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<div class="section boost_multiprecision_tut_primetest">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_multiprecision.tut.primetest"></a><a class="link" href="primetest.html" title="Primality Testing">Primality Testing</a>
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</h3></div></div></div>
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<p>
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The library implements a Miller-Rabin test for primality:
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</p>
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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">multiprecision</span><span class="special">/</span><span class="identifier">miller_rabin</span><span class="special">.</span><span class="identifier">hpp</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">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Engine</span><span class="special">></span>
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<span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">>&</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">,</span> <span class="identifier">Engine</span><span class="special">&</span> <span class="identifier">gen</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">Backend</span><span class="special">,</span> <span class="identifier">expression_template_option</span> <span class="identifier">ExpressionTemplates</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Engine</span><span class="special">></span>
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<span class="keyword">bool</span> <span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">number</span><span class="special"><</span><span class="identifier">Backend</span><span class="special">,</span> <span class="identifier">ExpressionTemplates</span><span class="special">>&</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">trials</span><span class="special">);</span>
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</pre>
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<p>
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These functions perform a Miller-Rabin test for primality, if the result
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is <code class="computeroutput"><span class="keyword">false</span></code> then <span class="emphasis"><em>n</em></span>
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is definitely composite, while if the result is <code class="computeroutput"><span class="keyword">true</span></code>
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then <span class="emphasis"><em>n</em></span> is prime with probability <span class="emphasis"><em>0.25^trials</em></span>.
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The algorithm used performs some trial divisions to exclude small prime factors,
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does one Fermat test to exclude many more composites, and then uses the Miller-Rabin
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algorithm straight out of Knuth Vol 2, which recommends 25 trials for a pretty
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strong likelihood that <span class="emphasis"><em>n</em></span> is prime.
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</p>
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<p>
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The third optional argument is for a Uniform Random Number Generator from
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Boost.Random. When not provided the <code class="computeroutput"><span class="identifier">mt19937</span></code>
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generator is used. Note that when producing random primes then you should
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probably use a different random number generator to produce candidate prime
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numbers for testing, than is used internally by <code class="computeroutput"><span class="identifier">miller_rabin_test</span></code>
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for determining whether the value is prime. It also helps of course to seed
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the generators with some source of randomness.
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</p>
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<p>
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The following example searches for a prime <code class="computeroutput"><span class="identifier">p</span></code>
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for which <code class="computeroutput"><span class="special">(</span><span class="identifier">p</span><span class="special">-</span><span class="number">1</span><span class="special">)/</span><span class="number">2</span></code> is also probably prime:
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</p>
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<p>
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</p>
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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">multiprecision</span><span class="special">/</span><span class="identifier">cpp_int</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">miller_rabin</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span>
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<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></span>
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<span class="keyword">int</span> <span class="identifier">main</span><span class="special">()</span>
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<span class="special">{</span>
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<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">random</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">;</span>
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<span class="keyword">typedef</span> <span class="identifier">cpp_int</span> <span class="identifier">int_type</span><span class="special">;</span>
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<span class="identifier">mt11213b</span> <span class="identifier">base_gen</span><span class="special">(</span><span class="identifier">clock</span><span class="special">());</span>
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<span class="identifier">independent_bits_engine</span><span class="special"><</span><span class="identifier">mt11213b</span><span class="special">,</span> <span class="number">256</span><span class="special">,</span> <span class="identifier">int_type</span><span class="special">></span> <span class="identifier">gen</span><span class="special">(</span><span class="identifier">base_gen</span><span class="special">);</span>
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<span class="comment">//</span>
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<span class="comment">// We must use a different generator for the tests and number generation, otherwise</span>
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<span class="comment">// we get false positives.</span>
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<span class="comment">//</span>
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<span class="identifier">mt19937</span> <span class="identifier">gen2</span><span class="special">(</span><span class="identifier">clock</span><span class="special">());</span>
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<span class="keyword">for</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span> <span class="special">=</span> <span class="number">0</span><span class="special">;</span> <span class="identifier">i</span> <span class="special"><</span> <span class="number">100000</span><span class="special">;</span> <span class="special">++</span><span class="identifier">i</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="identifier">int_type</span> <span class="identifier">n</span> <span class="special">=</span> <span class="identifier">gen</span><span class="special">();</span>
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<span class="keyword">if</span><span class="special">(</span><span class="identifier">miller_rabin_test</span><span class="special">(</span><span class="identifier">n</span><span class="special">,</span> <span class="number">25</span><span class="special">,</span> <span class="identifier">gen2</span><span class="special">))</span>
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<span class="special">{</span>
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<span class="comment">// Value n is probably prime, see if (n-1)/2 is also prime:</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">"We have a probable prime with value: "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">n</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
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<span class="keyword">if</span><span class="special">(</span><span class="identifier">miller_rabin_test</span><span class="special">((</span><span class="identifier">n</span><span class="special">-</span><span class="number">1</span><span class="special">)/</span><span class="number">2</span><span class="special">,</span> <span class="number">25</span><span class="special">,</span> <span class="identifier">gen2</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">"We have a safe prime with value: "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">hex</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">showbase</span> <span class="special"><<</span> <span class="identifier">n</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
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<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
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<span class="special">}</span>
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<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">"Ooops, no safe primes were found"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
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<span class="keyword">return</span> <span class="number">1</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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<td align="right"><div class="copyright-footer">Copyright © 2002-2012 John Maddock and Christopher Kormanyos<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="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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