/* crypto/ec/ec_mult.c */
/* ====================================================================
- * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
+ * Copyright (c) 1998-2002 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
/* TODO: optional precomputation of multiples of 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'.
+/* Determine the modified 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.
+ * where at most one of any w+1 consecutive digits is non-zero
+ * with the exception that the most significant digit may be only
+ * w-1 zeros away from that next non-zero digit.
*/
-static signed char *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len, BN_CTX *ctx)
+static signed char *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len)
{
- BIGNUM *c;
+ int window_val;
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);
next_bit = bit << 1; /* at most 256 */
mask = next_bit - 1; /* at most 255 */
- if (!BN_copy(c, scalar)) goto err;
- if (c->neg)
+ if (scalar->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);
+ len = BN_num_bits(scalar);
+ r = OPENSSL_malloc(len + 1); /* modified wNAF may be one digit longer than binary representation */
if (r == NULL) goto err;
+ if (scalar->d == NULL || scalar->top == 0)
+ {
+ ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
+ goto err;
+ }
+ window_val = scalar->d[0] & mask;
j = 0;
- while (!BN_is_zero(c))
+ while ((window_val != 0) || (j + w + 1 < len)) /* if j+w+1 >= len, window_val will not increase */
{
- int u = 0;
+ int digit = 0;
- if (BN_is_odd(c))
+ /* 0 <= window_val <= 2^(w+1) */
+
+ if (window_val & 1)
{
- if (c->d == NULL || c->top == 0)
+ /* 0 < window_val < 2^(w+1) */
+
+ if (window_val & bit)
{
- ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
- goto err;
+ digit = window_val - next_bit; /* -2^w < digit < 0 */
+
+#if 1 /* modified wNAF */
+ if (j + w + 1 >= len)
+ {
+ /* special case for generating modified wNAFs:
+ * no new bits will be added into window_val,
+ * so using a positive digit here will decrease
+ * the total length of the representation */
+
+ digit = window_val & (mask >> 1); /* 0 < digit < 2^w */
+ }
+#endif
}
- u = c->d[0] & mask;
- if (u & bit)
+ else
{
- u -= next_bit;
- /* u < 0 */
- if (!BN_add_word(c, -u)) goto err;
+ digit = window_val; /* 0 < digit < 2^w */
}
- else
+
+ if (digit <= -bit || digit >= bit || !(digit & 1))
{
- /* u > 0 */
- if (!BN_sub_word(c, u)) goto err;
+ ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
+ goto err;
}
- if (u <= -bit || u >= bit || !(u & 1) || c->neg)
+ window_val -= digit;
+
+ /* now window_val is 0 or 2^(w+1) in standard wNAF generation;
+ * for modified window NAFs, it may also be 2^w
+ */
+ if (window_val != 0 && window_val != next_bit && window_val != bit)
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
}
}
- r[j++] = sign * u;
-
- if (BN_is_odd(c))
+ r[j++] = sign * digit;
+
+ window_val >>= 1;
+ window_val += bit * BN_is_bit_set(scalar, j + w);
+
+ if (window_val > next_bit)
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
}
- if (!BN_rshift1(c, c)) goto err;
}
- if (j > len)
+ if (j > len + 1)
{
ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
goto err;
ok = 1;
err:
- BN_CTX_end(ctx);
if (!ok)
{
OPENSSL_free(r);
}
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);
+ wNAF[i] = compute_wNAF((i < num ? scalars[i] : scalar), wsize[i], &wNAF_len[i]);
if (wNAF[i] == NULL) goto err;
if (wNAF_len[i] > max_len)
max_len = wNAF_len[i];
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)
{
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