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97 lines
4.6 KiB
HTML
97 lines
4.6 KiB
HTML
4 years ago
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual describes how to install and use the GNU multiple precision
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arithmetic library, version 6.1.0.
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Copyright 1991, 1993-2015 Free Software Foundation, Inc.
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Permission is granted to copy, distribute and/or modify this document under
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the terms of the GNU Free Documentation License, Version 1.3 or any later
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version published by the Free Software Foundation; with no Invariant Sections,
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with the Front-Cover Texts being "A GNU Manual", and with the Back-Cover
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Texts being "You have freedom to copy and modify this GNU Manual, like GNU
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software". A copy of the license is included in
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GNU Free Documentation License. -->
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<!-- Created by GNU Texinfo 6.4, http://www.gnu.org/software/texinfo/ -->
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<head>
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<title>Prime Testing Algorithm (GNU MP 6.1.0)</title>
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<meta name="description" content="How to install and use the GNU multiple precision arithmetic library, version 6.1.0.">
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<meta name="keywords" content="Prime Testing Algorithm (GNU MP 6.1.0)">
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<link href="index.html#Top" rel="start" title="Top">
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<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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<link href="Other-Algorithms.html#Other-Algorithms" rel="up" title="Other Algorithms">
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<link href="Factorial-Algorithm.html#Factorial-Algorithm" rel="next" title="Factorial Algorithm">
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<link href="Other-Algorithms.html#Other-Algorithms" rel="prev" title="Other Algorithms">
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</head>
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<body lang="en">
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<a name="Prime-Testing-Algorithm"></a>
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<div class="header">
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<p>
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Next: <a href="Factorial-Algorithm.html#Factorial-Algorithm" accesskey="n" rel="next">Factorial Algorithm</a>, Previous: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="p" rel="prev">Other Algorithms</a>, Up: <a href="Other-Algorithms.html#Other-Algorithms" accesskey="u" rel="up">Other Algorithms</a> [<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<a name="Prime-Testing"></a>
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<h4 class="subsection">15.7.1 Prime Testing</h4>
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<a name="index-Prime-testing-algorithms"></a>
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<p>The primality testing in <code>mpz_probab_prime_p</code> (see <a href="Number-Theoretic-Functions.html#Number-Theoretic-Functions">Number Theoretic Functions</a>) first does some trial division by small factors and then uses the
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Miller-Rabin probabilistic primality testing algorithm, as described in Knuth
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section 4.5.4 algorithm P (see <a href="References.html#References">References</a>).
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</p>
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<p>For an odd input <em>n</em>, and with <em>n = q*2^k+1</em> where
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<em>q</em> is odd, this algorithm selects a random base <em>x</em> and tests
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whether <em>x^q mod n</em> is 1 or <em>-1</em>, or an <em>x^(q*2^j) mod n</em> is <em>1</em>, for <em>1<=j<=k</em>. If so then <em>n</em>
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is probably prime, if not then <em>n</em> is definitely composite.
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</p>
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<p>Any prime <em>n</em> will pass the test, but some composites do too. Such
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composites are known as strong pseudoprimes to base <em>x</em>. No <em>n</em> is
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a strong pseudoprime to more than <em>1/4</em> of all bases (see Knuth exercise
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22), hence with <em>x</em> chosen at random there’s no more than a <em>1/4</em>
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chance a “probable prime” will in fact be composite.
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</p>
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<p>In fact strong pseudoprimes are quite rare, making the test much more
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powerful than this analysis would suggest, but <em>1/4</em> is all that’s proven
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for an arbitrary <em>n</em>.
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</p>
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</body>
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</html>
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