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