/*----- Header files ------------------------------------------------------*/
#include <mLib/dstr.h>
+#include <mLib/macros.h>
#include "grand.h"
#include "mp.h"
/*----- Main code ---------------------------------------------------------*/
+/* Oh, just shut up. */
+CLANG_WARNING("-Wempty-body")
+
/* --- @strongprime_setup@ --- *
*
* Arguments: @const char *name@ = pointer to name root
* 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,
{
mp *s, *t, *q;
dstr dn = DSTR_INIT;
- size_t nb;
+ unsigned slop, nb, u, i;
mp *rr = d;
pgen_filterctx c;
pgen_jumpctx j;
rabin rb;
- /* --- The bitslop parameter --- *
+ /* --- Figure out how large the smaller primes should be --- *
+ *
+ * We want them to be `as large as possible', subject to the constraint
+ * that we produce a number of the requested size at the end. This is
+ * tricky, because the final prime search is going to involve quite large
+ * jumps from its starting point; the size of the jumps are basically
+ * determined by our choice here, and if they're too big then we won't find
+ * a prime in time.
*
- * There's quite a lot of prime searching to be done. The constant
- * @BITSLOP@ is a (low) approximation to the base-2 log of the expected
- * number of steps to find a prime number. Experimentation shows that
- * numbers around 10 seem to be good.
+ * Let's suppose we're trying to make an %$N$%-bit prime. The expected
+ * number of steps tends to increase linearly with size, i.e., we need to
+ * take about %2^k N$% steps for some %$k$%. If we're jumping by a
+ * %$J$%-bit quantity each time, from an %$N$%-bit starting point, then we
+ * will only be able to find a match if %$2^k N 2^{J-1} \le 2^{N-1}$%,
+ * i.e., if %$J \le N - (k + \log_2 N)$%.
+ *
+ * Experimentation shows that taking %$k + \log_2 N = 12$% works well for
+ * %$N = 1024$%, so %$k = 2$%. Add a few extra bits for luck.
*/
-#define BITSLOP 12
+ for (i = 1; i && nbits >> i; i <<= 1); assert(i);
+ for (slop = 6, nb = nbits; nb > 1; i >>= 1) {
+ u = nb >> i;
+ if (u) { slop += i; nb = u; }
+ }
+ if (nbits/2 <= slop) return (0);
/* --- Choose two primes %$s$% and %$t$% of half the required size --- */
- if (nbits/2 <= BITSLOP) return (0);
- nb = nbits/2 - BITSLOP;
+ nb = nbits/2 - slop;
c.step = 1;
rr = mprand(rr, nb, r, 1);
rabin_iters(nb), pgen_test, &rb)) == 0)
goto fail_t;
- /* --- Choose a suitable value for %$r = 2it + 1$% for some %$i$% --- */
+ /* --- Choose a suitable value for %$r = 2it + 1$% for some %$i$% --- *
+ *
+ * Then %$r \equiv 1 \pmod{t}$%, i.e., %$r - 1$% is a multiple of %$t$%.
+ */
rr = mp_lsl(rr, t, 1);
pfilt_create(&c.f, rr);
- rr = mp_lsl(rr, rr, BITSLOP - 1);
+ rr = mp_lsl(rr, rr, slop - 1);
rr = mp_add(rr, rr, MP_ONE);
DRESET(&dn); dstr_putf(&dn, "%s [r]", name);
j.j = &c.f;
q = pgen(dn.buf, MP_NEW, rr, event, ectx, n, pgen_jump, &j,
- rabin_iters(nb + BITSLOP), pgen_test, &rb);
+ rabin_iters(nb + slop), pgen_test, &rb);
pfilt_destroy(&c.f);
if (!q)
goto fail_r;
- /* --- Select a suitable starting-point for finding %$p$% --- *
+ /* --- Select a suitable congruence class for %$p$% --- *
*
- * This computes %$p_0 = 2 s (s^{r - 2} \bmod r) - 1$%.
+ * This computes %$p_0 = 2 s (s^{-1} \bmod r) - 1$%. Then %$p_0 + 1$% is
+ * clearly a multiple of %$s$%, and
+ *
+ * %$p_0 - 1 \equiv 2 s s^{-1} - 2 \equiv 0 \pmod{r}$%
+ *
+ * is a multiple of %$r$%.
*/
- {
- mpmont mm;
-
- mpmont_create(&mm, q);
- rr = mp_sub(rr, q, MP_TWO);
- rr = mpmont_exp(&mm, rr, s, rr);
- mpmont_destroy(&mm);
- rr = mp_mul(rr, rr, s);
- rr = mp_lsl(rr, rr, 1);
- rr = mp_sub(rr, rr, MP_ONE);
- }
+ rr = mp_modinv(rr, s, q);
+ rr = mp_mul(rr, rr, s);
+ rr = mp_lsl(rr, rr, 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);
}
mp_drop(rr);
dstr_destroy(&dn);
return (0);
-
-#undef BITSLOP
}
/* --- @strongprime@ --- *
* * %$p - 1$% has a large prime factor %$r$%,
* * %$p + 1$% has a large prime factor %$s$%, and
* * %$r - 1$% has a large prime factor %$t$%.
- *
- * The numbers produced may be slightly larger than requested,
- * by a few bits.
*/
mp *strongprime(const char *name, mp *d, unsigned nbits, grand *r,
j.j = &f;
p = pgen(name, p, p, event, ectx, n, pgen_jump, &j,
rabin_iters(nbits), pgen_test, &rb);
+ if (mp_bits(p) != nbits) { mp_drop(p); return (0); }
pfilt_destroy(&f);
mp_drop(d);
return (p);
~mdw / catacomb / blobdiff