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94 lines
4.5 KiB
HTML
94 lines
4.5 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>Extended GCD (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="Extended GCD (GNU MP 6.1.0)">
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<meta name="resource-type" content="document">
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<meta name="distribution" content="global">
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<meta name="Generator" content="makeinfo">
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<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
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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="Jacobi-Symbol.html#Jacobi-Symbol" rel="next" title="Jacobi Symbol">
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<link href="Subquadratic-GCD.html#Subquadratic-GCD" rel="prev" title="Subquadratic GCD">
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-->
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</style>
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</head>
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<body lang="en">
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<a name="Extended-GCD"></a>
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<div class="header">
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<p>
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Next: <a href="Jacobi-Symbol.html#Jacobi-Symbol" accesskey="n" rel="next">Jacobi Symbol</a>, Previous: <a href="Subquadratic-GCD.html#Subquadratic-GCD" accesskey="p" rel="prev">Subquadratic 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="Extended-GCD-1"></a>
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<h4 class="subsection">15.3.4 Extended GCD</h4>
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<p>The extended GCD function, or GCDEXT, calculates <em>gcd(a,b)</em> and also
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cofactors <em>x</em> and <em>y</em> satisfying <em>a*x+b*y=gcd(a,b)</em>. All the algorithms used for plain GCD are extended to
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handle this case. The binary algorithm is used only for single-limb GCDEXT.
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Lehmer’s algorithm is used for sizes up to <code>GCDEXT_DC_THRESHOLD</code>. Above
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this threshold, GCDEXT is implemented as a loop around HGCD, but with more
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book-keeping to keep track of the cofactors. This gives the same asymptotic
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running time as for GCD and HGCD, <em>O(M(N)*log(N))</em>
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</p>
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<p>One difference to plain GCD is that while the inputs <em>a</em> and <em>b</em> are
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reduced as the algorithm proceeds, the cofactors <em>x</em> and <em>y</em> grow in
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size. This makes the tuning of the chopping-point more difficult. The current
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code chops off the most significant half of the inputs for the call to HGCD in
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the first iteration, and the most significant two thirds for the remaining
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calls. This strategy could surely be improved. Also the stop condition for the
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loop, where Lehmer’s algorithm is invoked once the inputs are reduced below
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<code>GCDEXT_DC_THRESHOLD</code>, could maybe be improved by taking into account the
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current size of the cofactors.
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</p>
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</body>
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</html>
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