Mercurial > octave
annotate libinterp/corefcn/__isprimelarge__.cc @ 30351:4f4fa00edada stable
maint: __isprimelarge__.cc: apply Octave coding style
* libinterp/corefcn/__isprimelarge__.cc: apply Octave coding style
author | Arun Giridhar <arungiridhar@gmail.com> |
---|---|
date | Thu, 25 Nov 2021 14:18:37 +0900 |
parents | a49c635b179d |
children | 08f6bbd3ed23 7f246cfeceda |
rev | line source |
---|---|
30261
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
1 //////////////////////////////////////////////////////////////////////// |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
2 // |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
3 // Copyright (C) 1994-2021 The Octave Project Developers |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
4 // |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
5 // See the file COPYRIGHT.md in the top-level directory of this |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
6 // distribution or <https://octave.org/copyright/>. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
7 // |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
8 // This file is part of Octave. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
9 // |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
10 // Octave is free software: you can redistribute it and/or modify it |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
11 // under the terms of the GNU General Public License as published by |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
12 // the Free Software Foundation, either version 3 of the License, or |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
13 // (at your option) any later version. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
14 // |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
15 // Octave is distributed in the hope that it will be useful, but |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
16 // WITHOUT ANY WARRANTY; without even the implied warranty of |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
17 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
18 // GNU General Public License for more details. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
19 // |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
20 // You should have received a copy of the GNU General Public License |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
21 // along with Octave; see the file COPYING. If not, see |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
22 // <https://www.gnu.org/licenses/>. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
23 // |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
24 //////////////////////////////////////////////////////////////////////// |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
25 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
26 #include "defun.h" |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
27 #include "error.h" |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
28 #include "ovl.h" |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
29 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
30 OCTAVE_NAMESPACE_BEGIN |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
31 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
32 // This function implements the Schrage technique for modular multiplication. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
33 // The returned value is equivalent to "mod (a*b, modulus)" |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
34 // but calculated without overflow. |
30351
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
35 uint64_t |
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
36 safemultiply (uint64_t a, uint64_t b, uint64_t modulus) |
30261
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
37 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
38 if (! a || ! b) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
39 return 0; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
40 else if (b == 1) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
41 return a; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
42 else if (a == 1) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
43 return b; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
44 else if (a > b) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
45 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
46 uint64_t tmp = a; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
47 a = b; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
48 b = tmp; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
49 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
50 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
51 uint64_t q = modulus / a; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
52 uint64_t r = modulus - q * a; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
53 uint64_t term1 = a * (b % q); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
54 uint64_t term2 = (r < q) ? r * (b / q) : safemultiply (r, b / q, modulus); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
55 return (term1 > term2) ? (term1 - term2) : (term1 + modulus - term2); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
56 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
57 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
58 // This function returns "mod (a^b, modulus)" |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
59 // but calculated without overflow. |
30351
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
60 uint64_t |
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
61 safepower (uint64_t a, uint64_t b, uint64_t modulus) |
30261
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
62 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
63 uint64_t retval = 1; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
64 while (b > 0) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
65 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
66 if (b & 1) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
67 retval = safemultiply (retval, a, modulus); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
68 b >>= 1; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
69 a = safemultiply (a, a, modulus); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
70 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
71 return retval; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
72 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
73 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
74 // This function implements a single round of Miller-Rabin primality testing. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
75 // Returns false if composite, true if pseudoprime for this divisor. |
30351
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
76 bool |
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
77 millerrabin (uint64_t div, uint64_t d, uint64_t r, uint64_t n) |
30261
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
78 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
79 uint64_t x = safepower (div, d, n); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
80 if (x == 1 || x == n-1) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
81 return true; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
82 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
83 for (uint64_t j = 1; j < r; j++) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
84 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
85 x = safemultiply (x, x, n); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
86 if (x == n-1) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
87 return true; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
88 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
89 return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
90 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
91 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
92 // This function uses the Miller-Rabin test to find out whether the input is |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
93 // prime or composite. The input is required to be a scalar 64-bit integer. |
30351
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
94 bool |
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
95 isprimescalar (uint64_t n) |
30261
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
96 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
97 // Fast return for even numbers. n==2 is excluded by the time this function is called. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
98 if (! (n & 1)) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
99 return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
100 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
101 // n is now odd. Rewrite n as d * 2^r + 1, where d is odd. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
102 uint64_t d = n-1; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
103 uint64_t r = 0; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
104 while (! (d & 1)) |
30351
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
105 { |
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
106 d >>= 1; |
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
107 r++; |
4f4fa00edada
maint: __isprimelarge__.cc: apply Octave coding style
Arun Giridhar <arungiridhar@gmail.com>
parents:
30261
diff
changeset
|
108 } |
30261
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
109 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
110 // Miller-Rabin test with the first 12 primes. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
111 // If the number passes all 12 tests, then it is prime. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
112 // If it fails any, then it is composite. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
113 // The first 12 primes suffice to test all 64-bit integers. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
114 if (! millerrabin ( 2, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
115 if (! millerrabin ( 3, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
116 if (! millerrabin ( 5, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
117 if (! millerrabin ( 7, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
118 if (! millerrabin (11, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
119 if (! millerrabin (13, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
120 if (! millerrabin (17, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
121 if (! millerrabin (19, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
122 if (! millerrabin (23, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
123 if (! millerrabin (29, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
124 if (! millerrabin (31, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
125 if (! millerrabin (37, d, r, n)) return false; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
126 // If we are all the way here, then it is prime. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
127 return true; |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
128 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
129 /* |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
130 Mathematical references for the curious as to why we need only |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
131 the 12 smallest primes for testing all 64-bit numbers: |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
132 (1) https://oeis.org/A014233 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
133 Comment: a(12) > 2^64. Hence the primality of numbers < 2^64 can be |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
134 determined by asserting strong pseudoprimality to all prime bases <= 37 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
135 (=prime(12)). Testing to prime bases <=31 does not suffice, |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
136 as a(11) < 2^64 and a(11) is a strong pseudoprime |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
137 to all prime bases <= 31 (=prime(11)). - Joerg Arndt, Jul 04 2012 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
138 (2) https://arxiv.org/abs/1509.00864 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
139 Strong Pseudoprimes to Twelve Prime Bases |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
140 Jonathan P. Sorenson, Jonathan Webster |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
141 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
142 In addition, a source listed here: https://miller-rabin.appspot.com/ |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
143 reports that all 64-bit numbers can be covered with only 7 divisors, |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
144 namely 2, 325, 9375, 28178, 450775, 9780504, and 1795265022. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
145 There was no peer-reviewed article to back it up though, |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
146 so this code uses the 12 primes <= 37. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
147 */ |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
148 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
149 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
150 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
151 DEFUN (__isprimelarge__, args, , |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
152 doc: /* -*- texinfo -*- |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
153 @deftypefn {} {@var{x} =} __isprimelarge__ (@var{n}) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
154 Use the Miller-Rabin test to find out whether the elements of N are prime or |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
155 composite. The input N is required to be a vector or array of 64-bit integers. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
156 You should call isprime(N) instead of directly calling this function. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
157 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
158 @seealso{isprime, factor} |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
159 @end deftypefn */) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
160 { |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
161 int nargin = args.length (); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
162 if (nargin != 1) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
163 print_usage (); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
164 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
165 // This function is intended for internal use by isprime.m, |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
166 // so the following error handling should not be necessary. But it is |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
167 // probably good practice for any curious users calling it directly. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
168 uint64NDArray vec = args(0).xuint64_array_value |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
169 ("__isprimelarge__: unable to convert input. Call isprime() instead."); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
170 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
171 boolNDArray retval (vec.dims(), false); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
172 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
173 for (octave_idx_type i = vec.numel() - 1; i >= 0; i--) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
174 retval(i) = isprimescalar (vec(i)); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
175 // Note: If vec(i) <= 37, this function could go into an infinite loop. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
176 // That situation does not arise when calling this from isprime.m |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
177 // but it could arise if the user calls this function directly with low input |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
178 // or negative input. |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
179 // But it turns out that adding this validation: |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
180 // "if (vec(i) <= 37) then raise an error else call isprimescalar (vec(i))" |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
181 // slows this function down by over 20% for some inputs, |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
182 // so it is better to leave all the input validation in isprime.m |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
183 // and not add it here. The function DOCSTRING now explicitly says: |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
184 // "You should call isprime(N) instead of directly calling this function." |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
185 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
186 return ovl (retval); |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
187 } |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
188 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
189 /* |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
190 %!assert (__isprimelarge__ (41:50), logical ([1 0 1 0 0 0 1 0 0 0])) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
191 %!assert (__isprimelarge__ (uint64 (12345)), false) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
192 %!assert (__isprimelarge__ (uint64 (2147483647)), true) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
193 %!assert (__isprimelarge__ (uint64 (2305843009213693951)), true) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
194 %!assert (__isprimelarge__ (uint64 (18446744073709551557)), true) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
195 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
196 %!assert (__isprimelarge__ ([uint64(12345), uint64(2147483647), uint64(2305843009213693951), uint64(18446744073709551557)]), logical ([0 1 1 1])) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
197 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
198 %!error <unable to convert input> (__isprimelarge__ ({'foo'; 'bar'})) |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
199 */ |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
200 |
a49c635b179d
isprime.m: Speed up for large integers using Miller-Rabin test (bug #61312).
Arun Giridhar <arungiridhar@gmail.com>
parents:
diff
changeset
|
201 OCTAVE_NAMESPACE_END |