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@@ -105,6 +105,7 @@ $(function(){initNavTree('d8/d53/modular__inverse__fermat__little__theorem_8cpp.
<div class="header">
<div class="summary">
<a href="#namespaces">Namespaces</a> &#124;
<a href="#func-members">Functions</a> </div>
<div class="headertitle"><div class="title">modular_inverse_fermat_little_theorem.cpp File Reference</div></div>
</div><!--header-->
@@ -112,21 +113,40 @@ $(function(){initNavTree('d8/d53/modular__inverse__fermat__little__theorem_8cpp.
<p>C++ Program to find the modular inverse using <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" target="_blank">Fermat's Little Theorem</a>
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;iostream&gt;</code><br />
<code>#include &lt;vector&gt;</code><br />
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;cstdint&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for modular_inverse_fermat_little_theorem.cpp:</div>
<div class="dyncontent">
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</div>
</div><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="namespaces" name="namespaces"></a>
Namespaces</h2></td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">namespace &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../dd/d47/namespacemath.html">math</a></td></tr>
<tr class="memdesc:dd/d47/namespacemath"><td class="mdescLeft">&#160;</td><td class="mdescRight">for IO operations <br /></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:"><td class="memItemLeft" align="right" valign="top">namespace &#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d7/d50/namespacemodular__inverse__fermat.html">modular_inverse_fermat</a></td></tr>
<tr class="memdesc:d7/d50/namespacemodular__inverse__fermat"><td class="mdescLeft">&#160;</td><td class="mdescRight">Calculate modular inverse using Fermat's Little Theorem. <br /></td></tr>
<tr class="separator:"><td class="memSeparator" colspan="2">&#160;</td></tr>
</table><table class="memberdecls">
<tr class="heading"><td colspan="2"><h2 class="groupheader"><a id="func-members" name="func-members"></a>
Functions</h2></td></tr>
<tr class="memitem:a4c6eefd1254eab3e8d34bf02c205e0f4" id="r_a4c6eefd1254eab3e8d34bf02c205e0f4"><td class="memItemLeft" align="right" valign="top">int64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a4c6eefd1254eab3e8d34bf02c205e0f4">binExpo</a> (int64_t a, int64_t b, int64_t m)</td></tr>
<tr class="separator:a4c6eefd1254eab3e8d34bf02c205e0f4"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a09660096b134753128952246f4f4e4bd" id="r_a09660096b134753128952246f4f4e4bd"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a09660096b134753128952246f4f4e4bd">isPrime</a> (int64_t m)</td></tr>
<tr class="separator:a09660096b134753128952246f4f4e4bd"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a1c62a87e79f29b98d8fe5b12eacdd928" id="r_a1c62a87e79f29b98d8fe5b12eacdd928"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a>&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a1c62a87e79f29b98d8fe5b12eacdd928">math::modular_inverse_fermat::binExpo</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> a, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> b, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> m)</td></tr>
<tr class="memdesc:a1c62a87e79f29b98d8fe5b12eacdd928"><td class="mdescLeft">&#160;</td><td class="mdescRight">Calculate exponent with modulo using binary exponentiation in \(O(\log b)\) time. <br /></td></tr>
<tr class="separator:a1c62a87e79f29b98d8fe5b12eacdd928"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:aba4929409fee35c3cb559e962a544b3e" id="r_aba4929409fee35c3cb559e962a544b3e"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#aba4929409fee35c3cb559e962a544b3e">math::modular_inverse_fermat::isPrime</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> m)</td></tr>
<tr class="memdesc:aba4929409fee35c3cb559e962a544b3e"><td class="mdescLeft">&#160;</td><td class="mdescRight">Check if an integer is a prime number in \(O(\sqrt{m})\) time. <br /></td></tr>
<tr class="separator:aba4929409fee35c3cb559e962a544b3e"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae7e807e02c65c6fffd6162b4c66290c2" id="r_ae7e807e02c65c6fffd6162b4c66290c2"><td class="memItemLeft" align="right" valign="top"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a>&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ae7e807e02c65c6fffd6162b4c66290c2">math::modular_inverse_fermat::modular_inverse</a> (<a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> a, <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> m)</td></tr>
<tr class="memdesc:ae7e807e02c65c6fffd6162b4c66290c2"><td class="mdescLeft">&#160;</td><td class="mdescRight">calculates the modular inverse. <br /></td></tr>
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<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d" id="r_aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</td></tr>
<tr class="memdesc:aa8dca7b867074164d5f45b0f3851269d"><td class="mdescLeft">&#160;</td><td class="mdescRight">Self-test implementation. <br /></td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4" id="r_ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
<tr class="memdesc:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main function. <br /></td></tr>
<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memSeparator" colspan="2">&#160;</td></tr>
</table>
<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
@@ -153,7 +173,7 @@ Functions</h2></td></tr>
a^{m-2} &amp;≡&amp; a^{-1} \;\text{mod}\; m
\end{eqnarray*}
</p>
<p>We will find the exponent using binary exponentiation. Such that the algorithm works in \(O(\log m)\) time.</p>
<p>We will find the exponent using binary exponentiation such that the algorithm works in \(O(\log n)\) time.</p>
<p>Examples: -</p><ul>
<li>a = 3 and m = 7</li>
<li>\(a^{-1} \;\text{mod}\; m\) is equivalent to \(a^{m-2} \;\text{mod}\; m\)</li>
@@ -161,74 +181,104 @@ Functions</h2></td></tr>
Hence, \(3^{-1} \;\text{mod}\; 7 = 5\) or \(3 \times 5 \;\text{mod}\; 7 = 1 \;\text{mod}\; 7\) (as \(a\times a^{-1} = 1\)) </li>
</ul>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a4c6eefd1254eab3e8d34bf02c205e0f4" name="a4c6eefd1254eab3e8d34bf02c205e0f4"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a4c6eefd1254eab3e8d34bf02c205e0f4">&#9670;&#160;</a></span>binExpo()</h2>
<a id="a1c62a87e79f29b98d8fe5b12eacdd928" name="a1c62a87e79f29b98d8fe5b12eacdd928"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a1c62a87e79f29b98d8fe5b12eacdd928">&#9670;&#160;</a></span>binExpo()</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">int64_t binExpo </td>
<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> math::modular_inverse_fermat::binExpo </td>
<td>(</td>
<td class="paramtype">int64_t</td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">int64_t</td> <td class="paramname"><span class="paramname"><em>b</em></span>, </td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></td> <td class="paramname"><span class="paramname"><em>b</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">int64_t</td> <td class="paramname"><span class="paramname"><em>m</em></span>&#160;)</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></td> <td class="paramname"><span class="paramname"><em>m</em></span>&#160;)</td>
</tr>
</table>
</div><div class="memdoc">
<p>Recursive function to calculate exponent in \(O(\log n)\) using binary exponent. </p>
<div class="fragment"><div class="line"><span class="lineno"> 52</span> {</div>
<div class="line"><span class="lineno"> 53</span> a %= m;</div>
<div class="line"><span class="lineno"> 54</span> int64_t res = 1;</div>
<div class="line"><span class="lineno"> 55</span> <span class="keywordflow">while</span> (b &gt; 0) {</div>
<div class="line"><span class="lineno"> 56</span> <span class="keywordflow">if</span> (b % 2) {</div>
<div class="line"><span class="lineno"> 57</span> res = res * a % m;</div>
<div class="line"><span class="lineno"> 58</span> }</div>
<div class="line"><span class="lineno"> 59</span> a = a * a % m;</div>
<div class="line"><span class="lineno"> 60</span> <span class="comment">// Dividing b by 2 is similar to right shift.</span></div>
<div class="line"><span class="lineno"> 61</span> b &gt;&gt;= 1;</div>
<div class="line"><span class="lineno"> 62</span> }</div>
<div class="line"><span class="lineno"> 63</span> <span class="keywordflow">return</span> res;</div>
<div class="line"><span class="lineno"> 64</span>}</div>
</div><!-- fragment -->
</div>
</div>
<a id="a09660096b134753128952246f4f4e4bd" name="a09660096b134753128952246f4f4e4bd"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a09660096b134753128952246f4f4e4bd">&#9670;&#160;</a></span>isPrime()</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">bool isPrime </td>
<td>(</td>
<td class="paramtype">int64_t</td> <td class="paramname"><span class="paramname"><em>m</em></span></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Prime check in \(O(\sqrt{m})\) time. </p>
<div class="fragment"><div class="line"><span class="lineno"> 68</span> {</div>
<div class="line"><span class="lineno"> 69</span> <span class="keywordflow">if</span> (m &lt;= 1) {</div>
<div class="line"><span class="lineno"> 70</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 71</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 72</span> <span class="keywordflow">for</span> (int64_t i = 2; i * i &lt;= m; i++) {</div>
<div class="line"><span class="lineno"> 73</span> <span class="keywordflow">if</span> (m % i == 0) {</div>
<div class="line"><span class="lineno"> 74</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 75</span> }</div>
<div class="line"><span class="lineno"> 76</span> }</div>
<div class="line"><span class="lineno"> 77</span> }</div>
<div class="line"><span class="lineno"> 78</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<p>Calculate exponent with modulo using binary exponentiation in \(O(\log b)\) time. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">a</td><td>The base </td></tr>
<tr><td class="paramname">b</td><td>The exponent </td></tr>
<tr><td class="paramname">m</td><td>The modulo </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The result of \(a^{b} % m\) </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 67</span> {</div>
<div class="line"><span class="lineno"> 68</span> a %= m;</div>
<div class="line"><span class="lineno"> 69</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> res = 1;</div>
<div class="line"><span class="lineno"> 70</span> <span class="keywordflow">while</span> (b &gt; 0) {</div>
<div class="line"><span class="lineno"> 71</span> <span class="keywordflow">if</span> (b % 2 != 0) {</div>
<div class="line"><span class="lineno"> 72</span> res = res * a % m;</div>
<div class="line"><span class="lineno"> 73</span> }</div>
<div class="line"><span class="lineno"> 74</span> a = a * a % m;</div>
<div class="line"><span class="lineno"> 75</span> <span class="comment">// Dividing b by 2 is similar to right shift by 1 bit</span></div>
<div class="line"><span class="lineno"> 76</span> b &gt;&gt;= 1;</div>
<div class="line"><span class="lineno"> 77</span> }</div>
<div class="line"><span class="lineno"> 78</span> <span class="keywordflow">return</span> res;</div>
<div class="line"><span class="lineno"> 79</span>}</div>
</div><!-- fragment -->
<div class="ttc" id="ainteger_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#aba4929409fee35c3cb559e962a544b3e">&#9670;&#160;</a></span>isPrime()</h2>
<div class="memitem">
<div class="memproto">
<table class="memname">
<tr>
<td class="memname">bool math::modular_inverse_fermat::isPrime </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></td> <td class="paramname"><span class="paramname"><em>m</em></span></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Check if an integer is a prime number in \(O(\sqrt{m})\) time. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">m</td><td>An intger to check for primality </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>true if the number is prime </dd>
<dd>
false if the number is not prime </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 86</span> {</div>
<div class="line"><span class="lineno"> 87</span> <span class="keywordflow">if</span> (m &lt;= 1) {</div>
<div class="line"><span class="lineno"> 88</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 89</span> } </div>
<div class="line"><span class="lineno"> 90</span> <span class="keywordflow">for</span> (<a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> i = 2; i * i &lt;= m; i++) {</div>
<div class="line"><span class="lineno"> 91</span> <span class="keywordflow">if</span> (m % i == 0) {</div>
<div class="line"><span class="lineno"> 92</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><span class="lineno"> 93</span> }</div>
<div class="line"><span class="lineno"> 94</span> }</div>
<div class="line"><span class="lineno"> 95</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><span class="lineno"> 96</span>}</div>
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@@ -245,33 +295,108 @@ Hence, \(3^{-1} \;\text{mod}\; 7 = 5\) or \(3 \times 5 \;\text{mod}\; 7 = 1 \;\
</tr>
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<p>Main function </p>
<div class="fragment"><div class="line"><span class="lineno"> 84</span> {</div>
<div class="line"><span class="lineno"> 85</span> int64_t a, m;</div>
<div class="line"><span class="lineno"> 86</span> <span class="comment">// Take input of a and m.</span></div>
<div class="line"><span class="lineno"> 87</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Computing ((a^(-1))%(m)) using Fermat&#39;s Little Theorem&quot;</span>;</div>
<div class="line"><span class="lineno"> 88</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 89</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Give input &#39;a&#39; and &#39;m&#39; space separated : &quot;</span>;</div>
<div class="line"><span class="lineno"> 90</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; a &gt;&gt; m;</div>
<div class="line"><span class="lineno"> 91</span> <span class="keywordflow">if</span> (<a class="code hl_function" href="#a09660096b134753128952246f4f4e4bd">isPrime</a>(m)) {</div>
<div class="line"><span class="lineno"> 92</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;The modular inverse of a with mod m is (a^(m-2)) : &quot;</span>;</div>
<div class="line"><span class="lineno"> 93</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="code hl_function" href="#a4c6eefd1254eab3e8d34bf02c205e0f4">binExpo</a>(a, m - 2, m) &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 94</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 95</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;m must be a prime number.&quot;</span>;</div>
<div class="line"><span class="lineno"> 96</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><span class="lineno"> 97</span> }</div>
<div class="line"><span class="lineno"> 98</span>}</div>
<div class="ttc" id="abasic_istream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a></div></div>
<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
<div class="ttc" id="aendl_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div><div class="ttdeci">T endl(T... args)</div></div>
<div class="ttc" id="amodular__inverse__fermat__little__theorem_8cpp_html_a09660096b134753128952246f4f4e4bd"><div class="ttname"><a href="#a09660096b134753128952246f4f4e4bd">isPrime</a></div><div class="ttdeci">bool isPrime(int64_t m)</div><div class="ttdef"><b>Definition</b> modular_inverse_fermat_little_theorem.cpp:68</div></div>
<div class="ttc" id="amodular__inverse__fermat__little__theorem_8cpp_html_a4c6eefd1254eab3e8d34bf02c205e0f4"><div class="ttname"><a href="#a4c6eefd1254eab3e8d34bf02c205e0f4">binExpo</a></div><div class="ttdeci">int64_t binExpo(int64_t a, int64_t b, int64_t m)</div><div class="ttdef"><b>Definition</b> modular_inverse_fermat_little_theorem.cpp:52</div></div>
<p>Main function. </p>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 137</span> {</div>
<div class="line"><span class="lineno"> 138</span> <a class="code hl_function" href="#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// run self-test implementation</span></div>
<div class="line"><span class="lineno"> 139</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 140</span>}</div>
<div class="ttc" id="amodular__inverse__fermat__little__theorem_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdoc">Self-test implementation.</div><div class="ttdef"><b>Definition</b> modular_inverse_fermat_little_theorem.cpp:122</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae7e807e02c65c6fffd6162b4c66290c2">&#9670;&#160;</a></span>modular_inverse()</h2>
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<td class="memname"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a> math::modular_inverse_fermat::modular_inverse </td>
<td>(</td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype"><a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/types/integer.html">std::int64_t</a></td> <td class="paramname"><span class="paramname"><em>m</em></span>&#160;)</td>
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<p>calculates the modular inverse. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">a</td><td>Integer value for the base </td></tr>
<tr><td class="paramname">m</td><td>Integer value for modulo </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>The result that is the modular inverse of a modulo m </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 103</span> {</div>
<div class="line"><span class="lineno"> 104</span> <span class="keywordflow">while</span> (a &lt; 0) {</div>
<div class="line"><span class="lineno"> 105</span> a += m;</div>
<div class="line"><span class="lineno"> 106</span> }</div>
<div class="line"><span class="lineno"> 107</span> </div>
<div class="line"><span class="lineno"> 108</span> <span class="comment">// Check for invalid cases</span></div>
<div class="line"><span class="lineno"> 109</span> <span class="keywordflow">if</span> (!<a class="code hl_function" href="#aba4929409fee35c3cb559e962a544b3e">isPrime</a>(m) || a == 0) {</div>
<div class="line"><span class="lineno"> 110</span> <span class="keywordflow">return</span> -1; <span class="comment">// Invalid input</span></div>
<div class="line"><span class="lineno"> 111</span> }</div>
<div class="line"><span class="lineno"> 112</span> </div>
<div class="line"><span class="lineno"> 113</span> <span class="keywordflow">return</span> <a class="code hl_function" href="../../de/dcf/binary__exponent_8cpp.html#aeb48dce0725e63d19147944f41843c73">binExpo</a>(a, m - 2, m); <span class="comment">// Fermat&#39;s Little Theorem</span></div>
<div class="line"><span class="lineno"> 114</span>}</div>
<div class="ttc" id="abinary__exponent_8cpp_html_aeb48dce0725e63d19147944f41843c73"><div class="ttname"><a href="../../de/dcf/binary__exponent_8cpp.html#aeb48dce0725e63d19147944f41843c73">binExpo</a></div><div class="ttdeci">int binExpo(int a, int b)</div><div class="ttdef"><b>Definition</b> binary_exponent.cpp:28</div></div>
<div class="ttc" id="amodular__inverse__fermat__little__theorem_8cpp_html_aba4929409fee35c3cb559e962a544b3e"><div class="ttname"><a href="#aba4929409fee35c3cb559e962a544b3e">math::modular_inverse_fermat::isPrime</a></div><div class="ttdeci">bool isPrime(std::int64_t m)</div><div class="ttdoc">Check if an integer is a prime number in time.</div><div class="ttdef"><b>Definition</b> modular_inverse_fermat_little_theorem.cpp:86</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa8dca7b867074164d5f45b0f3851269d">&#9670;&#160;</a></span>test()</h2>
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<td class="memname">static void test </td>
<td>(</td>
<td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
<td></td>
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<p>Self-test implementation. </p>
<dl class="section return"><dt>Returns</dt><dd>void </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 122</span> {</div>
<div class="line"><span class="lineno"> 123</span> assert(math::modular_inverse_fermat::modular_inverse(0, 97) == -1);</div>
<div class="line"><span class="lineno"> 124</span> assert(math::modular_inverse_fermat::modular_inverse(15, -2) == -1);</div>
<div class="line"><span class="lineno"> 125</span> assert(math::modular_inverse_fermat::modular_inverse(3, 10) == -1);</div>
<div class="line"><span class="lineno"> 126</span> assert(math::modular_inverse_fermat::modular_inverse(3, 7) == 5);</div>
<div class="line"><span class="lineno"> 127</span> assert(math::modular_inverse_fermat::modular_inverse(1, 101) == 1);</div>
<div class="line"><span class="lineno"> 128</span> assert(math::modular_inverse_fermat::modular_inverse(-1337, 285179) == 165519);</div>
<div class="line"><span class="lineno"> 129</span> assert(math::modular_inverse_fermat::modular_inverse(123456789, 998244353) == 25170271);</div>
<div class="line"><span class="lineno"> 130</span> assert(math::modular_inverse_fermat::modular_inverse(-9876543210, 1000000007) == 784794281);</div>
<div class="line"><span class="lineno"> 131</span>}</div>
</div><!-- fragment -->
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