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[openssl.git] / crypto / bn / bn_sqrt.c

diff --git a/crypto/bn/bn_sqrt.c b/crypto/bn/bn_sqrt.c
index a54d9d29195a5c9c693b9b2c621cfca87b85f7ca..e2a1105dc838b2f29b3867f06f653d08a90f368f 100644 (file)
--- a/crypto/bn/bn_sqrt.c
+++ b/crypto/bn/bn_sqrt.c
@@ -65,6 +65,8 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
  * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course
  * in Algebraic Computational Number Theory", algorithm 1.5.1).
  * 'p' must be prime!
+ * If 'a' is not a square, this is not necessarily detected by
+ * the algorithms; a bogus result must be expected in this case.
  */
        {
        BIGNUM *ret = in;
@@ -133,21 +135,19 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
        e = 1;
        while (!BN_is_bit_set(p, e))
                e++;
-       if (e > 2)
-               /* we don't need this  q  if  e = 1 or 2 */
-               if (!BN_rshift(q, p, e)) goto end;
-       q->neg = 0;
+       /* we'll set  q  later (if needed) */
 
        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
+               /* 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.  (p-3)/4 + 1.
+                *     2 * (|p|+1)/4 == 1   (mod (|p|-1)/2),
+                * so we can use exponent  (|p|+1)/4,  i.e.  (|p|-3)/4 + 1.
                 */
                if (!BN_rshift(q, p, 2)) goto end;
+               q->neg = 0;
                if (!BN_add_word(q, 1)) goto end;
                if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
                err = 0;
@@ -156,16 +156,16 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
        
        if (e == 2)
                {
-               /* p == 5  (mod 8)
+               /* |p| == 5  (mod 8)
                 *
                 * In this case  2  is always a non-square since
                 * Legendre(2,p) = (-1)^((p^2-1)/8)  for any odd prime.
                 * So if  a  really is a square, then  2*a  is a non-square.
                 * Thus for
-                *      b := (2*a)^((p-5)/8),
+                *      b := (2*a)^((|p|-5)/8),
                 *      i := (2*a)*b^2
                 * we have
-                *     i^2 = (2*a)^((1 + (p-5)/4)*2)
+                *     i^2 = (2*a)^((1 + (|p|-5)/4)*2)
                 *         = (2*a)^((p-1)/2)
                 *         = -1;
                 * so if we set
@@ -192,8 +192,9 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
                /* t := 2*a */
                if (!BN_mod_lshift1_quick(t, a, p)) goto end;
 
-               /* b := (2*a)^((p-5)/8) */
+               /* b := (2*a)^((|p|-5)/8) */
                if (!BN_rshift(q, p, 3)) goto end;
+               q->neg = 0;
                if (!BN_mod_exp(b, t, q, p, ctx)) goto end;
 
                /* y := b^2 */
@@ -201,7 +202,7 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
 
                /* t := (2*a)*b^2 - 1*/
                if (!BN_mod_mul(t, t, y, p, ctx)) goto end;
-               if (!BN_sub_word(t, 1)) goto end; /* cannot become negative */
+               if (!BN_sub_word(t, 1)) goto end;
 
                /* x = a*b*t */
                if (!BN_mod_mul(x, a, b, p, ctx)) goto end;
@@ -214,6 +215,8 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
        
        /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
         * First, find some  y  that is not a square. */
+       if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
+       q->neg = 0;
        i = 2;
        do
                {
@@ -236,7 +239,7 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
                                if (!BN_set_word(y, i)) goto end;
                        }
                
-               r = BN_kronecker(y, p, ctx);
+               r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */
                if (r < -1) goto end;
                if (r == 0)
                        {
@@ -258,6 +261,8 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
                goto end;
                }
 
+       /* Here's our actual 'q': */
+       if (!BN_rshift(q, q, e)) goto end;
 
        /* Now that we have some non-square, we can find an element
         * of order  2^e  by computing its q'th power. */