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<div class="headertitle"><div class="title">modular_inverse_fermat_little_theorem.cpp File Reference</div></div>
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<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;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>
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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>
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<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>
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Functions</h2></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>
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<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>
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<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="memdesc:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main function. <br /></td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><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> </p>
<p>Fermat's Little Theorem state that </p><p class="formulaDsp">
\[ϕ(m) = m-1\]
</p>
<p> where \(m\) is a prime number. </p><p class="formulaDsp">
\begin{eqnarray*}
a \cdot x &amp;≡&amp; 1 \;\text{mod}\; m\\
x &amp;≡&amp; a^{-1} \;\text{mod}\; m
\end{eqnarray*}
</p>
<p> Using Euler's theorem we can modify the equation. </p><p class="formulaDsp">
\[
a^{ϕ(m)} ≡ 1 \;\text{mod}\; m
\]
</p>
<p> (Where '^' denotes the exponent operator)</p>
<p>Here 'ϕ' is Euler's Totient Function. For modular inverse existence 'a' and 'm' must be relatively primes numbers. To apply Fermat's Little Theorem is necessary that 'm' must be a prime number. Generally in many competitive programming competitions 'm' is either 1000000007 (1e9+7) or 998244353.</p>
<p>We considered m as large prime (1e9+7). \(a^{ϕ(m)} ≡ 1 \;\text{mod}\; m\) (Using Euler's Theorem) \(ϕ(m) = m-1\) using Fermat's Little Theorem. \(a^{m-1} ≡ 1 \;\text{mod}\; m\) Now multiplying both side by \(a^{-1}\). </p><p class="formulaDsp">
\begin{eqnarray*}
a^{m-1} \cdot a^{-1} &amp;≡&amp; a^{-1} \;\text{mod}\; m\\
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 \(O(\log n)\) 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>
<li>\(3^5 \;\text{mod}\; 7 = 243 \;\text{mod}\; 7 = 5\) <br />
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>
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<h2 class="memtitle"><span class="permalink"><a href="#a1c62a87e79f29b98d8fe5b12eacdd928">&#9670;&#160;</a></span>binExpo()</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::binExpo </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>
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<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>
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<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>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 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>
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<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>
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<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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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&#160;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">void</td> <td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
<td></td>
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<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>
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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>
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<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>
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<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>
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<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>
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<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>
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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>
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