#include "ec_lcl.h"
-/* TODO: width-m NAFs */
+/* TODO: optional precomputation of multiples of the generator */
-/* TODO: optional Lim-Lee precomputation for the generator */
-/* this is just BN_window_bits_for_exponent_size from bn_lcl.h for now;
- * the table should be updated for EC */ /* TODO */
+/*
+ * wNAF-based interleaving multi-exponentation method
+ * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp>)
+ */
+
+
+/* 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 = 0, 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,
+ * sometimes smaller windows will give better performance
+ * (thus the boundaries should be increased)
+ */
#define EC_window_bits_for_scalar_size(b) \
- ((b) > 671 ? 6 : \
- (b) > 239 ? 5 : \
- (b) > 79 ? 4 : \
- (b) > 23 ? 3 : 1)
+ ((b) >= 2000 ? 6 : \
+ (b) >= 800 ? 5 : \
+ (b) >= 300 ? 4 : \
+ (b) >= 70 ? 3 : \
+ (b) >= 20 ? 2 : \
+ 1)
/* Compute
- * \sum scalars[i]*points[i]
- * where
+ * \sum scalars[i]*points[i],
+ * also including
* scalar*generator
- * is included in the addition if scalar != NULL
+ * 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)
EC_POINT *tmp = NULL;
size_t totalnum;
size_t i, j;
- int k, t;
+ int k;
+ int r_is_inverted = 0;
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) */
+ 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;
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;
+ wNAF_len = OPENSSL_malloc(totalnum * sizeof wNAF_len[0]);
+ wNAF = OPENSSL_malloc((totalnum + 1) * sizeof wNAF[0]);
+ 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;
bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
wsize[i] = EC_window_bits_for_scalar_size(bits);
- num_val += 1 << (wsize[i] - 1);
- if (bits > max_bits)
- max_bits = bits;
- wbits[i] = 0;
- wpos[i] = 0;
+ num_val += 1u << (wsize[i] - 1);
}
/* all precomputed points go into a single array 'val',
for (i = 0; i < totalnum; i++)
{
val_sub[i] = v;
- for (j = 0; j < (1 << (wsize[i] - 1)); j++)
+ for (j = 0; j < (1u << (wsize[i] - 1)); j++)
{
*v = EC_POINT_new(group);
if (*v == NULL) goto err;
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 < (1 << (wsize[i] - 1)); j++)
+ 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, maybe we should only do this if total_num > 1 */
+#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--)
+ for (k = max_len - 1; k >= 0; k--)
{
if (!r_is_at_infinity)
{
for (i = 0; i < totalnum; i++)
{
- if (wbits[i] == 0)
+ if (wNAF_len[i] > (size_t)k)
{
- const BIGNUM *s;
-
- s = i < num ? scalars[i] : scalar;
+ int digit = wNAF[i][k];
+ int is_neg;
- if (BN_is_bit_set(s, k))
+ if (digit)
{
- /* 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--)
+ is_neg = digit < 0;
+
+ if (is_neg)
+ digit = -digit;
+
+ if (is_neg != r_is_inverted)
{
- wbits[i] <<= 1;
- if (BN_is_bit_set(s, t))
- wbits[i]++;
+ 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;
}
- /* 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;
+ }
+ else
+ {
+ if (r_is_inverted)
+ if (!EC_POINT_invert(group, r, ctx)) goto err;
+ }
ret = 1;
EC_POINT_free(tmp);
if (wsize != NULL)
OPENSSL_free(wsize);
- if (wbits != NULL)
- OPENSSL_free(wbits);
- if (wpos != NULL)
- OPENSSL_free(wpos);
+ 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++)
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