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annotate libinterp/corefcn/__isprimelarge__.cc @ 31051:5f015f2829b7

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maint: Merge stable to default
author Arun Giridhar <arungiridhar@gmail.com>
date Wed, 01 Jun 2022 17:20:22 -0400
parents b65e4c635343 3a93a77dd4cf
children 068342cc93b8
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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_NAMESPACE_BEGIN
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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 int nargin = args.length ();
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159 if (nargin != 1)
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160 print_usage ();
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161
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162 // This function is intended for internal use by isprime.m,
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163 // so the following error handling should not be necessary. But it is
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164 // probably good practice for any curious users calling it directly.
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165 uint64NDArray vec = args(0).xuint64_array_value
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166 ("__isprimelarge__: unable to convert input. Call isprime() instead.");
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167
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168 boolNDArray retval (vec.dims(), false);
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169
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170 for (octave_idx_type i = vec.numel() - 1; i >= 0; i--)
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171 retval(i) = isprimescalar (vec(i));
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172 // Note: If vec(i) <= 37, this function could go into an infinite loop.
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173 // That situation does not arise when calling this from isprime.m
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174 // but it could arise if the user calls this function directly with low input
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175 // or negative input.
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176 // But it turns out that adding this validation:
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177 // "if (vec(i) <= 37) then raise an error else call isprimescalar (vec(i))"
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178 // slows this function down by over 20% for some inputs,
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179 // so it is better to leave all the input validation in isprime.m
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180 // and not add it here. The function DOCSTRING now explicitly says:
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181 // "You should call isprime(N) instead of directly calling this function."
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182
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183 return ovl (retval);
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184 }
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185
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186 /*
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187 %!assert (__isprimelarge__ (41:50), logical ([1 0 1 0 0 0 1 0 0 0]))
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188 %!assert (__isprimelarge__ (uint64 (12345)), false)
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189 %!assert (__isprimelarge__ (uint64 (2147483647)), true)
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190 %!assert (__isprimelarge__ (uint64 (2305843009213693951)), true)
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191 %!assert (__isprimelarge__ (uint64 (18446744073709551557)), true)
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192
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193 %!assert (__isprimelarge__ ([uint64(12345), uint64(2147483647), ...
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194 %! uint64(2305843009213693951), ...
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195 %! uint64(18446744073709551557)]),
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196 %! logical ([0 1 1 1]))
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197
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198 %!error <unable to convert input> (__isprimelarge__ ({'foo'; 'bar'}))
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199 */
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200
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201 OCTAVE_NAMESPACE_END