* Use: Sets up for a strong prime search, so that primes with
* particular properties can be found. It's probably important
* to note that the number left in the filter context @f@ is
- * congruent to 2 (mod 4).
+ * congruent to 2 (mod 4); that the jump value is twice the
+ * product of two large primes; and that the starting point is
+ * at least %$3 \cdot 2^{N-2}$%. (Hence, if you multiply two
+ * such numbers, the product is at least
+ *
+ * %$9 \cdot 2^{2N-4} > 2^{2N-1}$%
+ *
+ * i.e., it will be (at least) a %$2 N$%-bit value.
*/
mp *strongprime_setup(const char *name, mp *d, pfilt *f, unsigned nbits,
if (!q)
goto fail_r;
- /* --- Select a suitable starting-point for finding %$p$% --- *
+ /* --- Select a suitable congruence class for %$p$% --- *
*
* This computes %$p_0 = 2 s (s^{r - 2} \bmod r) - 1$%.
*/
rr = mp_sub(rr, rr, MP_ONE);
}
- /* --- Now find %$p = p_0 + 2jrs$% for some %$j$% --- */
+ /* --- Pick a starting point for the search --- *
+ *
+ * Select %$3 \cdot 2^{N-2} < p_1 < 2^N$% at random, only with
+ * %$p_1 \equiv p_0 \pmod{2 r s}$.
+ */
{
mp *x, *y;
x = mp_mul(MP_NEW, q, s);
x = mp_lsl(x, x, 1);
- pfilt_create(f, x);
- y = mp_lsl(MP_NEW, MP_ONE, nbits - 1);
+ pfilt_create(f, x); /* %$2 r s$% */
+ y = mprand(MP_NEW, nbits, r, 0);
+ y = mp_setbit(y, y, nbits - 2);
rr = mp_leastcongruent(rr, y, rr, x);
mp_drop(x); mp_drop(y);
}