+/* -*-c-*-
+ *
+ * Grantham's Frobenius primality test
+ *
+ * (c) 2018 Straylight/Edgeware
+ */
+
+/*----- Licensing notice --------------------------------------------------*
+ *
+ * This file is part of Catacomb.
+ *
+ * Catacomb is free software: you can redistribute it and/or modify it
+ * under the terms of the GNU Library General Public License as published
+ * by the Free Software Foundation; either version 2 of the License, or
+ * (at your option) any later version.
+ *
+ * Catacomb is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ * Library General Public License for more details.
+ *
+ * You should have received a copy of the GNU Library General Public
+ * License along with Catacomb. If not, write to the Free Software
+ * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307,
+ * USA.
+ */
+
+/*----- Header files ------------------------------------------------------*/
+
+#include "mp.h"
+#include "mpmont.h"
+#include "mpscan.h"
+#include "pgen.h"
+
+#include "mptext.h"
+
+/*----- Main code ---------------------------------------------------------*/
+
+static int squarep(mp *n)
+{
+ mp *t = MP_NEW;
+ int rc;
+
+ if (MP_NEGP(n)) return (0);
+ t = mp_sqrt(t, n); t = mp_sqr(t, t);
+ rc = MP_EQ(t, n); mp_drop(t); return (rc);
+}
+
+/* --- @pgen_granfrob@ --- *
+ *
+ * Arguments: @mp *n@ = an integer to test
+ * @int a, b@ = coefficients; if @a@ is zero then choose
+ * automatically
+ *
+ * Returns: One of the @PGEN_...@ codes.
+ *
+ * Use: Performs a quadratic versoin of Grantham's Frobenius
+ * primality test, which is a simple extension of the standard
+ * Lucas test.
+ *
+ * If %$a^2 - 4 b$% is a perfect square then the test can't
+ * work; this function returns @PGEN_ABORT@ under these
+ * circumstances.
+ */
+
+int pgen_granfrob(mp *n, int a, int b)
+{
+ mp *v = MP_NEW, *w = MP_NEW, *aa = MP_NEW, *bb = MP_NEW, *bi = MP_NEW,
+ *k = MP_NEW, *x = MP_NEW, *y = MP_NEW, *z = MP_NEW, *t, *u;
+ mp ma; mpw wa;
+ mp mb; mpw wb;
+ mp md; mpw wd; int d;
+ mpmont mm;
+ mpscan msc;
+ int e, bit, rc;
+
+ /* Maybe this is a no-hoper. */
+ if (MP_NEGP(n)) return (PGEN_FAIL);
+ if (MP_EQ(n, MP_TWO)) return (PGEN_DONE);
+ if (!MP_ODDP(n)) return (PGEN_FAIL);
+
+ /* First, build the parameters as large integers. */
+ mp_build(&ma, &wa, &wa + 1); mp_build(&mb, &wb, &wb + 1);
+ mp_build(&md, &wd, &wd + 1);
+ mpmont_create(&mm, n);
+
+ /* Prepare the Lucas sequence parameters. Here, %$\Delta$% is the
+ * disciminant of the polynomial %$p(x) = x^2 - a x + b$%, i.e.,
+ * %$\Delta = a^2 - 4 b$%.
+ */
+ if (a) {
+ /* Explicit parameters. Just set them and check that they'll work. */
+
+ if (a >= 0) wa = a; else { wa = -a; ma.f |= MP_NEG; }
+ if (b >= 0) wb = b; else { wb = -b; mb.f |= MP_NEG; }
+ d = a*a - 4*b;
+ if (d >= 0) wd = d; else { wd = -d; md.f |= MP_NEG; }
+
+ /* Determine the quadratic character of %$\Delta$%. If %$(\Delta | n)$%
+ * is zero then we'll have a problem, but we'll catch that case with the
+ * GCD check below.
+ */
+ e = mp_jacobi(&md, n);
+
+ /* If %$\Delta$% is a perfect square then the test can't work. */
+ if (e == 1 && squarep(&md)) { rc = PGEN_ABORT; goto end; }
+ } else {
+ /* Determine parameters. Use Selfridge's `Method A': choose the first
+ * %$\Delta$% from the sequence %$5$%, %$-7$%, %%\dots%%, such that
+ * %$(\Delta | n) = -1$%.
+ */
+
+ wa = 1; wd = 5;
+ for (;;) {
+ e = mp_jacobi(&md, n); if (e != +1) break;
+ if (wd == 25 && squarep(n)) { rc = PGEN_FAIL; goto end; }
+ wd += 2; md.f ^= MP_NEG;
+ }
+ a = 1; d = wd;
+ if (md.f&MP_NEG) { wb = (wd + 1)/4; d = -d; }
+ else { wb = (wd - 1)/4; mb.f |= MP_NEG; }
+ b = (1 - d)/4;
+ }
+
+ /* The test won't work if %$\gcd(2 a b \Delta, n) \ne 1$%. */
+ x = mp_lsl(x, &ma, 1); x = mp_mul(x, x, &mb); x = mp_mul(x, x, &md);
+ mp_gcd(&y, 0, 0, x, n);
+ if (!MP_EQ(y, MP_ONE))
+ { rc = MP_EQ(y, n) ? PGEN_ABORT : PGEN_FAIL; goto end; }
+
+ /* Now we use binary a Lucas chain to evaluate %$V_{n-e}(a, b) \pmod{n}$%.
+ * Here,
+ *
+ * * %$U_{i+1}(a, b) = a U_i(a, b) - b U_{i-1}(a, b)$%, and
+ * * %$V_{i+1}(a, b) = a V_i(a, b) - b V_{i-1}(a, b)$%; with
+ * * %$U_0(a, b) = 0$%, $%U_1(a, b) = 1$%, %$V_0(a, b) = 2$%, and
+ * %$V_1(a, b) = a$%.
+ *
+ * To compute this, we use the handy identities
+ *
+ * %$V_{i+j}(a, b) = V_i(a, b) V_j(a, b) - b^i V_{j-i}(a, b)$%
+ *
+ * and
+ *
+ * %$U_i(a, b) = (2 V_{i+1}(a, b) - a V_i(a, b))/\Delta$%.
+ *
+ * Let %$k = n - e$%. Given %$V_i(a, b)$% and %$V_{i+1}(a, b)$%, we can
+ * determine either %$V_{2i}(a, b)$% and %$V_{2i+1}(a, b)$%, or
+ * %$V_{2i+1}(a, b)$% and %$V_{2i+2}(a, b)$%.
+ *
+ * To do this, suppose that %$n < 2^\ell$% and %$0 \le i \le \ell%; we'll
+ * start with %$i = 0$%. Divide %$n = n_i 2^{\ell-i} + n'_i$% with
+ * %$n'_i < 2^{\ell-i}$%. To do this, we maintain %$v_i = V_{n_i}(a, b)$%,
+ * %$w_i = V_{n_i+1}(a, b)$%, and %$b_i = b^{n_i}$%, all modulo %$n$%. If
+ * %$n'_i < 2^{\ell-i-1}$% then we have %$n'_{i+1} = n'_i$% and
+ * %$n_{i+i} = 2 n_i$%; otherwise %$n'_{i+1} = n'_i - 2^{\ell-i-1}$% and
+ * %$n_{i+i} = 2 n_i + 1$%.
+ */
+ k = mp_add(k, n, e > 0 ? MP_MONE : MP_ONE);
+ aa = mpmont_mul(&mm, aa, &ma, mm.r2);
+ bb = mpmont_mul(&mm, bb, &mb, mm.r2); bi = MP_COPY(mm.r);
+ v = mpmont_mul(&mm, v, MP_TWO, mm.r2); w = MP_COPY(aa);
+
+ for (mpscan_rinitx(&msc, k->v, k->vl); mpscan_rstep(&msc); ) {
+ bit = mpscan_rbit(&msc);
+
+ /* We will want %$x = V_{n_i+1}(a, b) = V_{n_i} V_{n_i+1} - a b^{n_i}$%,
+ * but we don't yet know whether this is %$v_{i+1}$% or %$w_{i+1}$%.
+ */
+ x = mpmont_mul(&mm, x, v, w);
+ if (a == 1) x = mp_sub(x, x, bi);
+ else { y = mpmont_mul(&mm, y, aa, bi); x = mp_sub(x, x, y); }
+ if (MP_NEGP(x)) x = mp_add(x, x, n);
+
+ if (!bit) {
+ /* We're in the former case: %$n_{i+i} = 2 n_i$%. So %$w_{i+1} = x$%,
+ * %$v_{i+1} = (v_i^2 - 2 b_i$%, and %$b_{i+1} = b_i^2$%.
+ */
+
+ y = mp_sqr(y, v); y = mpmont_reduce(&mm, y, y);
+ y = mp_sub(y, y, bi); if (MP_NEGP(y)) y = mp_add(y, y, n);
+ y = mp_sub(y, y, bi); if (MP_NEGP(y)) y = mp_add(y, y, n);
+ bi = mp_sqr(bi, bi); bi = mpmont_reduce(&mm, bi, bi);
+ t = v; u = w; v = y; w = x; x = t; y = u;
+ } else {
+ /* We're in the former case: %$n_{i+i} = 2 n_i + 1$%. So
+ * %$v_{i+1} = x$%, %$w_{i+1} = w_i^2 - 2 b b^i$%$%, and
+ * %$b_{i+1} = b b_i^2$%.
+ */
+
+ y = mp_sqr(y, w); y = mpmont_reduce(&mm, y, y);
+ z = mpmont_mul(&mm, z, bi, bb);
+ y = mp_sub(y, y, z); if (MP_NEGP(y)) y = mp_add(y, y, n);
+ y = mp_sub(y, y, z); if (MP_NEGP(y)) y = mp_add(y, y, n);
+ bi = mpmont_mul(&mm, bi, bi, z);
+ t = v; u = w; v = x; w = y; x = t; y = u;
+ }
+ }
+
+ /* The Lucas test is that %$U_k(a, b) \equiv 0 \pmod{n}$% if %$n$% is
+ * prime. I'm assured that
+ *
+ * %$U_k(a, b) = (2 V_{k+1}(a, b) - a V_k(a, b))/\Delta$%
+ *
+ * so this is just a matter of checking that %$2 w - a v \equiv 0$%.
+ */
+ x = mp_add(x, w, w); y = mp_sub(y, x, n);
+ if (!MP_NEGP(y)) { t = x; x = y; y = t; }
+ if (a == 1) x = mp_sub(x, x, v);
+ else { y = mpmont_mul(&mm, y, v, aa); x = mp_sub(x, x, y); }
+ if (MP_NEGP(x)) x = mp_add(x, x, n);
+ if (!MP_ZEROP(x)) { rc = PGEN_FAIL; goto end; }
+
+ /* Grantham's Frobenius test is that, also, %$V_k(a, b) v = \equiv 2 b$%
+ * if %$n$% is prime and %$(\Delta | n) = -1$%, or %$v \equiv 2$% if
+ * %$(\Delta | n) = +1$%.
+ */
+ if (MP_ODDP(v)) v = mp_add(v, v, n);
+ v = mp_lsr(v, v, 1);
+ if (!MP_EQ(v, e == +1 ? mm.r : bb)) { rc = PGEN_FAIL; goto end; }
+
+ /* Looks like we made it. */
+ rc = PGEN_PASS;
+end:
+ mp_drop(v); mp_drop(w); mp_drop(aa); mp_drop(bb); mp_drop(bi);
+ mp_drop(k); mp_drop(x); mp_drop(y); mp_drop(z);
+ mpmont_destroy(&mm);
+ return (rc);
+}
+
+/*----- Test rig ----------------------------------------------------------*/
+
+#ifdef TEST_RIG
+
+#include <mLib/testrig.h>
+
+#include "mptext.h"
+
+static int verify(dstr *v)
+{
+ mp *n = *(mp **)v[0].buf;
+ int a = *(int *)v[1].buf, b = *(int *)v[2].buf, xrc = *(int *)v[3].buf, rc;
+ int ok = 1;
+
+ rc = pgen_granfrob(n, a, b);
+ if (rc != xrc) {
+ fputs("\n*** pgen_granfrob failed", stdout);
+ fputs("\nn = ", stdout); mp_writefile(n, stdout, 10);
+ printf("\na = %d", a);
+ printf("\nb = %d", a);
+ printf("\nexp rc = %d", xrc);
+ printf("\ncalc rc = %d\n", rc);
+ ok = 0;
+ }
+
+ mp_drop(n);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
+static test_chunk tests[] = {
+ { "pgen-granfrob", verify,
+ { &type_mp, &type_int, &type_int, &type_int, 0 } },
+ { 0, 0, { 0 } }
+};
+
+int main(int argc, char *argv[])
+{
+ sub_init();
+ test_run(argc, argv, tests, SRCDIR "/t/pgen");
+ return (0);
+}
+
+#endif
+
+/*----- That's all, folks -------------------------------------------------*/
~mdw / catacomb / commitdiff