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94 lines
4.2 KiB
HTML
94 lines
4.2 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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<head>
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<title>Jacobi Symbol (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="Jacobi Symbol (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="Greatest-Common-Divisor-Algorithms.html#Greatest-Common-Divisor-Algorithms" rel="up" title="Greatest Common Divisor Algorithms">
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<link href="Powering-Algorithms.html#Powering-Algorithms" rel="next" title="Powering Algorithms">
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<link href="Extended-GCD.html#Extended-GCD" rel="prev" title="Extended GCD">
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</head>
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<body lang="en">
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<a name="Jacobi-Symbol"></a>
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<div class="header">
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<p>
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Previous: <a href="Extended-GCD.html#Extended-GCD" accesskey="p" rel="prev">Extended GCD</a>, Up: <a href="Greatest-Common-Divisor-Algorithms.html#Greatest-Common-Divisor-Algorithms" accesskey="u" rel="up">Greatest Common Divisor 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="Jacobi-Symbol-1"></a>
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<h4 class="subsection">15.3.5 Jacobi Symbol</h4>
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<a name="index-Jacobi-symbol-algorithm"></a>
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<p>[This section is obsolete. The current Jacobi code actually uses a very
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efficient algorithm.]
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</p>
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<p><code>mpz_jacobi</code> and <code>mpz_kronecker</code> are currently implemented with a
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simple binary algorithm similar to that described for the GCDs (see <a href="Binary-GCD.html#Binary-GCD">Binary GCD</a>). They’re not very fast when both inputs are large. Lehmer’s multi-step
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improvement or a binary based multi-step algorithm is likely to be better.
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</p>
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<p>When one operand fits a single limb, and that includes <code>mpz_kronecker_ui</code>
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and friends, an initial reduction is done with either <code>mpn_mod_1</code> or
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<code>mpn_modexact_1_odd</code>, followed by the binary algorithm on a single limb.
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The binary algorithm is well suited to a single limb, and the whole
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calculation in this case is quite efficient.
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</p>
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<p>In all the routines sign changes for the result are accumulated using some bit
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twiddling, avoiding table lookups or conditional jumps.
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</p>
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</body>
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</html>
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