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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="boost_integer.mod_inverse"></a><a class="link" href="mod_inverse.html" title="Modular Multiplicative Inverse">Modular Multiplicative Inverse</a>
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</h2></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="mod_inverse.html#boost_integer.mod_inverse.introduction">Introduction</a></span></dt>
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<dt><span class="section"><a href="mod_inverse.html#boost_integer.mod_inverse.synopsis">Synopsis</a></span></dt>
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<dt><span class="section"><a href="mod_inverse.html#boost_integer.mod_inverse.usage">Usage</a></span></dt>
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<dt><span class="section"><a href="mod_inverse.html#boost_integer.mod_inverse.references">References</a></span></dt>
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</dl></div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_integer.mod_inverse.introduction"></a><a class="link" href="mod_inverse.html#boost_integer.mod_inverse.introduction" title="Introduction">Introduction</a>
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</h3></div></div></div>
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<p>
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The modular multiplicative inverse of a number <span class="emphasis"><em>a</em></span> is
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that number <span class="emphasis"><em>x</em></span> which satisfies <span class="emphasis"><em>ax</em></span>
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= 1 mod <span class="emphasis"><em>p</em></span>. A fast algorithm for computing modular multiplicative
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inverses based on the extended Euclidean algorithm exists and is provided
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by Boost.
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_integer.mod_inverse.synopsis"></a><a class="link" href="mod_inverse.html#boost_integer.mod_inverse.synopsis" title="Synopsis">Synopsis</a>
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</h3></div></div></div>
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<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">mod_inverse</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<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>
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<span class="keyword">template</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">Z</span><span class="special">></span>
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<span class="identifier">Z</span> <span class="identifier">mod_inverse</span><span class="special">(</span><span class="identifier">Z</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Z</span> <span class="identifier">m</span><span class="special">);</span>
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<span class="special">}}</span>
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</pre>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_integer.mod_inverse.usage"></a><a class="link" href="mod_inverse.html#boost_integer.mod_inverse.usage" title="Usage">Usage</a>
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</h3></div></div></div>
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<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">x</span> <span class="special">=</span> <span class="identifier">mod_inverse</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">5</span><span class="special">);</span>
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<span class="comment">// prints x = 3:</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x = "</span> <span class="special"><<</span> <span class="identifier">x</span> <span class="special"><<</span> <span class="string">"\n"</span><span class="special">;</span>
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<span class="keyword">int</span> <span class="identifier">y</span> <span class="special">=</span> <span class="identifier">mod_inverse</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">4</span><span class="special">);</span>
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<span class="keyword">if</span> <span class="special">(</span><span class="identifier">y</span> <span class="special">==</span> <span class="number">0</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"There is no inverse of 2 mod 4\n"</span><span class="special">;</span>
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<span class="special">}</span>
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</pre>
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<p>
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Multiplicative modular inverses exist if and only if <span class="emphasis"><em>a</em></span>
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and <span class="emphasis"><em>m</em></span> are coprime. If <span class="emphasis"><em>a</em></span> and <span class="emphasis"><em>m</em></span>
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share a common factor, then <code class="computeroutput"><span class="identifier">mod_inverse</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span>
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<span class="identifier">m</span><span class="special">)</span></code>
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returns zero.
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</p>
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</div>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_integer.mod_inverse.references"></a><a class="link" href="mod_inverse.html#boost_integer.mod_inverse.references" title="References">References</a>
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</h3></div></div></div>
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<p>
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Wagstaff, Samuel S., <span class="emphasis"><em>The Joy of Factoring</em></span>, Vol. 68.
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American Mathematical Soc., 2013.
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</p>
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</div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2001-2009 Beman
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Dawes, Daryle Walker, Gennaro Prota, John Maddock<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="https://www.boost.org/LICENSE_1_0.txt" target="_top">https://www.boost.org/LICENSE_1_0.txt</a>)
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