/* TODO: width-m NAFs */
-/* TODO: optional Lim-Lee precomputation for the generator */
+/* TODO: optional precomputation of multiples of 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 */
#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)
+/* 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
+ */
+
+
/* 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)
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 (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
projects / openssl.git / blobdiff