BN_sqrt

8 files changed, 438 insertions, 21 deletions

diff --git a/crypto/bn/Makefile.ssl b/crypto/bn/Makefile.ssl
index b32fd97c41..962cda9bae 100644
--- a/crypto/bn/Makefile.ssl
+++ b/crypto/bn/Makefile.ssl
@@ -37,12 +37,12 @@ APPS=
 LIB=$(TOP)/libcrypto.a
 LIBSRC=	bn_add.c bn_div.c bn_exp.c bn_lib.c bn_ctx.c bn_mul.c bn_mod.c \
 	bn_print.c bn_rand.c bn_shift.c bn_word.c bn_blind.c \
-	bn_kron.c bn_gcd.c bn_prime.c bn_err.c bn_sqr.c bn_asm.c \
+	bn_kron.c bn_sqrt.c bn_gcd.c bn_prime.c bn_err.c bn_sqr.c bn_asm.c \
 	bn_recp.c bn_mont.c bn_mpi.c bn_exp2.c
 
 LIBOBJ=	bn_add.o bn_div.o bn_exp.o bn_lib.o bn_ctx.o bn_mul.o bn_mod.o \
 	bn_print.o bn_rand.o bn_shift.o bn_word.o bn_blind.o \
-	bn_kron.o bn_gcd.o bn_prime.o bn_err.o bn_sqr.o $(BN_ASM) \
+	bn_kron.o bn_sqrt.o bn_gcd.o bn_prime.o bn_err.o bn_sqr.o $(BN_ASM) \
 	bn_recp.o bn_mont.o bn_mpi.o bn_exp2.o
 
 SRC= $(LIBSRC)
diff --git a/crypto/bn/bn.h b/crypto/bn/bn.h
index c81f3de8be..b55ac95642 100644
--- a/crypto/bn/bn.h
+++ b/crypto/bn/bn.h
@@ -238,7 +238,7 @@ typedef struct bignum_st
 	} BIGNUM;
 
 /* Used for temp variables */
-#define BN_CTX_NUM	16
+#define BN_CTX_NUM	20
 #define BN_CTX_NUM_POS	12
 typedef struct bignum_ctx
 	{
@@ -357,6 +357,7 @@ int	BN_mod_sub(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *m, BN_
 int	BN_mod_sub_quick(BIGNUM *r, const BIGNUM *a, const BIGNUM *b, const BIGNUM *m);
 int	BN_mod_mul(BIGNUM *r, const BIGNUM *a, const BIGNUM *b,
 	const BIGNUM *m, BN_CTX *ctx);
+int	BN_mod_sqr(BIGNUM *r, const BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);
 int	BN_mod_lshift1(BIGNUM *r, const BIGNUM *a, const BIGNUM *m, BN_CTX *ctx);
 int	BN_mod_lshift1_quick(BIGNUM *r, const BIGNUM *a, const BIGNUM *m);
 int	BN_mod_lshift(BIGNUM *r, const BIGNUM *a, int n, const BIGNUM *m, BN_CTX *ctx);
@@ -414,6 +415,8 @@ int	BN_gcd(BIGNUM *r,const BIGNUM *a,const BIGNUM *b,BN_CTX *ctx);
 int	BN_kronecker(const BIGNUM *a,const BIGNUM *b,BN_CTX *ctx); /* returns -2 for error */
 BIGNUM *BN_mod_inverse(BIGNUM *ret,
 	const BIGNUM *a, const BIGNUM *n,BN_CTX *ctx);
+BIGNUM *BN_mod_sqrt(BIGNUM *ret,
+	const BIGNUM *a, const BIGNUM *n,BN_CTX *ctx);
 BIGNUM *BN_generate_prime(BIGNUM *ret,int bits,int safe,
 	const BIGNUM *add, const BIGNUM *rem,
 	void (*callback)(int,int,void *),void *cb_arg);
@@ -517,6 +520,7 @@ void bn_dump1(FILE *o, const char *a, const BN_ULONG *b,int n);
 #define BN_F_BN_MOD_INVERSE				 110
 #define BN_F_BN_MOD_LSHIFT_QUICK			 119
 #define BN_F_BN_MOD_MUL_RECIPROCAL			 111
+#define BN_F_BN_MOD_SQRT				 121
 #define BN_F_BN_MPI2BN					 112
 #define BN_F_BN_NEW					 113
 #define BN_F_BN_RAND					 114
@@ -531,8 +535,11 @@ void bn_dump1(FILE *o, const char *a, const BN_ULONG *b,int n);
 #define BN_R_EXPAND_ON_STATIC_BIGNUM_DATA		 105
 #define BN_R_INPUT_NOT_REDUCED				 110
 #define BN_R_INVALID_LENGTH				 106
+#define BN_R_NOT_A_SQUARE				 111
 #define BN_R_NOT_INITIALIZED				 107
 #define BN_R_NO_INVERSE					 108
+#define BN_R_P_IS_NOT_PRIME				 112
+#define BN_R_TOO_MANY_ITERATIONS			 113
 #define BN_R_TOO_MANY_TEMPORARY_VARIABLES		 109
 
 #ifdef  __cplusplus
diff --git a/crypto/bn/bn_err.c b/crypto/bn/bn_err.c
index 75a3458b11..afb9320322 100644
--- a/crypto/bn/bn_err.c
+++ b/crypto/bn/bn_err.c
@@ -83,6 +83,7 @@ static ERR_STRING_DATA BN_str_functs[]=
 {ERR_PACK(0,BN_F_BN_MOD_INVERSE,0),	"BN_mod_inverse"},
 {ERR_PACK(0,BN_F_BN_MOD_LSHIFT_QUICK,0),	"BN_mod_lshift_quick"},
 {ERR_PACK(0,BN_F_BN_MOD_MUL_RECIPROCAL,0),	"BN_mod_mul_reciprocal"},
+{ERR_PACK(0,BN_F_BN_MOD_SQRT,0),	"BN_mod_sqrt"},
 {ERR_PACK(0,BN_F_BN_MPI2BN,0),	"BN_mpi2bn"},
 {ERR_PACK(0,BN_F_BN_NEW,0),	"BN_new"},
 {ERR_PACK(0,BN_F_BN_RAND,0),	"BN_rand"},
@@ -100,8 +101,11 @@ static ERR_STRING_DATA BN_str_reasons[]=
 {BN_R_EXPAND_ON_STATIC_BIGNUM_DATA       ,"expand on static bignum data"},
 {BN_R_INPUT_NOT_REDUCED                  ,"input not reduced"},
 {BN_R_INVALID_LENGTH                     ,"invalid length"},
+{BN_R_NOT_A_SQUARE                       ,"not a square"},
 {BN_R_NOT_INITIALIZED                    ,"not initialized"},
 {BN_R_NO_INVERSE                         ,"no inverse"},
+{BN_R_P_IS_NOT_PRIME                     ,"p is not prime"},
+{BN_R_TOO_MANY_ITERATIONS                ,"too many iterations"},
 {BN_R_TOO_MANY_TEMPORARY_VARIABLES       ,"too many temporary variables"},
 {0,NULL}
 	};
diff --git a/crypto/bn/bn_exp.c b/crypto/bn/bn_exp.c
index 35ab56efc0..51c8282593 100644
--- a/crypto/bn/bn_exp.c
+++ b/crypto/bn/bn_exp.c
@@ -205,6 +205,8 @@ int BN_mod_exp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p, const BIGNUM *m,
 		if (a->top == 1 && !a->neg)
 			{
 			BN_ULONG A = a->d[0];
+			if (m->top == 1)
+				A %= m->d[0]; /* make sure that A is reduced */
 			ret=BN_mod_exp_mont_word(r,A,p,m,ctx,NULL);
 			}
 		else
@@ -235,8 +237,13 @@ int BN_mod_exp_recp(BIGNUM *r, const BIGNUM *a, const BIGNUM *p,
 
 	if (bits == 0)
 		{
-		BN_one(r);
-		return(1);
+		ret = BN_one(r);
+		return ret;
+		}
+	if (BN_is_zero(a))
+		{
+		ret = BN_zero(r);
+		return ret;
 		}
 
 	BN_CTX_start(ctx);
@@ -355,8 +362,13 @@ int BN_mod_exp_mont(BIGNUM *rr, const BIGNUM *a, const BIGNUM *p,
 	bits=BN_num_bits(p);
 	if (bits == 0)
 		{
-		BN_one(rr);
-		return(1);
+		ret = BN_one(rr);
+		return ret;
+		}
+	if (BN_is_zero(a))
+		{
+		ret = BN_zero(rr);
+		return ret;
 		}
 	BN_CTX_start(ctx);
 	d = BN_CTX_get(ctx);
@@ -500,9 +512,15 @@ int BN_mod_exp_mont_word(BIGNUM *rr, BN_ULONG a, const BIGNUM *p,
 	bits = BN_num_bits(p);
 	if (bits == 0)
 		{
-		BN_one(rr);
-		return(1);
+		ret = BN_one(rr);
+		return ret;
 		}
+	if (a == 0)
+		{
+		ret = BN_zero(rr);
+		return ret;
+		}
+
 	BN_CTX_start(ctx);
 	d = BN_CTX_get(ctx);
 	r = BN_CTX_get(ctx);
@@ -611,8 +629,13 @@ int BN_mod_exp_simple(BIGNUM *r,
 
 	if (bits == 0)
 		{
-		BN_one(r);
-		return(1);
+		ret = BN_one(r);
+		return ret;
+		}
+	if (BN_is_zero(a))
+		{
+		ret = BN_one(r);
+		return ret;
 		}
 
 	BN_CTX_start(ctx);
diff --git a/crypto/bn/bn_exp2.c b/crypto/bn/bn_exp2.c
index 70c4d83a79..56f1c959bd 100644
--- a/crypto/bn/bn_exp2.c
+++ b/crypto/bn/bn_exp2.c
@@ -141,9 +141,15 @@ int BN_mod_exp2_mont(BIGNUM *rr, const BIGNUM *a1, const BIGNUM *p1,
 	bits2=BN_num_bits(p2);
 	if ((bits1 == 0) && (bits2 == 0))
 		{
-		BN_one(rr);
-		return(1);
+		ret = BN_one(rr);
+		return ret;
 		}
+	if (BN_is_zero(a1) || BN_is_zero(a2))
+		{
+		ret = BN_zero(rr);
+		return ret;
+		}
+	
 	bits=(bits1 > bits2)?bits1:bits2;
 
 	BN_CTX_start(ctx);
diff --git a/crypto/bn/bn_kron.c b/crypto/bn/bn_kron.c
index 0dd8a194cb..49f75594ae 100644
--- a/crypto/bn/bn_kron.c
+++ b/crypto/bn/bn_kron.c
@@ -1,5 +1,3 @@
-/* totally untested */
-
 /* crypto/bn/bn_kron.c */
 /* ====================================================================
  * Copyright (c) 1998-2000 The OpenSSL Project.  All rights reserved.
diff --git a/crypto/bn/bn_sqrt.c b/crypto/bn/bn_sqrt.c
index dd6d86a43d..5176772e4e 100644
--- a/crypto/bn/bn_sqrt.c
+++ b/crypto/bn/bn_sqrt.c
@@ -1 +1,308 @@
-XXX
+/* crypto/bn/bn_mod.c */
+/* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
+ * and Bodo Moeller for the OpenSSL project. */
+/* ====================================================================
+ * Copyright (c) 1998-2000 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
+ * are met:
+ *
+ * 1. Redistributions of source code must retain the above copyright
+ *    notice, this list of conditions and the following disclaimer. 
+ *
+ * 2. Redistributions in binary form must reproduce the above copyright
+ *    notice, this list of conditions and the following disclaimer in
+ *    the documentation and/or other materials provided with the
+ *    distribution.
+ *
+ * 3. All advertising materials mentioning features or use of this
+ *    software must display the following acknowledgment:
+ *    "This product includes software developed by the OpenSSL Project
+ *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
+ *
+ * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
+ *    endorse or promote products derived from this software without
+ *    prior written permission. For written permission, please contact
+ *    openssl-core@openssl.org.
+ *
+ * 5. Products derived from this software may not be called "OpenSSL"
+ *    nor may "OpenSSL" appear in their names without prior written
+ *    permission of the OpenSSL Project.
+ *
+ * 6. Redistributions of any form whatsoever must retain the following
+ *    acknowledgment:
+ *    "This product includes software developed by the OpenSSL Project
+ *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
+ * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+ * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
+ * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+ * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
+ * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
+ * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
+ * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+ * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
+ * OF THE POSSIBILITY OF SUCH DAMAGE.
+ * ====================================================================
+ *
+ * This product includes cryptographic software written by Eric Young
+ * (eay@cryptsoft.com).  This product includes software written by Tim
+ * Hudson (tjh@cryptsoft.com).
+ *
+ */
+
+#include "cryptlib.h"
+#include "bn_lcl.h"
+
+
+BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 
+/* Returns 'ret' such that
+ *      ret^2 == a (mod p),
+ * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
+ * in Algebraic Computational Number Theory", algorithm 1.5.1).
+ * 'p' must be prime!
+ */
+	{
+	BIGNUM *ret = in;
+	int err = 1;
+	int r;
+	BIGNUM *b, *q, *t, *x, *y;
+	int e, i, j;
+	
+	if (!BN_is_odd(p) || BN_abs_is_word(p, 1))
+		{
+		if (BN_abs_is_word(p, 2))
+			{
+			if (ret == NULL)
+				ret = BN_new();
+			if (ret == NULL)
+				goto end;
+			if (!BN_set_word(ret, BN_is_bit_set(a, 0)))
+				{
+				BN_free(ret);
+				return NULL;
+				}
+			return ret;
+			}
+
+		BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
+		return(NULL);
+		}
+
+#if 0 /* if BN_mod_sqrt is used with correct input, this just wastes time */
+	r = BN_kronecker(a, p, ctx);
+	if (r < -1) return NULL;
+	if (r == -1)
+		{
+		BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
+		return(NULL);
+		}
+#endif
+
+	BN_CTX_start(ctx);
+	b = BN_CTX_get(ctx);
+	q = BN_CTX_get(ctx);
+	t = BN_CTX_get(ctx);
+	x = BN_CTX_get(ctx);
+	y = BN_CTX_get(ctx);
+	if (y == NULL) goto end;
+	
+	if (ret == NULL)
+		ret = BN_new();
+	if (ret == NULL) goto end;
+
+	/* now write  |p| - 1  as  2^e*q  where  q  is odd */
+	e = 1;
+	while (!BN_is_bit_set(p, e))
+		e++;
+	if (!BN_rshift(q, p, e)) goto end;
+	q->neg = 0;
+
+	if (e == 1)
+		{
+		/* The easy case:  (p-1)/2  is odd, so 2 has an inverse
+		 * modulo  (p-1)/2,  and square roots can be computed
+		 * directly by modular exponentiation.
+		 * We have
+		 *     2 * (p+1)/4 == 1   (mod (p-1)/2),
+		 * so we can use exponent  (p+1)/4,  i.e.  (q+1)/2.
+		 */
+		if (!BN_add_word(q,1)) goto end;
+		if (!BN_rshift1(q,q)) goto end;
+		if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
+		err = 0;
+		goto end;
+		}
+	
+	/* e > 1, so we really have to use the Tonelli/Shanks algorithm.
+	 * First, find some  y  that is not a square. */
+	i = 1;
+	do
+		{
+		/* For efficiency, try small numbers first;
+		 * if this fails, try random numbers.
+		 */
+		if (i < 20)
+			{
+			if (!BN_set_word(y, i)) goto end;
+			}
+		else
+			{
+			if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end;
+			if (BN_ucmp(y, p) >= 0)
+				{
+				if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end;
+				}
+			/* now 0 <= y < |p| */
+			if (BN_is_zero(y))
+				if (!BN_set_word(y, i)) goto end;
+			}
+		
+		r = BN_kronecker(y, p, ctx);
+		if (r < -1) goto end;
+		if (r == 0)
+			{
+			/* m divides p */
+			BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
+			goto end;
+			}
+		}
+	while (r == 1 && i++ < 80);
+	
+	if (r != -1)
+		{
+		/* Many rounds and still no non-square -- this is more likely
+		 * a bug than just bad luck.
+		 * Even if  p  is not prime, we should have found some  y
+		 * such that r == -1.
+		 */
+		BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS);
+		goto end;
+		}
+
+
+	/* Now that we have some non-square, we can find an element
+	 * of order  2^e  by computing its q'th power. */
+	if (!BN_mod_exp(y, y, q, p, ctx)) goto end;
+	if (BN_is_one(y))
+		{
+		BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME);
+		goto end;
+		}
+
+	/* Now we know that (if  p  is indeed prime) there is an integer
+	 * k,  0 <= k < 2^e,  such that
+	 *
+	 *      a^q * y^k == 1   (mod p).
+	 *
+	 * As  a^q  is a square and  y  is not,  k  must be even.
+	 * q+1  is even, too, so there is an element
+	 *
+	 *     X := a^((q+1)/2) * y^(k/2),
+	 *
+	 * and it satisfies
+	 *
+	 *     X^2 = a^q * a     * y^k
+	 *         = a,
+	 *
+	 * so it is the square root that we are looking for.
+	 */
+	
+	/* t := (q-1)/2  (note that  q  is odd) */
+	if (!BN_rshift1(t, q)) goto end;
+	
+	/* x := a^((q-1)/2) */
+	if (BN_is_zero(t)) /* special case: p = 2^e + 1 */
+		{
+		if (!BN_nnmod(t, a, p, ctx)) goto end;
+		if (BN_is_zero(t))
+			{
+			/* special case: a == 0  (mod p) */
+			if (!BN_zero(ret)) goto end;
+			err = 0;
+			goto end;
+			}
+		else
+			if (!BN_one(x)) goto end;
+		}
+	else
+		{
+		if (!BN_mod_exp(x, a, t, p, ctx)) goto end;
+		if (BN_is_zero(x))
+			{
+			/* special case: a == 0  (mod p) */
+			if (!BN_zero(ret)) goto end;
+			err = 0;
+			goto end;
+			}
+		}
+
+	/* b := a*x^2  (= a^q) */
+	if (!BN_mod_sqr(b, x, p, ctx)) goto end;
+	if (!BN_mod_mul(b, b, a, p, ctx)) goto end;
+	
+	/* x := a*x    (= a^((q+1)/2)) */
+	if (!BN_mod_mul(x, x, a, p, ctx)) goto end;
+
+	while (1)
+		{
+		/* Now  b  is  a^q * y^k  for some even  k  (0 <= k < 2^E
+		 * where  E  refers to the original value of  e,  which we
+		 * don't keep in a variable),  and  x  is  a^((q+1)/2) * y^(k/2).
+		 *
+		 * We have  a*b = x^2,
+		 *    y^2^(e-1) = -1,
+		 *    b^2^(e-1) = 1.
+		 */
+
+		if (BN_is_one(b))
+			{
+			if (!BN_copy(ret, x)) goto end;
+			err = 0;
+			goto end;
+			}
+
+
+		/* find smallest  i  such that  b^(2^i) = 1 */
+		i = 1;
+		if (!BN_mod_sqr(t, b, p, ctx)) goto end;
+		while (!BN_is_one(t))
+			{
+			i++;
+			if (i == e)
+				{
+				BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
+				goto end;
+				}
+			if (!BN_mod_mul(t, t, t, p, ctx)) goto end;
+			}
+		
+
+		/* t := y^2^(e - i - 1) */
+		if (!BN_copy(t, y)) goto end;
+		for (j = e - i - 1; j > 0; j--)
+			{
+			if (!BN_mod_sqr(t, t, p, ctx)) goto end;
+			}
+		if (!BN_mod_mul(y, t, t, p, ctx)) goto end;
+		if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
+		if (!BN_mod_mul(b, b, y, p, ctx)) goto end;
+		e = i;
+		}
+
+ end:
+	if (err)
+		{
+		if (ret != NULL && ret != in)
+			{
+			BN_clear_free(ret);
+			}
+		ret = NULL;
+		}
+	BN_CTX_end(ctx);
+	return ret;
+	}
diff --git a/crypto/bn/bntest.c b/crypto/bn/bntest.c
index a664f3b91a..8f5b72c7f2 100644
--- a/crypto/bn/bntest.c
+++ b/crypto/bn/bntest.c
@@ -92,6 +92,7 @@ int test_mod_mul(BIO *bp,BN_CTX *ctx);
 int test_mod_exp(BIO *bp,BN_CTX *ctx);
 int test_exp(BIO *bp,BN_CTX *ctx);
 int test_kron(BIO *bp,BN_CTX *ctx);
+int test_sqrt(BIO *bp,BN_CTX *ctx);
 int rand_neg(void);
 static int results=0;
 
@@ -233,6 +234,10 @@ int main(int argc, char *argv[])
 	if (!test_kron(out,ctx)) goto err;
 	BIO_flush(out);
 
+	message(out,"BN_mod_sqrt");
+	if (!test_sqrt(out,ctx)) goto err;
+	BIO_flush(out);
+
 	BN_CTX_free(ctx);
 	BIO_free(out);
 
@@ -940,11 +945,6 @@ int test_kron(BIO *bp, BN_CTX *ctx)
 
 	if (!BN_generate_prime(b, 512, 0, NULL, NULL, genprime_cb, NULL)) goto err;
 	putc('\n', stderr);
-	if (1 != BN_is_prime(b, 10, NULL, ctx, NULL))
-		{
-		fprintf(stderr, "BN_is_prime failed\n");
-		goto err;
-		}
 
 	for (i = 0; i < num0; i++)
 		{
@@ -998,6 +998,78 @@ int test_kron(BIO *bp, BN_CTX *ctx)
 	return ret;
 	}
 
+int test_sqrt(BIO *bp, BN_CTX *ctx)
+	{
+	BIGNUM *a,*p,*r;
+	int i, j;
+	int ret = 0;
+
+	a = BN_new();
+	p = BN_new();
+	r = BN_new();
+	if (a == NULL || p == NULL || r == NULL) goto err;
+	
+	for (i = 0; i < 16; i++)
+		{
+		if (i < 8)
+			{
+			unsigned primes[8] = { 2, 3, 7, 11, 13, 17, 19 };
+			
+			if (!BN_set_word(p, primes[i])) goto err;
+			}
+		else
+			{
+			if (!BN_set_word(a, 32)) goto err;
+			if (!BN_set_word(r, 2*i + 1)) goto err;
+		
+			if (!BN_generate_prime(p, 256, 0, a, r, genprime_cb, NULL)) goto err;
+			putc('\n', stderr);
+			}
+
+		for (j = 0; j < num2; j++)
+			{
+			/* construct 'a' such that it is a square modulo p,
+			 * but in general not a proper square and not reduced modulo p */
+			if (!BN_rand(r, 256, 0, 3)) goto err;
+			if (!BN_nnmod(r, r, p, ctx)) goto err;
+			if (!BN_mod_sqr(r, r, p, ctx)) goto err;
+			if (!BN_rand(a, 256, 0, 3)) goto err;
+			if (!BN_nnmod(a, a, p, ctx)) goto err;
+			if (!BN_mod_sqr(a, a, p, ctx)) goto err;
+			if (!BN_mul(a, a, r, ctx)) goto err;
+
+			if (!BN_mod_sqrt(r, a, p, ctx)) goto err;
+			if (!BN_mod_sqr(r, r, p, ctx)) goto err;
+
+			if (!BN_nnmod(a, a, p, ctx)) goto err;
+
+			if (BN_cmp(a, r) != 0)
+				{
+				fprintf(stderr, "BN_mod_sqrt failed: a = ");
+				BN_print_fp(stderr, a);
+				fprintf(stderr, ", r = ");
+				BN_print_fp(stderr, r);
+				fprintf(stderr, ", p = ");
+				BN_print_fp(stderr, p);
+				fprintf(stderr, "\n");
+				goto err;
+				}
+
+			putc('.', stderr);
+			fflush(stderr);
+			}
+		
+		putc('\n', stderr);
+		fflush(stderr);
+		}
+	ret = 1;
+ err:
+	if (a != NULL) BN_free(a);
+	if (p != NULL) BN_free(p);
+	if (r != NULL) BN_free(r);
+	return ret;
+	}
+
 int test_lshift(BIO *bp,BN_CTX *ctx,BIGNUM *a_)
 	{
 	BIGNUM *a,*b,*c,*d;