Change array representation of binary polynomials to make GF2m part of

the BN library more generally useful.

Submitted by: Douglas Stebila

CHANGES

@@ -4,6 +4,14 @@
 
  Changes between 0.9.8b and 0.9.9  [xx XXX xxxx]
 
+  *) Change the array representation of binary polynomials: the list
+     of degrees of non-zero coefficients is now terminated with -1.
+     Previously it was terminated with 0, which was also part of the
+     value; thus, the array representation was not applicable to
+     polynomials where t^0 has coefficient zero.  This change makes
+     the array representation useful in a more general context.
+     [Douglas Stebila]
+
   *) Various modifications and fixes to SSL/TLS cipher string
      handling.  For ECC, the code now distinguishes between fixed ECDH
      with RSA certificates on the one hand and with ECDSA certificates

crypto/bn/bn.h

@@ -558,24 +558,24 @@ int	BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
  *     t^p[0] + t^p[1] + ... + t^p[k]
  * where m = p[0] > p[1] > ... > p[k] = 0.
  */
-int	BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[]);
+int	BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[]);
 	/* r = a mod p */
 int	BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
-	const unsigned int p[], BN_CTX *ctx); /* r = (a * b) mod p */
-int	BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[],
+	const int p[], BN_CTX *ctx); /* r = (a * b) mod p */
+int	BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[],
 	BN_CTX *ctx); /* r = (a * a) mod p */
-int	BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *b, const unsigned int p[],
+int	BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *b, const int p[],
 	BN_CTX *ctx); /* r = (1 / b) mod p */
 int	BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
-	const unsigned int p[], BN_CTX *ctx); /* r = (a / b) mod p */
+	const int p[], BN_CTX *ctx); /* r = (a / b) mod p */
 int	BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
-	const unsigned int p[], BN_CTX *ctx); /* r = (a ^ b) mod p */
+	const int p[], BN_CTX *ctx); /* r = (a ^ b) mod p */
 int	BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a,
-	const unsigned int p[], BN_CTX *ctx); /* r = sqrt(a) mod p */
+	const int p[], BN_CTX *ctx); /* r = sqrt(a) mod p */
 int	BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a,
-	const unsigned int p[], BN_CTX *ctx); /* r^2 + r = a mod p */
-int	BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max);
-int	BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a);
+	const int p[], BN_CTX *ctx); /* r^2 + r = a mod p */
+int	BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max);
+int	BN_GF2m_arr2poly(const int p[], BIGNUM *a);
 
 /* faster mod functions for the 'NIST primes' 
  * 0 <= a < p^2 */

crypto/bn/bn_gf2m.c

@@ -258,7 +258,7 @@ int	BN_GF2m_add(BIGNUM *r, const BIGNUM *a, const BIGNUM *b)
 
 
 /* Performs modular reduction of a and store result in r.  r could be a. */
-int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[])
+int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const int p[])
 	{
 	int j, k;
 	int n, dN, d0, d1;
@@ -355,11 +355,11 @@ int BN_GF2m_mod_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[])
 int	BN_GF2m_mod(BIGNUM *r, const BIGNUM *a, const BIGNUM *p)
 	{
 	int ret = 0;
-	const int max = BN_num_bits(p);
-	unsigned int *arr=NULL;
+	const int max = BN_num_bits(p) + 1;
+	int *arr=NULL;
 	bn_check_top(a);
 	bn_check_top(p);
-	if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
+	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 	ret = BN_GF2m_poly2arr(p, arr, max);
 	if (!ret || ret > max)
 		{
@@ -377,7 +377,7 @@ err:
 /* Compute the product of two polynomials a and b, reduce modulo p, and store
  * the result in r.  r could be a or b; a could be b.
  */
-int	BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
+int	BN_GF2m_mod_mul_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
 	{
 	int zlen, i, j, k, ret = 0;
 	BIGNUM *s;
@@ -433,12 +433,12 @@ err:
 int	BN_GF2m_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
 	{
 	int ret = 0;
-	const int max = BN_num_bits(p);
-	unsigned int *arr=NULL;
+	const int max = BN_num_bits(p) + 1;
+	int *arr=NULL;
 	bn_check_top(a);
 	bn_check_top(b);
 	bn_check_top(p);
-	if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
+	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 	ret = BN_GF2m_poly2arr(p, arr, max);
 	if (!ret || ret > max)
 		{
@@ -454,7 +454,7 @@ err:
 
 
 /* Square a, reduce the result mod p, and store it in a.  r could be a. */
-int	BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
+int	BN_GF2m_mod_sqr_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
 	{
 	int i, ret = 0;
 	BIGNUM *s;
@@ -489,12 +489,12 @@ err:
 int	BN_GF2m_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 	{
 	int ret = 0;
-	const int max = BN_num_bits(p);
-	unsigned int *arr=NULL;
+	const int max = BN_num_bits(p) + 1;
+	int *arr=NULL;
 
 	bn_check_top(a);
 	bn_check_top(p);
-	if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
+	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 	ret = BN_GF2m_poly2arr(p, arr, max);
 	if (!ret || ret > max)
 		{
@@ -576,7 +576,7 @@ err:
  * function is only provided for convenience; for best performance, use the 
  * BN_GF2m_mod_inv function.
  */
-int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
+int BN_GF2m_mod_inv_arr(BIGNUM *r, const BIGNUM *xx, const int p[], BN_CTX *ctx)
 	{
 	BIGNUM *field;
 	int ret = 0;
@@ -702,7 +702,7 @@ err:
  * function is only provided for convenience; for best performance, use the 
  * BN_GF2m_mod_div function.
  */
-int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const unsigned int p[], BN_CTX *ctx)
+int BN_GF2m_mod_div_arr(BIGNUM *r, const BIGNUM *yy, const BIGNUM *xx, const int p[], BN_CTX *ctx)
 	{
 	BIGNUM *field;
 	int ret = 0;
@@ -727,7 +727,7 @@ err:
  * the result in r.  r could be a.
  * Uses simple square-and-multiply algorithm A.5.1 from IEEE P1363.
  */
-int	BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const unsigned int p[], BN_CTX *ctx)
+int	BN_GF2m_mod_exp_arr(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const int p[], BN_CTX *ctx)
 	{
 	int ret = 0, i, n;
 	BIGNUM *u;
@@ -773,12 +773,12 @@ err:
 int BN_GF2m_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *p, BN_CTX *ctx)
 	{
 	int ret = 0;
-	const int max = BN_num_bits(p);
-	unsigned int *arr=NULL;
+	const int max = BN_num_bits(p) + 1;
+	int *arr=NULL;
 	bn_check_top(a);
 	bn_check_top(b);
 	bn_check_top(p);
-	if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
+	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 	ret = BN_GF2m_poly2arr(p, arr, max);
 	if (!ret || ret > max)
 		{
@@ -796,7 +796,7 @@ err:
  * the result in r.  r could be a.
  * Uses exponentiation as in algorithm A.4.1 from IEEE P1363.
  */
-int	BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const unsigned int p[], BN_CTX *ctx)
+int	BN_GF2m_mod_sqrt_arr(BIGNUM *r, const BIGNUM *a, const int p[], BN_CTX *ctx)
 	{
 	int ret = 0;
 	BIGNUM *u;
@@ -832,11 +832,11 @@ err:
 int BN_GF2m_mod_sqrt(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 	{
 	int ret = 0;
-	const int max = BN_num_bits(p);
-	unsigned int *arr=NULL;
+	const int max = BN_num_bits(p) + 1;
+	int *arr=NULL;
 	bn_check_top(a);
 	bn_check_top(p);
-	if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) * max)) == NULL) goto err;
+	if ((arr = (int *)OPENSSL_malloc(sizeof(int) * max)) == NULL) goto err;
 	ret = BN_GF2m_poly2arr(p, arr, max);
 	if (!ret || ret > max)
 		{
@@ -853,7 +853,7 @@ err:
 /* Find r such that r^2 + r = a mod p.  r could be a. If no r exists returns 0.
  * Uses algorithms A.4.7 and A.4.6 from IEEE P1363.
  */
-int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const unsigned int p[], BN_CTX *ctx)
+int BN_GF2m_mod_solve_quad_arr(BIGNUM *r, const BIGNUM *a_, const int p[], BN_CTX *ctx)
 	{
 	int ret = 0, count = 0;
 	unsigned int j;
@@ -951,11 +951,11 @@ err:
 int BN_GF2m_mod_solve_quad(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 	{
 	int ret = 0;
-	const int max = BN_num_bits(p);
-	unsigned int *arr=NULL;
+	const int max = BN_num_bits(p) + 1;
+	int *arr=NULL;
 	bn_check_top(a);
 	bn_check_top(p);
-	if ((arr = (unsigned int *)OPENSSL_malloc(sizeof(unsigned int) *
+	if ((arr = (int *)OPENSSL_malloc(sizeof(int) *
 						max)) == NULL) goto err;
 	ret = BN_GF2m_poly2arr(p, arr, max);
 	if (!ret || ret > max)
@@ -971,20 +971,17 @@ err:
 	}
 
 /* Convert the bit-string representation of a polynomial
- * ( \sum_{i=0}^n a_i * x^i , where a_0 is *not* zero) into an array
- * of integers corresponding to the bits with non-zero coefficient.
+ * ( \sum_{i=0}^n a_i * x^i) into an array of integers corresponding 
+ * to the bits with non-zero coefficient.  Array is terminated with -1.
  * Up to max elements of the array will be filled.  Return value is total
- * number of coefficients that would be extracted if array was large enough.
+ * number of array elements that would be filled if array was large enough.
  */
-int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max)
+int BN_GF2m_poly2arr(const BIGNUM *a, int p[], int max)
 	{
 	int i, j, k = 0;
 	BN_ULONG mask;
 
-	if (BN_is_zero(a) || !BN_is_bit_set(a, 0))
-		/* a_0 == 0 => return error (the unsigned int array
-		 * must be terminated by 0)
-		 */
+	if (BN_is_zero(a))
 		return 0;
 
 	for (i = a->top - 1; i >= 0; i--)
@@ -1004,24 +1001,28 @@ int BN_GF2m_poly2arr(const BIGNUM *a, unsigned int p[], int max)
 			}
 		}
 
+	if (k < max) {
+		p[k] = -1;
+		k++;
+	}
+
 	return k;
 	}
 
 /* Convert the coefficient array representation of a polynomial to a 
- * bit-string.  The array must be terminated by 0.
+ * bit-string.  The array must be terminated by -1.
  */
-int BN_GF2m_arr2poly(const unsigned int p[], BIGNUM *a)
+int BN_GF2m_arr2poly(const int p[], BIGNUM *a)
 	{
 	int i;
 
 	bn_check_top(a);
 	BN_zero(a);
-	for (i = 0; p[i] != 0; i++)
+	for (i = 0; p[i] != -1; i++)
 		{
 		if (BN_set_bit(a, p[i]) == 0)
 			return 0;
 		}
-	BN_set_bit(a, 0);
 	bn_check_top(a);
 
 	return 1;

crypto/bn/bntest.c

@@ -1118,8 +1118,8 @@ int test_gf2m_mod(BIO *bp)
 	{
 	BIGNUM *a,*b[2],*c,*d,*e;
 	int i, j, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();
@@ -1176,8 +1176,8 @@ int test_gf2m_mod_mul(BIO *bp,BN_CTX *ctx)
 	{
 	BIGNUM *a,*b[2],*c,*d,*e,*f,*g,*h;
 	int i, j, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();
@@ -1247,8 +1247,8 @@ int test_gf2m_mod_sqr(BIO *bp,BN_CTX *ctx)
 	{
 	BIGNUM *a,*b[2],*c,*d;
 	int i, j, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();
@@ -1306,8 +1306,8 @@ int test_gf2m_mod_inv(BIO *bp,BN_CTX *ctx)
 	{
 	BIGNUM *a,*b[2],*c,*d;
 	int i, j, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();
@@ -1361,8 +1361,8 @@ int test_gf2m_mod_div(BIO *bp,BN_CTX *ctx)
 	{
 	BIGNUM *a,*b[2],*c,*d,*e,*f;
 	int i, j, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();
@@ -1424,8 +1424,8 @@ int test_gf2m_mod_exp(BIO *bp,BN_CTX *ctx)
 	{
 	BIGNUM *a,*b[2],*c,*d,*e,*f;
 	int i, j, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();
@@ -1495,8 +1495,8 @@ int test_gf2m_mod_sqrt(BIO *bp,BN_CTX *ctx)
 	{
 	BIGNUM *a,*b[2],*c,*d,*e,*f;
 	int i, j, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();
@@ -1554,8 +1554,8 @@ int test_gf2m_mod_solve_quad(BIO *bp,BN_CTX *ctx)
 	{
 	BIGNUM *a,*b[2],*c,*d,*e;
 	int i, j, s = 0, t, ret = 0;
-	unsigned int p0[] = {163,7,6,3,0};
-	unsigned int p1[] = {193,15,0};
+	int p0[] = {163,7,6,3,0,-1};
+	int p1[] = {193,15,0,-1};
 
 	a=BN_new();
 	b[0]=BN_new();

crypto/ec/ec2_smpl.c

@@ -157,6 +157,7 @@ void ec_GF2m_simple_group_clear_finish(EC_GROUP *group)
 	group->poly[2] = 0;
 	group->poly[3] = 0;
 	group->poly[4] = 0;
+	group->poly[5] = -1;
 	}
 
 
@@ -174,6 +175,7 @@ int ec_GF2m_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
 	dest->poly[2] = src->poly[2];
 	dest->poly[3] = src->poly[3];
 	dest->poly[4] = src->poly[4];
+	dest->poly[5] = src->poly[5];
 	bn_wexpand(&dest->a, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2);
 	bn_wexpand(&dest->b, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2);
 	for (i = dest->a.top; i < dest->a.dmax; i++) dest->a.d[i] = 0;
@@ -190,7 +192,7 @@ int ec_GF2m_simple_group_set_curve(EC_GROUP *group,
 
 	/* group->field */
 	if (!BN_copy(&group->field, p)) goto err;
-	i = BN_GF2m_poly2arr(&group->field, group->poly, 5);
+	i = BN_GF2m_poly2arr(&group->field, group->poly, 6) - 1;
 	if ((i != 5) && (i != 3))
 		{
 		ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_SET_CURVE, EC_R_UNSUPPORTED_FIELD);

crypto/ec/ec_lcl.h

@@ -205,11 +205,14 @@ struct ec_group_st {
 	               * irreducible polynomial defining the field.
 	               */
 
-	unsigned int poly[5]; /* Field specification for curves over GF(2^m).
-	                       * The irreducible f(t) is then of the form:
-	                       *     t^poly[0] + t^poly[1] + ... + t^poly[k]
-	                       * where m = poly[0] > poly[1] > ... > poly[k] = 0.
-	                       */
+	int poly[6]; /* Field specification for curves over GF(2^m).
+	              * The irreducible f(t) is then of the form:
+	              *     t^poly[0] + t^poly[1] + ... + t^poly[k]
+	              * where m = poly[0] > poly[1] > ... > poly[k] = 0.
+	              * The array is terminated with poly[k+1]=-1.
+	              * All elliptic curve irreducibles have at most 5
+	              * non-zero terms.
+	              */
 
 	BIGNUM a, b; /* Curve coefficients.
 	              * (Here the assumption is that BIGNUMs can be used