Improve EC efficiency.

CHANGES

@@ -12,6 +12,10 @@
          *) applies to 0.9.6a/0.9.6b/0.9.6c and 0.9.7
          +) applies to 0.9.7 only
 
+  +) Use wNAFs in EC_POINTs_mul() for improved efficiency (about 10%
+     better than before for single multiplications over P-192 or P-224).
+     [Bodo Moeller]
+
   -) [In 0.9.6c-engine release:]
      Add support for Broadcom crypto accelerator cards, backported
      from 0.9.7.
@@ -943,9 +947,12 @@ des-cbc           3624.96k     5258.21k     5530.91k     5624.30k     5628.26k
      don't write to the wrong index in ERR_set_error_data.
      [Bodo Moeller]
 
-  +) Function EC_POINTs_mul for simultaneous scalar multiplication
-     of an arbitrary number of elliptic curve points, optionally
-     including the generator defined for the EC_GROUP.
+  +) Function EC_POINTs_mul for multiple scalar multiplication
+     of an arbitrary number of elliptic curve points
+          \sum scalars[i]*points[i],
+     optionally including the generator defined for the EC_GROUP:
+          scalar*generator +  \sum scalars[i]*points[i].
+
      EC_POINT_mul is a simple wrapper function for the typical case
      that the point list has just one item (besides the optional
      generator).

crypto/ec/ec.h

@@ -177,6 +177,7 @@ void ERR_load_EC_strings(void);
 /* Error codes for the EC functions. */
 
 /* Function codes. */
+#define EC_F_COMPUTE_WNAF				 143
 #define EC_F_EC_GFP_MONT_FIELD_DECODE			 133
 #define EC_F_EC_GFP_MONT_FIELD_ENCODE			 134
 #define EC_F_EC_GFP_MONT_FIELD_MUL			 131

crypto/ec/ec_err.c

@@ -66,6 +66,7 @@
 #ifndef OPENSSL_NO_ERR
 static ERR_STRING_DATA EC_str_functs[]=
 	{
+{ERR_PACK(0,EC_F_COMPUTE_WNAF,0),	"COMPUTE_WNAF"},
 {ERR_PACK(0,EC_F_EC_GFP_MONT_FIELD_DECODE,0),	"ec_GFp_mont_field_decode"},
 {ERR_PACK(0,EC_F_EC_GFP_MONT_FIELD_ENCODE,0),	"ec_GFp_mont_field_encode"},
 {ERR_PACK(0,EC_F_EC_GFP_MONT_FIELD_MUL,0),	"ec_GFp_mont_field_mul"},

crypto/ec/ec_mult.c

@@ -58,11 +58,369 @@
 #include "ec_lcl.h"
 
 
-/* TODO: width-m NAFs */
-
 /* TODO: optional precomputation of multiples of the generator */
 
 
+#if 1
+/*
+ * wNAF-based interleaving multi-exponentation method
+ */
+
+
+
+/* Determine the width-(w+1) Non-Adjacent Form 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) /* 'unsigned 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 */
+#define EC_window_bits_for_scalar_size(b) \
+		((b) >= 2000 ? 6 : \
+		 (b) >=  800 ? 5 : \
+		 (b) >=  300 ? 4 : \
+		 (b) >=   70 ? 3 : \
+		 (b) >=   20 ? 2 : \
+		  1)
+
+/* Compute
+ *      \sum scalars[i]*points[i],
+ * also including
+ *      scalar*generator
+ * 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)
+	{
+	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 */
+	size_t *wNAF_len = NULL;
+	size_t max_len = 0;
+	signed char **wNAF = NULL; /* individual wNAFs */
+	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 : \
@@ -143,14 +501,6 @@
  *     w = 1  if   19 >= b
  */
 
-
-
-/* Compute
- *      \sum scalars[i]*points[i],
- * also including
- *      scalar*generator
- * 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)
 	{
@@ -369,6 +719,7 @@ int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *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)