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annotate libinterp/corefcn/__isprimelarge__.cc @ 31605:e88a07dec498 stable

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maint: Use macros to begin/end C++ namespaces. * oct-conf-post-public.in.h: Define two macros (OCTAVE_BEGIN_NAMESPACE, OCTAVE_END_NAMESPACE) that can be used to start/end a namespace. * mk-opts.pl, build-env.h, build-env.in.cc, __betainc__.cc, __contourc__.cc, __dsearchn__.cc, __eigs__.cc, __expint__.cc, __ftp__.cc, __gammainc__.cc, __ichol__.cc, __ilu__.cc, __isprimelarge__.cc, __lin_interpn__.cc, __magick_read__.cc, __pchip_deriv__.cc, __qp__.cc, amd.cc, auto-shlib.cc, auto-shlib.h, balance.cc, base-text-renderer.cc, base-text-renderer.h, besselj.cc, bitfcns.cc, bsxfun.cc, c-file-ptr-stream.cc, c-file-ptr-stream.h, call-stack.cc, call-stack.h, ccolamd.cc, cellfun.cc, chol.cc, colamd.cc, colloc.cc, conv2.cc, daspk.cc, dasrt.cc, dassl.cc, data.cc, data.h, debug.cc, defaults.cc, defaults.h, defun-int.h, defun.cc, det.cc, dirfns.cc, display.cc, display.h, dlmread.cc, dmperm.cc, dot.cc, dynamic-ld.cc, dynamic-ld.h, eig.cc, ellipj.cc, environment.cc, environment.h, error.cc, error.h, errwarn.h, event-manager.cc, event-manager.h, event-queue.cc, event-queue.h, fcn-info.cc, fcn-info.h, fft.cc, fft2.cc, fftn.cc, file-io.cc, filter.cc, find.cc, ft-text-renderer.cc, ft-text-renderer.h, gcd.cc, getgrent.cc, getpwent.cc, getrusage.cc, givens.cc, gl-render.cc, gl-render.h, gl2ps-print.cc, gl2ps-print.h, graphics-toolkit.cc, graphics-toolkit.h, graphics.cc, graphics.in.h, gsvd.cc, gtk-manager.cc, gtk-manager.h, hash.cc, help.cc, help.h, hess.cc, hex2num.cc, hook-fcn.cc, hook-fcn.h, input.cc, input.h, interpreter-private.cc, interpreter-private.h, interpreter.cc, interpreter.h, inv.cc, jsondecode.cc, jsonencode.cc, kron.cc, latex-text-renderer.cc, latex-text-renderer.h, load-path.cc, load-path.h, load-save.cc, load-save.h, lookup.cc, ls-ascii-helper.cc, ls-ascii-helper.h, ls-oct-text.cc, ls-utils.cc, ls-utils.h, lsode.cc, lu.cc, mappers.cc, matrix_type.cc, max.cc, mex-private.h, mex.cc, mgorth.cc, nproc.cc, oct-fstrm.cc, oct-fstrm.h, oct-hdf5-types.cc, oct-hdf5-types.h, oct-hist.cc, oct-hist.h, oct-iostrm.cc, oct-iostrm.h, oct-opengl.h, oct-prcstrm.cc, oct-prcstrm.h, oct-procbuf.cc, oct-procbuf.h, oct-process.cc, oct-process.h, oct-stdstrm.h, oct-stream.cc, oct-stream.h, oct-strstrm.cc, oct-strstrm.h, oct-tex-lexer.in.ll, oct-tex-parser.yy, ordqz.cc, ordschur.cc, pager.cc, pager.h, pinv.cc, pow2.cc, pr-flt-fmt.cc, pr-output.cc, procstream.cc, procstream.h, psi.cc, qr.cc, quad.cc, quadcc.cc, qz.cc, rand.cc, rcond.cc, regexp.cc, schur.cc, settings.cc, settings.h, sighandlers.cc, sighandlers.h, sparse-xdiv.cc, sparse-xdiv.h, sparse-xpow.cc, sparse-xpow.h, sparse.cc, spparms.cc, sqrtm.cc, stack-frame.cc, stack-frame.h, stream-euler.cc, strfind.cc, strfns.cc, sub2ind.cc, svd.cc, sylvester.cc, symbfact.cc, syminfo.cc, syminfo.h, symrcm.cc, symrec.cc, symrec.h, symscope.cc, symscope.h, symtab.cc, symtab.h, syscalls.cc, sysdep.cc, sysdep.h, text-engine.cc, text-engine.h, text-renderer.cc, text-renderer.h, time.cc, toplev.cc, tril.cc, tsearch.cc, typecast.cc, url-handle-manager.cc, url-handle-manager.h, urlwrite.cc, utils.cc, utils.h, variables.cc, variables.h, xdiv.cc, xdiv.h, xnorm.cc, xnorm.h, xpow.cc, xpow.h, __delaunayn__.cc, __fltk_uigetfile__.cc, __glpk__.cc, __init_fltk__.cc, __init_gnuplot__.cc, __ode15__.cc, __voronoi__.cc, audiodevinfo.cc, audioread.cc, convhulln.cc, fftw.cc, gzip.cc, mk-build-env-features.sh, mk-builtins.pl, cdef-class.cc, cdef-class.h, cdef-fwd.h, cdef-manager.cc, cdef-manager.h, cdef-method.cc, cdef-method.h, cdef-object.cc, cdef-object.h, cdef-package.cc, cdef-package.h, cdef-property.cc, cdef-property.h, cdef-utils.cc, cdef-utils.h, ov-base.cc, ov-base.h, ov-bool-mat.cc, ov-builtin.h, ov-cell.cc, ov-class.cc, ov-class.h, ov-classdef.cc, ov-classdef.h, ov-complex.cc, ov-fcn-handle.cc, ov-fcn-handle.h, ov-fcn.h, ov-java.cc, ov-java.h, ov-mex-fcn.h, ov-null-mat.cc, ov-oncleanup.cc, ov-struct.cc, ov-typeinfo.cc, ov-typeinfo.h, ov-usr-fcn.cc, ov-usr-fcn.h, ov.cc, ov.h, octave.cc, octave.h, mk-ops.sh, op-b-b.cc, op-b-bm.cc, op-b-sbm.cc, op-bm-b.cc, op-bm-bm.cc, op-bm-sbm.cc, op-cdm-cdm.cc, op-cell.cc, op-chm.cc, op-class.cc, op-cm-cm.cc, op-cm-cs.cc, op-cm-m.cc, op-cm-s.cc, op-cm-scm.cc, op-cm-sm.cc, op-cs-cm.cc, op-cs-cs.cc, op-cs-m.cc, op-cs-s.cc, op-cs-scm.cc, op-cs-sm.cc, op-dm-dm.cc, op-dm-scm.cc, op-dm-sm.cc, op-dm-template.cc, op-dms-template.cc, op-fcdm-fcdm.cc, op-fcm-fcm.cc, op-fcm-fcs.cc, op-fcm-fm.cc, op-fcm-fs.cc, op-fcn.cc, op-fcs-fcm.cc, op-fcs-fcs.cc, op-fcs-fm.cc, op-fcs-fs.cc, op-fdm-fdm.cc, op-fm-fcm.cc, op-fm-fcs.cc, op-fm-fm.cc, op-fm-fs.cc, op-fs-fcm.cc, op-fs-fcs.cc, op-fs-fm.cc, op-fs-fs.cc, op-i16-i16.cc, op-i32-i32.cc, op-i64-i64.cc, op-i8-i8.cc, op-int-concat.cc, op-m-cm.cc, op-m-cs.cc, op-m-m.cc, op-m-s.cc, op-m-scm.cc, op-m-sm.cc, op-mi.cc, op-pm-pm.cc, op-pm-scm.cc, op-pm-sm.cc, op-pm-template.cc, op-range.cc, op-s-cm.cc, op-s-cs.cc, op-s-m.cc, op-s-s.cc, op-s-scm.cc, op-s-sm.cc, op-sbm-b.cc, op-sbm-bm.cc, op-sbm-sbm.cc, op-scm-cm.cc, op-scm-cs.cc, op-scm-m.cc, op-scm-s.cc, op-scm-scm.cc, op-scm-sm.cc, op-sm-cm.cc, op-sm-cs.cc, op-sm-m.cc, op-sm-s.cc, op-sm-scm.cc, op-sm-sm.cc, op-str-m.cc, op-str-s.cc, op-str-str.cc, op-struct.cc, op-ui16-ui16.cc, op-ui32-ui32.cc, op-ui64-ui64.cc, op-ui8-ui8.cc, ops.h, anon-fcn-validator.cc, anon-fcn-validator.h, bp-table.cc, bp-table.h, comment-list.cc, comment-list.h, filepos.h, lex.h, lex.ll, oct-lvalue.cc, oct-lvalue.h, oct-parse.yy, parse.h, profiler.cc, profiler.h, pt-anon-scopes.cc, pt-anon-scopes.h, pt-arg-list.cc, pt-arg-list.h, pt-args-block.cc, pt-args-block.h, pt-array-list.cc, pt-array-list.h, pt-assign.cc, pt-assign.h, pt-binop.cc, pt-binop.h, pt-bp.cc, pt-bp.h, pt-cbinop.cc, pt-cbinop.h, pt-cell.cc, pt-cell.h, pt-check.cc, pt-check.h, pt-classdef.cc, pt-classdef.h, pt-cmd.h, pt-colon.cc, pt-colon.h, pt-const.cc, pt-const.h, pt-decl.cc, pt-decl.h, pt-eval.cc, pt-eval.h, pt-except.cc, pt-except.h, pt-exp.cc, pt-exp.h, pt-fcn-handle.cc, pt-fcn-handle.h, pt-id.cc, pt-id.h, pt-idx.cc, pt-idx.h, pt-jump.h, pt-loop.cc, pt-loop.h, pt-mat.cc, pt-mat.h, pt-misc.cc, pt-misc.h, pt-pr-code.cc, pt-pr-code.h, pt-select.cc, pt-select.h, pt-spmd.cc, pt-spmd.h, pt-stmt.cc, pt-stmt.h, pt-tm-const.cc, pt-tm-const.h, pt-unop.cc, pt-unop.h, pt-vm-eval.cc, pt-walk.cc, pt-walk.h, pt.cc, pt.h, token.cc, token.h, Range.cc, Range.h, idx-vector.cc, idx-vector.h, range-fwd.h, CollocWt.cc, CollocWt.h, aepbalance.cc, aepbalance.h, chol.cc, chol.h, gepbalance.cc, gepbalance.h, gsvd.cc, gsvd.h, hess.cc, hess.h, lo-mappers.cc, lo-mappers.h, lo-specfun.cc, lo-specfun.h, lu.cc, lu.h, oct-convn.cc, oct-convn.h, oct-fftw.cc, oct-fftw.h, oct-norm.cc, oct-norm.h, oct-rand.cc, oct-rand.h, oct-spparms.cc, oct-spparms.h, qr.cc, qr.h, qrp.cc, qrp.h, randgamma.cc, randgamma.h, randmtzig.cc, randmtzig.h, randpoisson.cc, randpoisson.h, schur.cc, schur.h, sparse-chol.cc, sparse-chol.h, sparse-lu.cc, sparse-lu.h, sparse-qr.cc, sparse-qr.h, svd.cc, svd.h, child-list.cc, child-list.h, dir-ops.cc, dir-ops.h, file-ops.cc, file-ops.h, file-stat.cc, file-stat.h, lo-sysdep.cc, lo-sysdep.h, lo-sysinfo.cc, lo-sysinfo.h, mach-info.cc, mach-info.h, oct-env.cc, oct-env.h, oct-group.cc, oct-group.h, oct-password.cc, oct-password.h, oct-syscalls.cc, oct-syscalls.h, oct-time.cc, oct-time.h, oct-uname.cc, oct-uname.h, action-container.cc, action-container.h, base-list.h, cmd-edit.cc, cmd-edit.h, cmd-hist.cc, cmd-hist.h, f77-fcn.h, file-info.cc, file-info.h, lo-array-errwarn.cc, lo-array-errwarn.h, lo-hash.cc, lo-hash.h, lo-ieee.h, lo-regexp.cc, lo-regexp.h, lo-utils.cc, lo-utils.h, oct-base64.cc, oct-base64.h, oct-glob.cc, oct-glob.h, oct-inttypes.h, oct-mutex.cc, oct-mutex.h, oct-refcount.h, oct-shlib.cc, oct-shlib.h, oct-sparse.cc, oct-sparse.h, oct-string.h, octave-preserve-stream-state.h, pathsearch.cc, pathsearch.h, quit.cc, quit.h, unwind-prot.cc, unwind-prot.h, url-transfer.cc, url-transfer.h : Use new macros to begin/end C++ namespaces.
author Rik <rik@octave.org>
date Thu, 01 Dec 2022 14:23:45 -0800
parents 23520a50d74d
children aac27ad79be6
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1 ////////////////////////////////////////////////////////////////////////
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2 //
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3 // Copyright (C) 2021-2022 The Octave Project Developers
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4 //
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5 // See the file COPYRIGHT.md in the top-level directory of this
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6 // distribution or <https://octave.org/copyright/>.
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7 //
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8 // This file is part of Octave.
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9 //
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10 // Octave is free software: you can redistribute it and/or modify it
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11 // under the terms of the GNU General Public License as published by
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12 // the Free Software Foundation, either version 3 of the License, or
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13 // (at your option) any later version.
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14 //
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15 // Octave is distributed in the hope that it will be useful, but
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16 // WITHOUT ANY WARRANTY; without even the implied warranty of
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17 // MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 // GNU General Public License for more details.
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19 //
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20 // You should have received a copy of the GNU General Public License
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21 // along with Octave; see the file COPYING. If not, see
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22 // <https://www.gnu.org/licenses/>.
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23 //
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24 ////////////////////////////////////////////////////////////////////////
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25
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26 #if defined (HAVE_CONFIG_H)
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27 # include "config.h"
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28 #endif
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29
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30 #include "defun.h"
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31 #include "error.h"
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32 #include "ovl.h"
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33
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34 OCTAVE_BEGIN_NAMESPACE(octave)
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35
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36 // This function implements the Schrage technique for modular multiplication.
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37 // The returned value is equivalent to "mod (a*b, modulus)"
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38 // but calculated without overflow.
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39 uint64_t
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40 safemultiply (uint64_t a, uint64_t b, uint64_t modulus)
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41 {
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42 if (! a || ! b)
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43 return 0;
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44 else if (b == 1)
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45 return a;
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46 else if (a == 1)
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47 return b;
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48 else if (a > b)
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49 {
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50 uint64_t tmp = a;
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51 a = b;
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52 b = tmp;
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53 }
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54
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55 uint64_t q = modulus / a;
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56 uint64_t r = modulus - q * a;
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57 uint64_t term1 = a * (b % q);
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58 uint64_t term2 = (r < q) ? r * (b / q) : safemultiply (r, b / q, modulus);
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59 return (term1 > term2) ? (term1 - term2) : (term1 + modulus - term2);
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60 }
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61
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62 // This function returns "mod (a^b, modulus)"
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63 // but calculated without overflow.
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64 uint64_t
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65 safepower (uint64_t a, uint64_t b, uint64_t modulus)
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66 {
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67 uint64_t retval = 1;
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68 while (b > 0)
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69 {
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70 if (b & 1)
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71 retval = safemultiply (retval, a, modulus);
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72 b >>= 1;
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73 a = safemultiply (a, a, modulus);
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74 }
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75 return retval;
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76 }
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77
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78 // This function implements a single round of Miller-Rabin primality testing.
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79 // Returns false if composite, true if pseudoprime for this divisor.
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80 bool
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81 millerrabin (uint64_t div, uint64_t d, uint64_t r, uint64_t n)
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82 {
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83 uint64_t x = safepower (div, d, n);
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84 if (x == 1 || x == n-1)
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85 return true;
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86
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87 for (uint64_t j = 1; j < r; j++)
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88 {
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89 x = safemultiply (x, x, n);
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90 if (x == n-1)
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91 return true;
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92 }
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93 return false;
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94 }
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95
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96 // This function uses the Miller-Rabin test to find out whether the input is
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97 // prime or composite. The input is required to be a scalar 64-bit integer.
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98 bool
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99 isprimescalar (uint64_t n)
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100 {
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101 // Fast return for even numbers.
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102 // n==2 is excluded by the time this function is called.
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103 if (! (n & 1))
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104 return false;
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105
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106 // n is now odd. Rewrite n as d * 2^r + 1, where d is odd.
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107 uint64_t d = n-1;
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108 uint64_t r = 0;
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109 while (! (d & 1))
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110 {
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111 d >>= 1;
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112 r++;
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113 }
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114
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115 // Miller-Rabin test with the first 12 primes.
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116 // If the number passes all 12 tests, then it is prime.
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117 // If it fails any, then it is composite.
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118 // The first 12 primes suffice to test all 64-bit integers.
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119 return millerrabin ( 2, d, r, n) && millerrabin ( 3, d, r, n)
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120 && millerrabin ( 5, d, r, n) && millerrabin ( 7, d, r, n)
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121 && millerrabin (11, d, r, n) && millerrabin (13, d, r, n)
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122 && millerrabin (17, d, r, n) && millerrabin (19, d, r, n)
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123 && millerrabin (23, d, r, n) && millerrabin (29, d, r, n)
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124 && millerrabin (31, d, r, n) && millerrabin (37, d, r, n);
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125
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126 /*
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127 Mathematical references for the curious as to why we need only
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128 the 12 smallest primes for testing all 64-bit numbers:
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129 (1) https://oeis.org/A014233
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130 Comment: a(12) > 2^64. Hence the primality of numbers < 2^64 can be
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131 determined by asserting strong pseudoprimality to all prime bases <= 37
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132 (=prime(12)). Testing to prime bases <=31 does not suffice,
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133 as a(11) < 2^64 and a(11) is a strong pseudoprime
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134 to all prime bases <= 31 (=prime(11)). - Joerg Arndt, Jul 04 2012
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135 (2) https://arxiv.org/abs/1509.00864
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136 Strong Pseudoprimes to Twelve Prime Bases
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137 Jonathan P. Sorenson, Jonathan Webster
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138
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139 In addition, a source listed here: https://miller-rabin.appspot.com/
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140 reports that all 64-bit numbers can be covered with only 7 divisors,
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141 namely 2, 325, 9375, 28178, 450775, 9780504, and 1795265022.
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142 There was no peer-reviewed article to back it up though,
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143 so this code uses the 12 primes <= 37.
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144 */
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145
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146 }
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147
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148 DEFUN (__isprimelarge__, args, ,
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149 doc: /* -*- texinfo -*-
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150 @deftypefn {} {@var{x} =} __isprimelarge__ (@var{n})
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151 Use the Miller-Rabin test to find out whether the elements of N are prime or
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152 composite. The input N is required to be a vector or array of 64-bit integers.
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153 You should call isprime(N) instead of directly calling this function.
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154
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155 @seealso{isprime, factor}
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156 @end deftypefn */)
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157 {
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158 if (args.length () != 1)
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159 print_usage ();
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160
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161 // This function is intended for internal use by isprime.m,
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162 // so the following error handling should not be necessary. But it is
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163 // probably good practice for any curious users calling it directly.
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164 uint64NDArray vec = args(0).xuint64_array_value
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165 ("__isprimelarge__: unable to convert input. Call isprime() instead.");
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166
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167 boolNDArray retval (vec.dims(), false);
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168
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169 for (octave_idx_type i = vec.numel() - 1; i >= 0; i--)
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170 retval(i) = isprimescalar (vec(i));
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171 // Note: If vec(i) <= 37, this function could go into an infinite loop.
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172 // That situation does not arise when calling this from isprime.m
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173 // but it could arise if the user calls this function directly with low input
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174 // or negative input.
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175 // But it turns out that adding this validation:
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176 // "if (vec(i) <= 37) then raise an error else call isprimescalar (vec(i))"
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177 // slows this function down by over 20% for some inputs,
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178 // so it is better to leave all the input validation in isprime.m
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179 // and not add it here. The function DOCSTRING now explicitly says:
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180 // "You should call isprime(N) instead of directly calling this function."
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181
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182 return ovl (retval);
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183 }
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184
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185 /*
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186 %!assert (__isprimelarge__ (41:50), logical ([1 0 1 0 0 0 1 0 0 0]))
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187 %!assert (__isprimelarge__ (uint64 (12345)), false)
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188 %!assert (__isprimelarge__ (uint64 (2147483647)), true)
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189 %!assert (__isprimelarge__ (uint64 (2305843009213693951)), true)
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190 %!assert (__isprimelarge__ (uint64 (18446744073709551557)), true)
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191
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192 %!assert (__isprimelarge__ ([uint64(12345), uint64(2147483647), ...
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193 %! uint64(2305843009213693951), ...
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194 %! uint64(18446744073709551557)]),
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195 %! logical ([0 1 1 1]))
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196
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197 %!error <unable to convert input> (__isprimelarge__ ({'foo'; 'bar'}))
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198 */
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199
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200
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201 // This function implements a fast, private GCD function
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202 // optimized for uint64_t. No input validation by design.
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203 inline
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204 uint64_t
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205 localgcd (uint64_t a, uint64_t b)
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206 {
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207 return (a <= b) ? ( (b % a == 0) ? a : localgcd (a, b % a) )
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208 : ( (a % b == 0) ? b : localgcd (a % b, b) );
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209 }
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210
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211 // This function implements a textbook version of the Pollard Rho
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212 // factorization algorithm with Brent update.
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213 // The code is short and simple, but the math behind it is complicated.
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214 uint64_t
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215 pollardrho (uint64_t n, uint64_t c = 1)
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216 {
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217 uint64_t i = 1, j = 2; // cycle index values
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218 uint64_t x = (c+1) % n; // can also be rand () % n
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219 uint64_t y = x; // other value in the chain
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220 uint64_t g = 0; // GCD
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221
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222 while (true)
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223 {
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224 i++;
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225
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226 // Calculate x = mod (x^2 + c, n) without overflow.
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227 x = safemultiply (x, x, n) + c;
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228 if (x >= n)
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229 x -= n;
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230
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231 // Calculate GCD (abs (x-y), n).
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232 g = (x > y) ? localgcd (x - y, n) : (x < y) ? localgcd (y - x, n) : 0;
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233
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234 if (i == j) // cycle detected ==> double j
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235 {
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236 y = x;
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237 j <<= 1;
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238 }
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239
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240 if (g == n || i > 1000000) // cut losses, restart with a different c
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241 return pollardrho (n, c + 2);
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242
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243 if (g > 1) // found GCD ==> exit loop properly
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244 {
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245 error_unless (n % g == 0); // theoretical possibility of GCD error
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246 return g;
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247 }
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248 }
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249 }
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250
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251 DEFUN (__pollardrho__, args, ,
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252 doc: /* -*- texinfo -*-
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253 @deftypefn {} {@var{x} =} __pollardrho__ (@var{n})
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254 Private function. Use the Pollard Rho test to find a factor of @var{n}.
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255 The input @var{n} is required to be a composite 64-bit integer.
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256 Do not pass it a prime input! You should call factor(@var{n}) instead
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257 of directly calling this function.
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258
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259 @seealso{isprime, factor}
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260 @end deftypefn */)
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261 {
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262 if (args.length () != 1)
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263 print_usage ();
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264
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265 octave_uint64 inp = args(0).xuint64_scalar_value
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266 ("__pollardrho__: unable to convert input. Call factor() instead.");
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267
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268 uint64_t n = inp;
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269 octave_uint64 retval = pollardrho (n);
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270
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271 return ovl (retval);
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272 }
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273
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274 /*
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275 %!assert (__pollardrho__ (uint64 (78567695338254293)), uint64 (443363))
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276 %!assert (__pollardrho__ (1084978968791), uint64 (832957))
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277
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278 %!error <unable to convert input> (__pollardrho__ ({'foo'; 'bar'}))
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279 */
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280
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281 OCTAVE_END_NAMESPACE(octave)