* 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;
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 (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
/* 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;
/* 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
{
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)
{
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. */