
if (j > len)
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
}
len = j;
ok = 1;
err:
BN_CTX_end(ctx);
if (!ok)
{
OPENSSL_free(r);
r = NULL;
}
if (ok)
*ret_len = len;
return r;
}
/* TODO: table should be optimised for the wNAF-based implementation */
#define EC_window_bits_for_scalar_size(b) \
((b) >= 2000 ? 6 : \
(b) >= 800 ? 5 : \
(b) >= 300 ? 4 : \
(b) >= 70 ? 3 : \
(b) >= 20 ? 2 : \
1)
/* Compute
* \sum scalars[i]*points[i],
* also including
* scalar*generator
* in the addition if scalar != NULL
*/
int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
{
BN_CTX *new_ctx = NULL;
EC_POINT *generator = NULL;
EC_POINT *tmp = NULL;
size_t totalnum;
size_t i, j;
int k;
int r_is_inverted = 0;
int r_is_at_infinity = 1;
size_t *wsize = NULL; /* individual window sizes */
signed char **wNAF = NULL; /* individual wNAFs */
size_t *wNAF_len = NULL;
size_t max_len = 0;
size_t num_val;
EC_POINT **val = NULL; /* precomputation */
EC_POINT **v;
EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
int ret = 0;
if (scalar != NULL)
{
generator = EC_GROUP_get0_generator(group);
if (generator == NULL)
{
ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
return 0;
}
}
for (i = 0; i < num; i++)
{
if (group->meth != points[i]->meth)
{
ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS);
return 0;
}
}
totalnum = num + (scalar != NULL);
wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
wNAF_len = OPENSSL_malloc(totalnum * sizeof wNAF_len[0]);
wNAF = OPENSSL_malloc(totalnum * sizeof wNAF[0] + 1);
if (wNAF != NULL)
{
wNAF[0] = NULL; /* preliminary pivot */
}
if (wsize == NULL || wNAF_len == NULL || wNAF == NULL) goto err;
/* num_val := total number of points to precompute */
num_val = 0;
for (i = 0; i < totalnum; i++)
{
size_t bits;
bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
wsize[i] = EC_window_bits_for_scalar_size(bits);
num_val += 1u << (wsize[i] - 1);
}
/* all precomputed points go into a single array 'val',
* 'val_sub[i]' is a pointer to the subarray for the i-th point */
val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
if (val == NULL) goto err;
val[num_val] = NULL; /* pivot element */
val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
if (val_sub == NULL) goto err;
/* allocate points for precomputation */
v = val;
for (i = 0; i < totalnum; i++)
{
val_sub[i] = v;
for (j = 0; j < (1u << (wsize[i] - 1)); j++)
{
*v = EC_POINT_new(group);
if (*v == NULL) goto err;
v++;
}
}
if (!(v == val + num_val))
{
ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR);
goto err;
}
if (ctx == NULL)
{
ctx = new_ctx = BN_CTX_new();
if (ctx == NULL)
goto err;
}
tmp = EC_POINT_new(group);
if (tmp == NULL) goto err;
/* prepare precomputed values:
* val_sub[i][0] := points[i]
* val_sub[i][1] := 3 * points[i]
* val_sub[i][2] := 5 * points[i]
* ...
*/
for (i = 0; i < totalnum; i++)
{
if (i < num)
{
if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
}
else
{
if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
}
if (wsize[i] > 1)
{
if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
for (j = 1; j < (1u << (wsize[i] - 1)); j++)
{
if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
}
}
wNAF[i + 1] = NULL; /* make sure we always have a pivot */
wNAF[i] = compute_wNAF((i < num ? scalars[i] : scalar), wsize[i], &wNAF_len[i], ctx);
if (wNAF[i] == NULL) goto err;
if (wNAF_len[i] > max_len)
max_len = wNAF_len[i];
}
#if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err;
#endif
r_is_at_infinity = 1;
for (k = max_len - 1; k >= 0; k--)
{
if (!r_is_at_infinity)
{
if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
}
for (i = 0; i < totalnum; i++)
{
if (wNAF_len[i] > k)
{
int digit = wNAF[i][k];
int is_neg;
if (digit)
{
is_neg = digit < 0;
if (is_neg)
digit = -digit;
if (is_neg != r_is_inverted)
{
if (!r_is_at_infinity)
{
if (!EC_POINT_invert(group, r, ctx)) goto err;
}
r_is_inverted = !r_is_inverted;
}
/* digit > 0 */
if (r_is_at_infinity)
{
if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) goto err;
r_is_at_infinity = 0;
}
else
{
if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) goto err;
}
}
}
}
}
if (r_is_at_infinity)
{
if (!EC_POINT_set_to_infinity(group, r)) goto err;
}
else
{
if (r_is_inverted)
if (!EC_POINT_invert(group, r, ctx)) goto err;
}
ret = 1;
err:
if (new_ctx != NULL)
BN_CTX_free(new_ctx);
if (tmp != NULL)
EC_POINT_free(tmp);
if (wsize != NULL)
OPENSSL_free(wsize);
if (wNAF_len != NULL)
OPENSSL_free(wNAF_len);
if (wNAF != NULL)
{
signed char **w;
for (w = wNAF; *w != NULL; w++)
OPENSSL_free(*w);
OPENSSL_free(wNAF);
}
if (val != NULL)
{
for (v = val; *v != NULL; v++)
EC_POINT_clear_free(*v);
OPENSSL_free(val);
}
if (val_sub != NULL)
{
OPENSSL_free(val_sub);
}
return ret;
}
#else
/*
* Basic interleaving multi-exponentation method
*/
#define EC_window_bits_for_scalar_size(b) \
((b) >= 2000 ? 6 : \
(b) >= 800 ? 5 : \
(b) >= 300 ? 4 : \
(b) >= 70 ? 3 : \
(b) >= 20 ? 2 : \
1)
/* For window size 'w' (w >= 2), we compute the odd multiples
* 1*P .. (2^w-1)*P.
* This accounts for 2^(w-1) point additions (neglecting constants),
* each of which requires 16 field multiplications (4 squarings
* and 12 general multiplications) in the case of curves defined
* over GF(p), which are the only curves we have so far.
*
* Converting these precomputed points into affine form takes
* three field multiplications for inverting Z and one squaring
* and three multiplications for adjusting X and Y, i.e.
* 7 multiplications in total (1 squaring and 6 general multiplications),
* again except for constants.
*
* The average number of windows for a 'b' bit scalar is roughly
* b/(w+1).
* Each of these windows (except possibly for the first one, but
* we are ignoring constants anyway) requires one point addition.
* As the precomputed table stores points in affine form, these
* additions take only 11 field multiplications each (3 squarings
* and 8 general multiplications).
*
* So the total workload, except for constants, is
*
* 2^(w-1)*[5 squarings + 18 multiplications]
* + (b/(w+1))*[3 squarings + 8 multiplications]
*
* If we assume that 10 squarings are as costly as 9 multiplications,
* our task is to find the 'w' that, given 'b', minimizes
*
* 2^(w-1)*(5*9 + 18*10) + (b/(w+1))*(3*9 + 8*10)
* = 2^(w-1)*225 + (b/(w+1))*107.
*
* Thus optimal window sizes should be roughly as follows:
*
* w >= 6 if b >= 1414
* w = 5 if 1413 >= b >= 505
* w = 4 if 504 >= b >= 169
* w = 3 if 168 >= b >= 51
* w = 2 if 50 >= b >= 13
* w = 1 if 12 >= b
*
* If we assume instead that squarings are exactly as costly as
* multiplications, we have to minimize
* 2^(w-1)*23 + (b/(w+1))*11.
*
* This gives us the following (nearly unchanged) table of optimal
* windows sizes:
*
* w >= 6 if b >= 1406
* w = 5 if 1405 >= b >= 502
* w = 4 if 501 >= b >= 168
* w = 3 if 167 >= b >= 51
* w = 2 if 50 >= b >= 13
* w = 1 if 12 >= b
*
* Note that neither table tries to take into account memory usage
* (allocation overhead, code locality etc.). Actual timings with
* NIST curves P-192, P-224, and P-256 with scalars of 192, 224,
* and 256 bits, respectively, show that w = 3 (instead of 4) is
* preferrable; timings with NIST curve P-384 and 384-bit scalars
* confirm that w = 4 is optimal for this case; and timings with
* NIST curve P-521 and 521-bit scalars show that w = 4 (instead
* of 5) is preferrable. So we generously round up all the
* boundaries and use the following table:
*
* w >= 6 if b >= 2000
* w = 5 if 1999 >= b >= 800
* w = 4 if 799 >= b >= 300
* w = 3 if 299 >= b >= 70
* w = 2 if 69 >= b >= 20
* w = 1 if 19 >= b
*/
int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
{
BN_CTX *new_ctx = NULL;
EC_POINT *generator = NULL;
EC_POINT *tmp = NULL;
size_t totalnum;
size_t i, j;
int k, t;
int r_is_at_infinity = 1;
size_t max_bits = 0;
size_t *wsize = NULL; /* individual window sizes */
unsigned long *wbits = NULL; /* individual window contents */
int *wpos = NULL; /* position of bottom bit of current individual windows
* (wpos[i] is valid if wbits[i] != 0) */
size_t num_val;
EC_POINT **val = NULL; /* precomputation */
EC_POINT **v;
EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
int ret = 0;
if (scalar != NULL)
{
generator = EC_GROUP_get0_generator(group);
if (generator == NULL)
{
ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
return 0;
}
}
for (i = 0; i < num; i++)
{
if (group->meth != points[i]->meth)
{
ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS);
return 0;
}
}
totalnum = num + (scalar != NULL);
wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
wbits = OPENSSL_malloc(totalnum * sizeof wbits[0]);
wpos = OPENSSL_malloc(totalnum * sizeof wpos[0]);
if (wsize == NULL || wbits == NULL || wpos == NULL) goto err;
/* num_val := total number of points to precompute */
num_val = 0;
for (i = 0; i < totalnum; i++)
{
size_t bits;
bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
wsize[i] = EC_window_bits_for_scalar_size(bits);
num_val += 1u << (wsize[i] - 1);
if (bits > max_bits)
max_bits = bits;
wbits[i] = 0;
wpos[i] = 0;
}
/* all precomputed points go into a single array 'val',
* 'val_sub[i]' is a pointer to the subarray for the i-th point */
val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
if (val == NULL) goto err;
val[num_val] = NULL; /* pivot element */
val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
if (val_sub == NULL) goto err;
/* allocate points for precomputation */
v = val;
for (i = 0; i < totalnum; i++)
{
val_sub[i] = v;
for (j = 0; j < (1u << (wsize[i] - 1)); j++)
{
*v = EC_POINT_new(group);
if (*v == NULL) goto err;
v++;
}
}
if (!(v == val + num_val))
{
ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR);
goto err;
}
if (ctx == NULL)
{
ctx = new_ctx = BN_CTX_new();
if (ctx == NULL)
goto err;
}
tmp = EC_POINT_new(group);
if (tmp == NULL) goto err;
/* prepare precomputed values:
* val_sub[i][0] := points[i]
* val_sub[i][1] := 3 * points[i]
* val_sub[i][2] := 5 * points[i]
* ...
*/
for (i = 0; i < totalnum; i++)
{
if (i < num)
{
if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
if (scalars[i]->neg)
{
if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
}
}
else
{
if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
if (scalar->neg)
{
if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
}
}
if (wsize[i] > 1)
{
if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
for (j = 1; j < (1u << (wsize[i] - 1)); j++)
{
if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
}
}
}
#if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err;
#endif
r_is_at_infinity = 1;
for (k = max_bits - 1; k >= 0; k--)
{
if (!r_is_at_infinity)
{
if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
}
for (i = 0; i < totalnum; i++)
{
if (wbits[i] == 0)
{
const BIGNUM *s;
s = i < num ? scalars[i] : scalar;
if (BN_is_bit_set(s, k))
{
/* look at bits k - wsize[i] + 1 .. k for this window */
t = k - wsize[i] + 1;
while (!BN_is_bit_set(s, t)) /* BN_is_bit_set is false for t < 0 */
t++;
wpos[i] = t;
wbits[i] = 1;
for (t = k - 1; t >= wpos[i]; t--)
{
wbits[i] <<= 1;
if (BN_is_bit_set(s, t))
wbits[i]++;
}
/* now wbits[i] is the odd bit pattern at bits wpos[i] .. k */
}
}
if ((wbits[i] != 0) && (wpos[i] == k))
{
if (r_is_at_infinity)
{
if (!EC_POINT_copy(r, val_sub[i][wbits[i] >> 1])) goto err;
r_is_at_infinity = 0;
}
else
{
if (!EC_POINT_add(group, r, r, val_sub[i][wbits[i] >> 1], ctx)) goto err;
}
wbits[i] = 0;
}
}
}
if (r_is_at_infinity)
if (!EC_POINT_set_to_infinity(group, r)) goto err;
ret = 1;
err:
if (new_ctx != NULL)
BN_CTX_free(new_ctx);
if (tmp != NULL)
EC_POINT_free(tmp);
if (wsize != NULL)
OPENSSL_free(wsize);
if (wbits != NULL)
OPENSSL_free(wbits);
if (wpos != NULL)
OPENSSL_free(wpos);
if (val != NULL)
{
for (v = val; *v != NULL; v++)
EC_POINT_clear_free(*v);
OPENSSL_free(val);
}
if (val_sub != NULL)
{
OPENSSL_free(val_sub);
}
return ret;
}
#endif
int EC_POINT_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar, const EC_POINT *point, const BIGNUM *p_scalar, BN_CTX *ctx)
{
const EC_POINT *points[1];
const BIGNUM *scalars[1];
points[0] = point;
scalars[0] = p_scalar;
return EC_POINTs_mul(group, r, g_scalar, (point != NULL && p_scalar != NULL), points, scalars, ctx);
}
int EC_GROUP_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
{
const EC_POINT *generator;
BN_CTX *new_ctx = NULL;
BIGNUM *order;
int ret = 0;
generator = EC_GROUP_get0_generator(group);
if (generator == NULL)
{
ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR);
return 0;
}
if (ctx == NULL)
{
ctx = new_ctx = BN_CTX_new();
if (ctx == NULL)
return 0;
}
BN_CTX_start(ctx);
order = BN_CTX_get(ctx);
if (order == NULL) goto err;
if (!EC_GROUP_get_order(group, order, ctx)) return 0;
if (BN_is_zero(order))
{
ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER);
goto err;
}
/* TODO */
ret = 1;
err:
BN_CTX_end(ctx);
if (new_ctx != NULL)
BN_CTX_free(new_ctx);
return ret;
}
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