* 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;
int err = 1;
int r;
- BIGNUM *b, *q, *t, *x, *y;
+ BIGNUM *A, *b, *q, *t, *x, *y;
int e, i, j;
if (!BN_is_odd(p) || BN_abs_is_word(p, 1))
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)
+ if (BN_is_zero(a) || BN_is_one(a))
{
- BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
- return(NULL);
+ 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;
}
-#endif
BN_CTX_start(ctx);
+ A = BN_CTX_get(ctx);
b = BN_CTX_get(ctx);
q = BN_CTX_get(ctx);
t = BN_CTX_get(ctx);
ret = BN_new();
if (ret == NULL) goto end;
+ /* A = a mod p */
+ if (!BN_nnmod(A, a, p, ctx)) 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;
+ /* 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. (q+1)/2.
+ * 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_add_word(q,1)) goto end;
- if (!BN_rshift1(q,q)) goto end;
- if (!BN_mod_exp(ret, a, q, p, ctx)) goto end;
+ 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;
- goto end;
+ goto vrfy;
+ }
+
+ 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.)
+ */
+
+ /* 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;
+ q->neg = 0;
+ 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;
+
+ /* 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 vrfy;
}
- /* e > 1, so we really have to use the Tonelli/Shanks algorithm.
+ /* e > 2, so we really have to use the Tonelli/Shanks algorithm.
* First, find some y that is not a square. */
- i = 1;
+ if (!BN_copy(q, p)) goto end; /* use 'q' as temp */
+ q->neg = 0;
+ i = 2;
do
{
/* For efficiency, try small numbers first;
* if this fails, try random numbers.
*/
- if (i < 20)
+ if (i < 22)
{
if (!BN_set_word(y, i)) goto end;
}
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;
}
}
- while (r == 1 && i++ < 80);
+ while (r == 1 && ++i < 82);
if (r != -1)
{
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. */
/* 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_nnmod(t, A, p, ctx)) goto end;
if (BN_is_zero(t))
{
/* special case: a == 0 (mod p) */
}
else
{
- if (!BN_mod_exp(x, a, t, p, ctx)) goto end;
+ if (!BN_mod_exp(x, A, t, p, ctx)) goto end;
if (BN_is_zero(x))
{
/* special case: a == 0 (mod p) */
/* 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;
+ 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;
+ if (!BN_mod_mul(x, x, A, p, ctx)) goto end;
while (1)
{
{
if (!BN_copy(ret, x)) goto end;
err = 0;
- goto end;
+ goto vrfy;
}
e = i;
}
+ vrfy:
+ if (!err)
+ {
+ /* verify the result -- the input might have been not a square
+ * (test added in 0.9.8) */
+
+ if (!BN_mod_sqr(x, ret, p, ctx))
+ err = 1;
+
+ if (!err && 0 != BN_cmp(x, A))
+ {
+ BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE);
+ err = 1;
+ }
+ }
+
end:
if (err)
{
projects / openssl.git / blobdiff