+ 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;