Faster BN_mod_sqrt algorithm for p == 5 (8).

diff --git a/CHANGES b/CHANGES
index 4197e739e1d154f57732d08a6a8d0edfe4e88d9e..ab334b87e1135bc27485f03d5f49cbdb08559edd 100644 (file)
--- a/CHANGES
+++ b/CHANGES
@@ -32,7 +32,7 @@
      [Richard Levitte]
 
   *) New function BN_mod_sqrt for computing square roots modulo a prime
-     (Tonelli-Shanks algorithm).
+     (Tonelli-Shanks algorithm unless  p == 3 (mod 4)  or  p == 5 (mod 8)).
      [Lenka Fibikova <fibikova@exp-math.uni-essen.de>, Bodo Moeller]
 
   *) Store verify_result within SSL_SESSION also for client side to

diff --git a/crypto/bn/bn_sqrt.c b/crypto/bn/bn_sqrt.c
index 2a72c189cbad97b99694f99635dd1b175212b4de..a54d9d29195a5c9c693b9b2c621cfca87b85f7ca 100644 (file)
--- a/crypto/bn/bn_sqrt.c
+++ b/crypto/bn/bn_sqrt.c
@@ -93,6 +93,20 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
                return(NULL);
                }
 
+       if (BN_is_zero(a) || BN_is_one(a))
+               {
+               if (ret == NULL)
+                       ret = BN_new();
+               if (ret == NULL)
+                       goto end;
+               if (!BN_set_word(ret, BN_is_one(a)))
+                       {
+                       BN_free(ret);
+                       return NULL;
+                       }
+               return ret;
+               }
+
 #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;
@@ -119,7 +133,9 @@ 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 (!BN_rshift(q, p, e)) goto end;
+       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;
 
        if (e == 1)
@@ -129,16 +145,74 @@ BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx)
                 * 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.
+                * so we can use exponent  (p+1)/4,  i.e.  (p-3)/4 + 1.
                 */
-               if (!BN_add_word(q,1)) goto end;
-               if (!BN_rshift1(q,q)) goto end;
+               if (!BN_rshift(q, p, 2)) goto end;
+               if (!BN_add_word(q, 1)) 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.
+       if (e == 2)
+               {
+               /* 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),
+                *      i := (2*a)*b^2
+                * we have
+                *     i^2 = (2*a)^((1 + (p-5)/4)*2)
+                *         = (2*a)^((p-1)/2)
+                *         = -1;
+                * so if we set
+                *      x := a*b*(i-1),
+                * then
+                *     x^2 = a^2 * b^2 * (i^2 - 2*i + 1)
+                *         = a^2 * b^2 * (-2*i)
+                *         = a*(-i)*(2*a*b^2)
+                *         = a*(-i)*i
+                *         = a.
+                *
+                * (This is due to A.O.L. Atkin, 
+                * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>,
+                * November 1992.)
+                */
+
+               /* make sure that  a  is reduced modulo p */
+               if (a->neg || BN_ucmp(a, p) >= 0)
+                       {
+                       if (!BN_nnmod(x, a, p, ctx)) goto end;
+                       a = x; /* use x as temporary variable */
+                       }
+
+               /* t := 2*a */
+               if (!BN_mod_lshift1_quick(t, a, p)) goto end;
+
+               /* b := (2*a)^((p-5)/8) */
+               if (!BN_rshift(q, p, 3)) goto end;
+               if (!BN_mod_exp(b, t, q, p, ctx)) goto end;
+
+               /* y := b^2 */
+               if (!BN_mod_sqr(y, b, p, ctx)) goto end;
+
+               /* 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 */
+
+               /* x = a*b*t */
+               if (!BN_mod_mul(x, a, b, p, ctx)) goto end;
+               if (!BN_mod_mul(x, x, t, p, ctx)) goto end;
+
+               if (!BN_copy(ret, x)) goto end;
+               err = 0;
+               goto end;
+               }
+       
+       /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
         * First, find some  y  that is not a square. */
        i = 2;
        do