Mercurial > octave
view libinterp/corefcn/__isprimelarge__.cc @ 31603:23520a50d74d stable
maint: Review C++ files for style and coding conventions.
* __ftp__.cc (F__ftp_cwd__, F__ftp_mget__),
__isprimelarge__.cc (F__isprimelarge__, F__pollardrho__),
debug.cc (Fdbclear), file-io.cc (Ftempdir):
Don't bother to define nargin if it is used only once in function.
Instead, just call args.length () for the one instance.
* data.cc (fill_matrix): Eliminate useless break statement after error() which
throws exception.
* qr.cc (Fqr),
__ode15__.cc (IDA::integrate, IDA::event, IDA::interpolate, IDA::outputfun):
Use true or false rather than 0 or 1 when assigning to bool variables.
maint: Review C++ files for style and coding conventions.
author | Rik <rik@octave.org> |
---|---|
date | Wed, 30 Nov 2022 20:27:16 -0800 |
parents | 88d2395500e7 |
children | e88a07dec498 |
line wrap: on
line source
//////////////////////////////////////////////////////////////////////// // // Copyright (C) 2021-2022 The Octave Project Developers // // See the file COPYRIGHT.md in the top-level directory of this // distribution or <https://octave.org/copyright/>. // // This file is part of Octave. // // Octave is free software: you can redistribute it and/or modify it // under the terms of the GNU General Public License as published by // the Free Software Foundation, either version 3 of the License, or // (at your option) any later version. // // Octave 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 General Public License for more details. // // You should have received a copy of the GNU General Public License // along with Octave; see the file COPYING. If not, see // <https://www.gnu.org/licenses/>. // //////////////////////////////////////////////////////////////////////// #if defined (HAVE_CONFIG_H) # include "config.h" #endif #include "defun.h" #include "error.h" #include "ovl.h" OCTAVE_NAMESPACE_BEGIN // This function implements the Schrage technique for modular multiplication. // The returned value is equivalent to "mod (a*b, modulus)" // but calculated without overflow. uint64_t safemultiply (uint64_t a, uint64_t b, uint64_t modulus) { if (! a || ! b) return 0; else if (b == 1) return a; else if (a == 1) return b; else if (a > b) { uint64_t tmp = a; a = b; b = tmp; } uint64_t q = modulus / a; uint64_t r = modulus - q * a; uint64_t term1 = a * (b % q); uint64_t term2 = (r < q) ? r * (b / q) : safemultiply (r, b / q, modulus); return (term1 > term2) ? (term1 - term2) : (term1 + modulus - term2); } // This function returns "mod (a^b, modulus)" // but calculated without overflow. uint64_t safepower (uint64_t a, uint64_t b, uint64_t modulus) { uint64_t retval = 1; while (b > 0) { if (b & 1) retval = safemultiply (retval, a, modulus); b >>= 1; a = safemultiply (a, a, modulus); } return retval; } // This function implements a single round of Miller-Rabin primality testing. // Returns false if composite, true if pseudoprime for this divisor. bool millerrabin (uint64_t div, uint64_t d, uint64_t r, uint64_t n) { uint64_t x = safepower (div, d, n); if (x == 1 || x == n-1) return true; for (uint64_t j = 1; j < r; j++) { x = safemultiply (x, x, n); if (x == n-1) return true; } return false; } // This function uses the Miller-Rabin test to find out whether the input is // prime or composite. The input is required to be a scalar 64-bit integer. bool isprimescalar (uint64_t n) { // Fast return for even numbers. // n==2 is excluded by the time this function is called. if (! (n & 1)) return false; // n is now odd. Rewrite n as d * 2^r + 1, where d is odd. uint64_t d = n-1; uint64_t r = 0; while (! (d & 1)) { d >>= 1; r++; } // Miller-Rabin test with the first 12 primes. // If the number passes all 12 tests, then it is prime. // If it fails any, then it is composite. // The first 12 primes suffice to test all 64-bit integers. return millerrabin ( 2, d, r, n) && millerrabin ( 3, d, r, n) && millerrabin ( 5, d, r, n) && millerrabin ( 7, d, r, n) && millerrabin (11, d, r, n) && millerrabin (13, d, r, n) && millerrabin (17, d, r, n) && millerrabin (19, d, r, n) && millerrabin (23, d, r, n) && millerrabin (29, d, r, n) && millerrabin (31, d, r, n) && millerrabin (37, d, r, n); /* Mathematical references for the curious as to why we need only the 12 smallest primes for testing all 64-bit numbers: (1) https://oeis.org/A014233 Comment: a(12) > 2^64. Hence the primality of numbers < 2^64 can be determined by asserting strong pseudoprimality to all prime bases <= 37 (=prime(12)). Testing to prime bases <=31 does not suffice, as a(11) < 2^64 and a(11) is a strong pseudoprime to all prime bases <= 31 (=prime(11)). - Joerg Arndt, Jul 04 2012 (2) https://arxiv.org/abs/1509.00864 Strong Pseudoprimes to Twelve Prime Bases Jonathan P. Sorenson, Jonathan Webster In addition, a source listed here: https://miller-rabin.appspot.com/ reports that all 64-bit numbers can be covered with only 7 divisors, namely 2, 325, 9375, 28178, 450775, 9780504, and 1795265022. There was no peer-reviewed article to back it up though, so this code uses the 12 primes <= 37. */ } DEFUN (__isprimelarge__, args, , doc: /* -*- texinfo -*- @deftypefn {} {@var{x} =} __isprimelarge__ (@var{n}) Use the Miller-Rabin test to find out whether the elements of N are prime or composite. The input N is required to be a vector or array of 64-bit integers. You should call isprime(N) instead of directly calling this function. @seealso{isprime, factor} @end deftypefn */) { if (args.length () != 1) print_usage (); // This function is intended for internal use by isprime.m, // so the following error handling should not be necessary. But it is // probably good practice for any curious users calling it directly. uint64NDArray vec = args(0).xuint64_array_value ("__isprimelarge__: unable to convert input. Call isprime() instead."); boolNDArray retval (vec.dims(), false); for (octave_idx_type i = vec.numel() - 1; i >= 0; i--) retval(i) = isprimescalar (vec(i)); // Note: If vec(i) <= 37, this function could go into an infinite loop. // That situation does not arise when calling this from isprime.m // but it could arise if the user calls this function directly with low input // or negative input. // But it turns out that adding this validation: // "if (vec(i) <= 37) then raise an error else call isprimescalar (vec(i))" // slows this function down by over 20% for some inputs, // so it is better to leave all the input validation in isprime.m // and not add it here. The function DOCSTRING now explicitly says: // "You should call isprime(N) instead of directly calling this function." return ovl (retval); } /* %!assert (__isprimelarge__ (41:50), logical ([1 0 1 0 0 0 1 0 0 0])) %!assert (__isprimelarge__ (uint64 (12345)), false) %!assert (__isprimelarge__ (uint64 (2147483647)), true) %!assert (__isprimelarge__ (uint64 (2305843009213693951)), true) %!assert (__isprimelarge__ (uint64 (18446744073709551557)), true) %!assert (__isprimelarge__ ([uint64(12345), uint64(2147483647), ... %! uint64(2305843009213693951), ... %! uint64(18446744073709551557)]), %! logical ([0 1 1 1])) %!error <unable to convert input> (__isprimelarge__ ({'foo'; 'bar'})) */ // This function implements a fast, private GCD function // optimized for uint64_t. No input validation by design. inline uint64_t localgcd (uint64_t a, uint64_t b) { return (a <= b) ? ( (b % a == 0) ? a : localgcd (a, b % a) ) : ( (a % b == 0) ? b : localgcd (a % b, b) ); } // This function implements a textbook version of the Pollard Rho // factorization algorithm with Brent update. // The code is short and simple, but the math behind it is complicated. uint64_t pollardrho (uint64_t n, uint64_t c = 1) { uint64_t i = 1, j = 2; // cycle index values uint64_t x = (c+1) % n; // can also be rand () % n uint64_t y = x; // other value in the chain uint64_t g = 0; // GCD while (true) { i++; // Calculate x = mod (x^2 + c, n) without overflow. x = safemultiply (x, x, n) + c; if (x >= n) x -= n; // Calculate GCD (abs (x-y), n). g = (x > y) ? localgcd (x - y, n) : (x < y) ? localgcd (y - x, n) : 0; if (i == j) // cycle detected ==> double j { y = x; j <<= 1; } if (g == n || i > 1000000) // cut losses, restart with a different c return pollardrho (n, c + 2); if (g > 1) // found GCD ==> exit loop properly { error_unless (n % g == 0); // theoretical possibility of GCD error return g; } } } DEFUN (__pollardrho__, args, , doc: /* -*- texinfo -*- @deftypefn {} {@var{x} =} __pollardrho__ (@var{n}) Private function. Use the Pollard Rho test to find a factor of @var{n}. The input @var{n} is required to be a composite 64-bit integer. Do not pass it a prime input! You should call factor(@var{n}) instead of directly calling this function. @seealso{isprime, factor} @end deftypefn */) { if (args.length () != 1) print_usage (); octave_uint64 inp = args(0).xuint64_scalar_value ("__pollardrho__: unable to convert input. Call factor() instead."); uint64_t n = inp; octave_uint64 retval = pollardrho (n); return ovl (retval); } /* %!assert (__pollardrho__ (uint64 (78567695338254293)), uint64 (443363)) %!assert (__pollardrho__ (1084978968791), uint64 (832957)) %!error <unable to convert input> (__pollardrho__ ({'foo'; 'bar'})) */ OCTAVE_NAMESPACE_END