#include "ec_lcl.h"
-/* TODO: width-m NAFs */
+/* TODO: optional precomputation of multiples of the generator */
-/* TODO: optional Lim-Lee precomputation for the generator */
+#if 1
+/*
+ * 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, 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;
+ }
+ }
-/* this is just BN_window_bits_for_exponent_size from bn_lcl.h for now;
- * the table should be updated for EC */ /* TODO */
+ 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) > 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, BIGNUM *scalar,
- size_t num, EC_POINT *points[], BIGNUM *scalars[], BN_CTX *ctx)
+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;
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);
+ num_val += 1u << (wsize[i] - 1);
if (bits > max_bits)
max_bits = bits;
wbits[i] = 0;
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;
}
}
}
-#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
{
if (wbits[i] == 0)
{
- BIGNUM *s;
+ const BIGNUM *s;
s = i < num ? scalars[i] : scalar;
}
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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