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      1 /* mpz_lucas_mod -- Helper function for the strong Lucas
      2    primality test.
      3 
      4    THE FUNCTIONS IN THIS FILE ARE FOR INTERNAL USE ONLY.  THEY'RE ALMOST
      5    CERTAIN TO BE SUBJECT TO INCOMPATIBLE CHANGES OR DISAPPEAR COMPLETELY IN
      6    FUTURE GNU MP RELEASES.
      7 
      8 Copyright 2018 Free Software Foundation, Inc.
      9 
     10 Contributed by Marco Bodrato.
     11 
     12 This file is part of the GNU MP Library.
     13 
     14 The GNU MP Library is free software; you can redistribute it and/or modify
     15 it under the terms of either:
     16 
     17   * the GNU Lesser General Public License as published by the Free
     18     Software Foundation; either version 3 of the License, or (at your
     19     option) any later version.
     20 
     21 or
     22 
     23   * the GNU General Public License as published by the Free Software
     24     Foundation; either version 2 of the License, or (at your option) any
     25     later version.
     26 
     27 or both in parallel, as here.
     28 
     29 The GNU MP Library is distributed in the hope that it will be useful, but
     30 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
     31 or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
     32 for more details.
     33 
     34 You should have received copies of the GNU General Public License and the
     35 GNU Lesser General Public License along with the GNU MP Library.  If not,
     36 see https://www.gnu.org/licenses/.  */
     37 
     38 #include "gmp-impl.h"
     39 
     40 /* Computes V_{k+1}, Q^{k+1} (mod n) for the Lucas' sequence	*/
     41 /* with P=1, Q=Q; k = n>>b0.	*/
     42 /* Requires n > 4; b0 > 0; -2*Q must not overflow a long.	*/
     43 /* If U_{k+1}==0 (mod n) or V_{k+1}==0 (mod n), it returns 1,	*/
     44 /* otherwise it returns 0 and sets V=V_{k+1} and Qk=Q^{k+1}.	*/
     45 /* V will never grow beyond SIZ(n), Qk not beyond 2*SIZ(n).	*/
     46 int
     47 mpz_lucas_mod (mpz_ptr V, mpz_ptr Qk, long Q,
     48 	       mp_bitcnt_t b0, mpz_srcptr n, mpz_ptr T1, mpz_ptr T2)
     49 {
     50   mp_bitcnt_t bs;
     51   int res;
     52 
     53   ASSERT (b0 > 0);
     54   ASSERT (SIZ (n) > 1 || SIZ (n) > 0 && PTR (n) [0] > 4);
     55 
     56   mpz_set_ui (V, 1); /* U1 = 1 */
     57   bs = mpz_sizeinbase (n, 2) - 2;
     58   if (UNLIKELY (bs < b0))
     59     {
     60       /* n = 2^b0 - 1, should we use Lucas-Lehmer instead? */
     61       ASSERT (bs == b0 - 2);
     62       mpz_set_si (Qk, Q);
     63       return 0;
     64     }
     65   mpz_set_ui (Qk, 1); /* U2 = 1 */
     66 
     67   do
     68     {
     69       /* We use the iteration suggested in "Elementary Number Theory"	*/
     70       /* by Peter Hackman (November 1, 2009), section "L.XVII Scalar	*/
     71       /* Formulas", from http://hackmat.se/kurser/TATM54/booktot.pdf	*/
     72       /* U_{2k} = 2*U_{k+1}*U_k - P*U_k^2	*/
     73       /* U_{2k+1} = U_{k+1}^2  - Q*U_k^2	*/
     74       /* U_{2k+2} = P*U_{k+1}^2 - 2*Q*U_{k+1}*U_k	*/
     75       /* We note that U_{2k+2} = P*U_{2k+1} - Q*U_{2k}	*/
     76       /* The formulas are specialized for P=1, and only squares:	*/
     77       /* U_{2k}   = U_{k+1}^2 - |U_{k+1} - U_k|^2	*/
     78       /* U_{2k+1} = U_{k+1}^2 - Q*U_k^2		*/
     79       /* U_{2k+2} = U_{2k+1}  - Q*U_{2k}	*/
     80       mpz_mul (T1, Qk, Qk);	/* U_{k+1}^2		*/
     81       mpz_sub (Qk, V, Qk);	/* |U_{k+1} - U_k|	*/
     82       mpz_mul (T2, Qk, Qk);	/* |U_{k+1} - U_k|^2	*/
     83       mpz_mul (Qk, V, V);	/* U_k^2		*/
     84       mpz_sub (T2, T1, T2);	/* U_{k+1}^2 - (U_{k+1} - U_k)^2	*/
     85       if (Q > 0)		/* U_{k+1}^2 - Q U_k^2 = U_{2k+1}	*/
     86 	mpz_submul_ui (T1, Qk, Q);
     87       else
     88 	mpz_addmul_ui (T1, Qk, NEG_CAST (unsigned long, Q));
     89 
     90       /* A step k->k+1 is performed if the bit in $n$ is 1	*/
     91       if (mpz_tstbit (n, bs))
     92 	{
     93 	  /* U_{2k+2} = U_{2k+1} - Q*U_{2k}	*/
     94 	  mpz_mul_si (T2, T2, Q);
     95 	  mpz_sub (T2, T1, T2);
     96 	  mpz_swap (T1, T2);
     97 	}
     98       mpz_tdiv_r (Qk, T1, n);
     99       mpz_tdiv_r (V, T2, n);
    100     } while (--bs >= b0);
    101 
    102   res = SIZ (Qk) == 0;
    103   if (!res) {
    104     mpz_mul_si (T1, V, -2*Q);
    105     mpz_add (T1, Qk, T1);	/* V_k = U_k - 2Q*U_{k-1} */
    106     mpz_tdiv_r (V, T1, n);
    107     res = SIZ (V) == 0;
    108     if (!res && b0 > 1) {
    109       /* V_k and Q^k will be needed for further check, compute them.	*/
    110       /* FIXME: Here we compute V_k^2 and store V_k, but the former	*/
    111       /* will be recomputed by the calling function, shoul we store	*/
    112       /* that instead?							*/
    113       mpz_mul (T2, T1, T1);	/* V_k^2 */
    114       mpz_mul (T1, Qk, Qk);	/* P^2 U_k^2 = U_k^2 */
    115       mpz_sub (T2, T2, T1);
    116       ASSERT (SIZ (T2) == 0 || PTR (T2) [0] % 4 == 0);
    117       mpz_tdiv_q_2exp (T2, T2, 2);	/* (V_k^2 - P^2 U_k^2) / 4 */
    118       if (Q > 0)		/* (V_k^2 - (P^2 -4Q) U_k^2) / 4 = Q^k */
    119 	mpz_addmul_ui (T2, T1, Q);
    120       else
    121 	mpz_submul_ui (T2, T1, NEG_CAST (unsigned long, Q));
    122       mpz_tdiv_r (Qk, T2, n);
    123     }
    124   }
    125 
    126   return res;
    127 }
    128