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121 lines
5.8 KiB
HTML
121 lines
5.8 KiB
HTML
<!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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software". A copy of the license is included in
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GNU Free Documentation License. -->
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<head>
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<title>Random Number Algorithms (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="Random Number Algorithms (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="Assembly-Coding.html#Assembly-Coding" rel="next" title="Assembly Coding">
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<link href="Lucas-Numbers-Algorithm.html#Lucas-Numbers-Algorithm" rel="prev" title="Lucas Numbers Algorithm">
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</head>
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<body lang="en">
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<a name="Random-Number-Algorithms"></a>
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<div class="header">
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<p>
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Previous: <a href="Lucas-Numbers-Algorithm.html#Lucas-Numbers-Algorithm" accesskey="p" rel="prev">Lucas Numbers Algorithm</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="Random-Numbers"></a>
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<h4 class="subsection">15.7.6 Random Numbers</h4>
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<a name="index-Random-number-algorithms"></a>
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<p>For the <code>urandomb</code> functions, random numbers are generated simply by
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concatenating bits produced by the generator. As long as the generator has
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good randomness properties this will produce well-distributed <em>N</em> bit
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numbers.
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</p>
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<p>For the <code>urandomm</code> functions, random numbers in a range <em>0<=R<N</em>
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are generated by taking values <em>R</em> of <em>ceil(log2(N))</em> bits each until one satisfies <em>R<N</em>. This will normally
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require only one or two attempts, but the attempts are limited in case the
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generator is somehow degenerate and produces only 1 bits or similar.
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</p>
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<a name="index-Mersenne-twister-algorithm"></a>
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<p>The Mersenne Twister generator is by Matsumoto and Nishimura
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(see <a href="References.html#References">References</a>). It has a non-repeating period of <em>2^19937<!-- /@w -->-1</em>,
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which is a Mersenne prime, hence the name of the generator. The state is 624
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words of 32-bits each, which is iterated with one XOR and shift for each
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32-bit word generated, making the algorithm very fast. Randomness properties
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are also very good and this is the default algorithm used by GMP.
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</p>
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<a name="index-Linear-congruential-algorithm"></a>
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<p>Linear congruential generators are described in many text books, for instance
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Knuth volume 2 (see <a href="References.html#References">References</a>). With a modulus <em>M</em> and parameters
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<em>A</em> and <em>C</em>, an integer state <em>S</em> is iterated by the formula
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<em>S <- A*S+C mod M</em>. At each step the new
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state is a linear function of the previous, mod <em>M</em>, hence the name of
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the generator.
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</p>
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<p>In GMP only moduli of the form <em>2^N</em> are supported, and the current
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implementation is not as well optimized as it could be. Overheads are
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significant when <em>N</em> is small, and when <em>N</em> is large clearly the
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multiply at each step will become slow. This is not a big concern, since the
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Mersenne Twister generator is better in every respect and is therefore
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recommended for all normal applications.
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</p>
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<p>For both generators the current state can be deduced by observing enough
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output and applying some linear algebra (over GF(2) in the case of the
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Mersenne Twister). This generally means raw output is unsuitable for
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cryptographic applications without further hashing or the like.
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</p>
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<hr>
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<div class="header">
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<p>
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Previous: <a href="Lucas-Numbers-Algorithm.html#Lucas-Numbers-Algorithm" accesskey="p" rel="prev">Lucas Numbers Algorithm</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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</body>
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</html>
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