/* 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
#include "ec_lcl.h"
-/* TODO: width-m NAFs */
-
/* TODO: optional precomputation of multiples of the generator */
+
+/*
+ * wNAF-based interleaving multi-exponentation method
+ * (<URL:http://www.informatik.tu-darmstadt.de/TI/Mitarbeiter/moeller.html#multiexp>)
+ */
+
+
+/* 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
+ * 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)
+ {
+ int window_val;
+ int ok = 0;
+ signed char *r = NULL;
+ int sign = 1;
+ int bit, next_bit, mask;
+ size_t len = 0, j;
+
+ 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 (scalar->neg)
+ {
+ sign = -1;
+ }
+
+ 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 ((window_val != 0) || (j + w + 1 < len)) /* if j+w+1 >= len, window_val will not increase */
+ {
+ int digit = 0;
+
+ /* 0 <= window_val <= 2^(w+1) */
+
+ if (window_val & 1)
+ {
+ /* 0 < window_val < 2^(w+1) */
+
+ if (window_val & bit)
+ {
+ 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
+ }
+ else
+ {
+ digit = window_val; /* 0 < digit < 2^w */
+ }
+
+ if (digit <= -bit || digit >= bit || !(digit & 1))
+ {
+ ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
+ goto err;
+ }
+
+ 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 * 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 (j > len + 1)
+ {
+ ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR);
+ goto err;
+ }
+ len = j;
+ ok = 1;
+
+ err:
+ 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) >= 2000 ? 6 : \
(b) >= 800 ? 5 : \
(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],
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 += 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',
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_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]);
+ 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 */
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;
+ int digit = wNAF[i][k];
+ int is_neg;
- s = i < num ? scalars[i] : scalar;
-
- 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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