282 lines
11 KiB
C
282 lines
11 KiB
C
|
#include "tommath_private.h"
|
||
|
|
#ifdef MP_PRIME_STRONG_LUCAS_SELFRIDGE_C
|
||
|
|
|
||
|
|
/* LibTomMath, multiple-precision integer library -- Tom St Denis */
|
||
|
|
/* SPDX-License-Identifier: Unlicense */
|
||
|
|
|
||
|
|
/*
|
||
|
|
* See file mp_prime_is_prime.c or the documentation in doc/bn.tex for the details
|
||
|
|
*/
|
||
|
|
#ifndef LTM_USE_ONLY_MR
|
||
|
|
|
||
|
|
/*
|
||
|
|
* multiply bigint a with int d and put the result in c
|
||
|
|
* Like mp_mul_d() but with a signed long as the small input
|
||
|
|
*/
|
||
|
|
static mp_err s_mul_si(const mp_int *a, int32_t d, mp_int *c)
|
||
|
|
{
|
||
|
|
mp_int t;
|
||
|
|
mp_err err;
|
||
|
|
|
||
|
|
if ((err = mp_init(&t)) != MP_OKAY) {
|
||
|
|
return err;
|
||
|
|
}
|
||
|
|
|
||
|
|
/*
|
||
|
|
* mp_digit might be smaller than a long, which excludes
|
||
|
|
* the use of mp_mul_d() here.
|
||
|
|
*/
|
||
|
|
mp_set_i32(&t, d);
|
||
|
|
err = mp_mul(a, &t, c);
|
||
|
|
mp_clear(&t);
|
||
|
|
return err;
|
||
|
|
}
|
||
|
|
/*
|
||
|
|
Strong Lucas-Selfridge test.
|
||
|
|
returns true if it is a strong L-S prime, false if it is composite
|
||
|
|
|
||
|
|
Code ported from Thomas Ray Nicely's implementation of the BPSW test
|
||
|
|
at http://www.trnicely.net/misc/bpsw.html
|
||
|
|
|
||
|
|
Freeware copyright (C) 2016 Thomas R. Nicely <http://www.trnicely.net>.
|
||
|
|
Released into the public domain by the author, who disclaims any legal
|
||
|
|
liability arising from its use
|
||
|
|
|
||
|
|
The multi-line comments are made by Thomas R. Nicely and are copied verbatim.
|
||
|
|
Additional comments marked "CZ" (without the quotes) are by the code-portist.
|
||
|
|
|
||
|
|
(If that name sounds familiar, he is the guy who found the fdiv bug in the
|
||
|
|
Pentium (P5x, I think) Intel processor)
|
||
|
|
*/
|
||
|
|
mp_err mp_prime_strong_lucas_selfridge(const mp_int *a, bool *result)
|
||
|
|
{
|
||
|
|
/* CZ TODO: choose better variable names! */
|
||
|
|
mp_int Dz, gcd, Np1, Uz, Vz, U2mz, V2mz, Qmz, Q2mz, Qkdz, T1z, T2z, T3z, T4z, Q2kdz;
|
||
|
|
int32_t D, Ds, J, sign, P, Q, r, s, u, Nbits;
|
||
|
|
mp_err err;
|
||
|
|
bool oddness;
|
||
|
|
|
||
|
|
*result = false;
|
||
|
|
/*
|
||
|
|
Find the first element D in the sequence {5, -7, 9, -11, 13, ...}
|
||
|
|
such that Jacobi(D,N) = -1 (Selfridge's algorithm). Theory
|
||
|
|
indicates that, if N is not a perfect square, D will "nearly
|
||
|
|
always" be "small." Just in case, an overflow trap for D is
|
||
|
|
included.
|
||
|
|
*/
|
||
|
|
|
||
|
|
if ((err = mp_init_multi(&Dz, &gcd, &Np1, &Uz, &Vz, &U2mz, &V2mz, &Qmz, &Q2mz, &Qkdz, &T1z, &T2z, &T3z, &T4z, &Q2kdz,
|
||
|
|
NULL)) != MP_OKAY) {
|
||
|
|
return err;
|
||
|
|
}
|
||
|
|
|
||
|
|
D = 5;
|
||
|
|
sign = 1;
|
||
|
|
|
||
|
|
for (;;) {
|
||
|
|
Ds = sign * D;
|
||
|
|
sign = -sign;
|
||
|
|
mp_set_u32(&Dz, (uint32_t)D);
|
||
|
|
if ((err = mp_gcd(a, &Dz, &gcd)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
|
||
|
|
/* if 1 < GCD < N then N is composite with factor "D", and
|
||
|
|
Jacobi(D,N) is technically undefined (but often returned
|
||
|
|
as zero). */
|
||
|
|
if ((mp_cmp_d(&gcd, 1uL) == MP_GT) && (mp_cmp(&gcd, a) == MP_LT)) {
|
||
|
|
goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
if (Ds < 0) {
|
||
|
|
Dz.sign = MP_NEG;
|
||
|
|
}
|
||
|
|
if ((err = mp_kronecker(&Dz, a, &J)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
|
||
|
|
if (J == -1) {
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
D += 2;
|
||
|
|
|
||
|
|
if (D > (INT_MAX - 2)) {
|
||
|
|
err = MP_VAL;
|
||
|
|
goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
P = 1; /* Selfridge's choice */
|
||
|
|
Q = (1 - Ds) / 4; /* Required so D = P*P - 4*Q */
|
||
|
|
|
||
|
|
/* NOTE: The conditions (a) N does not divide Q, and
|
||
|
|
(b) D is square-free or not a perfect square, are included by
|
||
|
|
some authors; e.g., "Prime numbers and computer methods for
|
||
|
|
factorization," Hans Riesel (2nd ed., 1994, Birkhauser, Boston),
|
||
|
|
p. 130. For this particular application of Lucas sequences,
|
||
|
|
these conditions were found to be immaterial. */
|
||
|
|
|
||
|
|
/* Now calculate N - Jacobi(D,N) = N + 1 (even), and calculate the
|
||
|
|
odd positive integer d and positive integer s for which
|
||
|
|
N + 1 = 2^s*d (similar to the step for N - 1 in Miller's test).
|
||
|
|
The strong Lucas-Selfridge test then returns N as a strong
|
||
|
|
Lucas probable prime (slprp) if any of the following
|
||
|
|
conditions is met: U_d=0, V_d=0, V_2d=0, V_4d=0, V_8d=0,
|
||
|
|
V_16d=0, ..., etc., ending with V_{2^(s-1)*d}=V_{(N+1)/2}=0
|
||
|
|
(all equalities mod N). Thus d is the highest index of U that
|
||
|
|
must be computed (since V_2m is independent of U), compared
|
||
|
|
to U_{N+1} for the standard Lucas-Selfridge test; and no
|
||
|
|
index of V beyond (N+1)/2 is required, just as in the
|
||
|
|
standard Lucas-Selfridge test. However, the quantity Q^d must
|
||
|
|
be computed for use (if necessary) in the latter stages of
|
||
|
|
the test. The result is that the strong Lucas-Selfridge test
|
||
|
|
has a running time only slightly greater (order of 10 %) than
|
||
|
|
that of the standard Lucas-Selfridge test, while producing
|
||
|
|
only (roughly) 30 % as many pseudoprimes (and every strong
|
||
|
|
Lucas pseudoprime is also a standard Lucas pseudoprime). Thus
|
||
|
|
the evidence indicates that the strong Lucas-Selfridge test is
|
||
|
|
more effective than the standard Lucas-Selfridge test, and a
|
||
|
|
Baillie-PSW test based on the strong Lucas-Selfridge test
|
||
|
|
should be more reliable. */
|
||
|
|
|
||
|
|
if ((err = mp_add_d(a, 1uL, &Np1)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
s = mp_cnt_lsb(&Np1);
|
||
|
|
|
||
|
|
/* CZ
|
||
|
|
* This should round towards zero because
|
||
|
|
* Thomas R. Nicely used GMP's mpz_tdiv_q_2exp()
|
||
|
|
* and mp_div_2d() is equivalent. Additionally:
|
||
|
|
* dividing an even number by two does not produce
|
||
|
|
* any leftovers.
|
||
|
|
*/
|
||
|
|
if ((err = mp_div_2d(&Np1, s, &Dz, NULL)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
/* We must now compute U_d and V_d. Since d is odd, the accumulated
|
||
|
|
values U and V are initialized to U_1 and V_1 (if the target
|
||
|
|
index were even, U and V would be initialized instead to U_0=0
|
||
|
|
and V_0=2). The values of U_2m and V_2m are also initialized to
|
||
|
|
U_1 and V_1; the FOR loop calculates in succession U_2 and V_2,
|
||
|
|
U_4 and V_4, U_8 and V_8, etc. If the corresponding bits
|
||
|
|
(1, 2, 3, ...) of t are on (the zero bit having been accounted
|
||
|
|
for in the initialization of U and V), these values are then
|
||
|
|
combined with the previous totals for U and V, using the
|
||
|
|
composition formulas for addition of indices. */
|
||
|
|
|
||
|
|
mp_set(&Uz, 1uL); /* U=U_1 */
|
||
|
|
mp_set(&Vz, (mp_digit)P); /* V=V_1 */
|
||
|
|
mp_set(&U2mz, 1uL); /* U_1 */
|
||
|
|
mp_set(&V2mz, (mp_digit)P); /* V_1 */
|
||
|
|
|
||
|
|
mp_set_i32(&Qmz, Q);
|
||
|
|
if ((err = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
/* Initializes calculation of Q^d */
|
||
|
|
mp_set_i32(&Qkdz, Q);
|
||
|
|
|
||
|
|
Nbits = mp_count_bits(&Dz);
|
||
|
|
|
||
|
|
for (u = 1; u < Nbits; u++) { /* zero bit off, already accounted for */
|
||
|
|
/* Formulas for doubling of indices (carried out mod N). Note that
|
||
|
|
* the indices denoted as "2m" are actually powers of 2, specifically
|
||
|
|
* 2^(ul-1) beginning each loop and 2^ul ending each loop.
|
||
|
|
*
|
||
|
|
* U_2m = U_m*V_m
|
||
|
|
* V_2m = V_m*V_m - 2*Q^m
|
||
|
|
*/
|
||
|
|
|
||
|
|
if ((err = mp_mul(&U2mz, &V2mz, &U2mz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mod(&U2mz, a, &U2mz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_sqr(&V2mz, &V2mz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_sub(&V2mz, &Q2mz, &V2mz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mod(&V2mz, a, &V2mz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
|
||
|
|
/* Must calculate powers of Q for use in V_2m, also for Q^d later */
|
||
|
|
if ((err = mp_sqr(&Qmz, &Qmz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
|
||
|
|
/* prevents overflow */ /* CZ still necessary without a fixed prealloc'd mem.? */
|
||
|
|
if ((err = mp_mod(&Qmz, a, &Qmz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
|
||
|
|
if (s_mp_get_bit(&Dz, u)) {
|
||
|
|
/* Formulas for addition of indices (carried out mod N);
|
||
|
|
*
|
||
|
|
* U_(m+n) = (U_m*V_n + U_n*V_m)/2
|
||
|
|
* V_(m+n) = (V_m*V_n + D*U_m*U_n)/2
|
||
|
|
*
|
||
|
|
* Be careful with division by 2 (mod N)!
|
||
|
|
*/
|
||
|
|
if ((err = mp_mul(&U2mz, &Vz, &T1z)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mul(&Uz, &V2mz, &T2z)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mul(&V2mz, &Vz, &T3z)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mul(&U2mz, &Uz, &T4z)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = s_mul_si(&T4z, Ds, &T4z)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_add(&T1z, &T2z, &Uz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if (mp_isodd(&Uz)) {
|
||
|
|
if ((err = mp_add(&Uz, a, &Uz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
/* CZ
|
||
|
|
* This should round towards negative infinity because
|
||
|
|
* Thomas R. Nicely used GMP's mpz_fdiv_q_2exp().
|
||
|
|
* But mp_div_2() does not do so, it is truncating instead.
|
||
|
|
*/
|
||
|
|
oddness = mp_isodd(&Uz);
|
||
|
|
if ((err = mp_div_2(&Uz, &Uz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if (mp_isneg(&Uz) && oddness) {
|
||
|
|
if ((err = mp_sub_d(&Uz, 1uL, &Uz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
if ((err = mp_add(&T3z, &T4z, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if (mp_isodd(&Vz)) {
|
||
|
|
if ((err = mp_add(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
oddness = mp_isodd(&Vz);
|
||
|
|
if ((err = mp_div_2(&Vz, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if (mp_isneg(&Vz) && oddness) {
|
||
|
|
if ((err = mp_sub_d(&Vz, 1uL, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
if ((err = mp_mod(&Uz, a, &Uz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mod(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
|
||
|
|
/* Calculating Q^d for later use */
|
||
|
|
if ((err = mp_mul(&Qkdz, &Qmz, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* If U_d or V_d is congruent to 0 mod N, then N is a prime or a
|
||
|
|
strong Lucas pseudoprime. */
|
||
|
|
if (mp_iszero(&Uz) || mp_iszero(&Vz)) {
|
||
|
|
*result = true;
|
||
|
|
goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* NOTE: Ribenboim ("The new book of prime number records," 3rd ed.,
|
||
|
|
1995/6) omits the condition V0 on p.142, but includes it on
|
||
|
|
p. 130. The condition is NECESSARY; otherwise the test will
|
||
|
|
return false negatives---e.g., the primes 29 and 2000029 will be
|
||
|
|
returned as composite. */
|
||
|
|
|
||
|
|
/* Otherwise, we must compute V_2d, V_4d, V_8d, ..., V_{2^(s-1)*d}
|
||
|
|
by repeated use of the formula V_2m = V_m*V_m - 2*Q^m. If any of
|
||
|
|
these are congruent to 0 mod N, then N is a prime or a strong
|
||
|
|
Lucas pseudoprime. */
|
||
|
|
|
||
|
|
/* Initialize 2*Q^(d*2^r) for V_2m */
|
||
|
|
if ((err = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
|
||
|
|
for (r = 1; r < s; r++) {
|
||
|
|
if ((err = mp_sqr(&Vz, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_sub(&Vz, &Q2kdz, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mod(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if (mp_iszero(&Vz)) {
|
||
|
|
*result = true;
|
||
|
|
goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
/* Calculate Q^{d*2^r} for next r (final iteration irrelevant). */
|
||
|
|
if (r < (s - 1)) {
|
||
|
|
if ((err = mp_sqr(&Qkdz, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
if ((err = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) goto LBL_LS_ERR;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
LBL_LS_ERR:
|
||
|
|
mp_clear_multi(&Q2kdz, &T4z, &T3z, &T2z, &T1z, &Qkdz, &Q2mz, &Qmz, &V2mz, &U2mz, &Vz, &Uz, &Np1, &gcd, &Dz, NULL);
|
||
|
|
return err;
|
||
|
|
}
|
||
|
|
#endif
|
||
|
|
#endif
|