1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Extended Euclidean Algorithm</title> 5<link rel="stylesheet" href="../../../../../doc/src/boostbook.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Boost.Integer"> 8<link rel="up" href="../index.html" title="Boost.Integer"> 9<link rel="prev" href="gcd_lcm.html" title="Greatest Common Divisor and Least Common Multiple"> 10<link rel="next" href="mod_inverse.html" title="Modular Multiplicative Inverse"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="gcd_lcm.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../index.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="mod_inverse.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="boost_integer.extended_euclidean"></a><a class="link" href="extended_euclidean.html" title="Extended Euclidean Algorithm">Extended Euclidean Algorithm</a> 28</h2></div></div></div> 29<div class="toc"><dl class="toc"> 30<dt><span class="section"><a href="extended_euclidean.html#boost_integer.extended_euclidean.introduction">Introduction</a></span></dt> 31<dt><span class="section"><a href="extended_euclidean.html#boost_integer.extended_euclidean.synopsis">Synopsis</a></span></dt> 32<dt><span class="section"><a href="extended_euclidean.html#boost_integer.extended_euclidean.usage">Usage</a></span></dt> 33<dt><span class="section"><a href="extended_euclidean.html#boost_integer.extended_euclidean.references">References</a></span></dt> 34</dl></div> 35<div class="section"> 36<div class="titlepage"><div><div><h3 class="title"> 37<a name="boost_integer.extended_euclidean.introduction"></a><a class="link" href="extended_euclidean.html#boost_integer.extended_euclidean.introduction" title="Introduction">Introduction</a> 38</h3></div></div></div> 39<p> 40 The extended Euclidean algorithm solves the integer relation <span class="emphasis"><em>mx 41 + ny</em></span> = gcd(<span class="emphasis"><em>m</em></span>, <span class="emphasis"><em>n</em></span>) for 42 <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>y</em></span>. 43 </p> 44</div> 45<div class="section"> 46<div class="titlepage"><div><div><h3 class="title"> 47<a name="boost_integer.extended_euclidean.synopsis"></a><a class="link" href="extended_euclidean.html#boost_integer.extended_euclidean.synopsis" title="Synopsis">Synopsis</a> 48</h3></div></div></div> 49<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">integer</span><span class="special">/</span><span class="identifier">extended_euclidean</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 50 51<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">integer</span> <span class="special">{</span> 52 53<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Z</span><span class="special">></span> 54<span class="keyword">struct</span> <span class="identifier">euclidean_result_t</span> <span class="special">{</span> 55 <span class="identifier">Z</span> <span class="identifier">gcd</span><span class="special">;</span> 56 <span class="identifier">Z</span> <span class="identifier">x</span><span class="special">;</span> 57 <span class="identifier">Z</span> <span class="identifier">y</span><span class="special">;</span> 58<span class="special">};</span> 59 60 61<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Z</span><span class="special">></span> 62<span class="identifier">euclidean_result_t</span><span class="special"><</span><span class="identifier">Z</span><span class="special">></span> <span class="identifier">extended_euclidean</span><span class="special">(</span><span class="identifier">Z</span> <span class="identifier">m</span><span class="special">,</span> <span class="identifier">Z</span> <span class="identifier">n</span><span class="special">);</span> 63 64<span class="special">}}</span> 65</pre> 66</div> 67<div class="section"> 68<div class="titlepage"><div><div><h3 class="title"> 69<a name="boost_integer.extended_euclidean.usage"></a><a class="link" href="extended_euclidean.html#boost_integer.extended_euclidean.usage" title="Usage">Usage</a> 70</h3></div></div></div> 71<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">m</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span> 72<span class="keyword">int</span> <span class="identifier">n</span> <span class="special">=</span> <span class="number">15</span><span class="special">;</span> 73<span class="keyword">auto</span> <span class="identifier">res</span> <span class="special">=</span> <span class="identifier">extended_euclidean</span><span class="special">(</span><span class="identifier">m</span><span class="special">,</span> <span class="identifier">n</span><span class="special">);</span> 74 75<span class="keyword">int</span> <span class="identifier">gcd</span> <span class="special">=</span> <span class="identifier">res</span><span class="special">.</span><span class="identifier">gcd</span><span class="special">;</span> 76<span class="keyword">int</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">res</span><span class="special">.</span><span class="identifier">x</span><span class="special">;</span> 77<span class="keyword">int</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">res</span><span class="special">.</span><span class="identifier">y</span><span class="special">;</span> 78<span class="comment">// mx + ny = gcd(m,n) should now hold</span> 79</pre> 80</div> 81<div class="section"> 82<div class="titlepage"><div><div><h3 class="title"> 83<a name="boost_integer.extended_euclidean.references"></a><a class="link" href="extended_euclidean.html#boost_integer.extended_euclidean.references" title="References">References</a> 84</h3></div></div></div> 85<p> 86 Wagstaff, Samuel S., <span class="emphasis"><em>The Joy of Factoring</em></span>, Vol. 68. 87 American Mathematical Soc., 2013. 88 </p> 89</div> 90</div> 91<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 92<td align="left"></td> 93<td align="right"><div class="copyright-footer">Copyright © 2001-2009 Beman 94 Dawes, Daryle Walker, Gennaro Prota, John Maddock<p> 95 Distributed under the Boost Software License, Version 1.0. (See accompanying 96 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 97 </p> 98</div></td> 99</tr></table> 100<hr> 101<div class="spirit-nav"> 102<a accesskey="p" href="gcd_lcm.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../index.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="mod_inverse.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 103</div> 104</body> 105</html> 106
