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<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">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><div class="textblock"><div class="dynheader">
Include dependency graph for modular_inverse_fermat_little_theorem.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:a4c6eefd1254eab3e8d34bf02c205e0f4"><td class="memItemLeft" align="right" valign="top">int64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html#a4c6eefd1254eab3e8d34bf02c205e0f4">binExpo</a> (int64_t <a class="el" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, int64_t b, int64_t m)</td></tr>
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<tr class="memitem:a09660096b134753128952246f4f4e4bd"><td class="memItemLeft" align="right" valign="top">bool&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html#a09660096b134753128952246f4f4e4bd">isPrime</a> (int64_t m)</td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</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">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 \(O(\log m)\) 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="#a4c6eefd1254eab3e8d34bf02c205e0f4">&#9670;&nbsp;</a></span>binExpo()</h2>
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<td class="memname">int64_t binExpo </td>
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<td class="paramtype">int64_t&#160;</td>
<td class="paramname"><em>a</em>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">int64_t&#160;</td>
<td class="paramname"><em>b</em>, </td>
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<td class="paramtype">int64_t&#160;</td>
<td class="paramname"><em>m</em>&#160;</td>
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<p>Recursive function to calculate exponent in \(O(\log n)\) using binary exponent. </p>
<div class="fragment"><div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; {</div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> %= m;</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; int64_t res = 1;</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; <span class="keywordflow">while</span> (b &gt; 0) {</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; <span class="keywordflow">if</span> (b % 2) {</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; res = res * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> % m;</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; }</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> = <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> * <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> % m;</div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; <span class="comment">// Dividing b by 2 is similar to right shift.</span></div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; b &gt;&gt;= 1;</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; }</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; <span class="keywordflow">return</span> res;</div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#a09660096b134753128952246f4f4e4bd">&#9670;&nbsp;</a></span>isPrime()</h2>
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<td class="memname">bool isPrime </td>
<td>(</td>
<td class="paramtype">int64_t&#160;</td>
<td class="paramname"><em>m</em></td><td>)</td>
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<p>Prime check in \(O(\sqrt{m})\) time. </p>
<div class="fragment"><div class="line"><a name="l00068"></a><span class="lineno"> 68</span>&#160; {</div>
<div class="line"><a name="l00069"></a><span class="lineno"> 69</span>&#160; <span class="keywordflow">if</span> (m &lt;= 1) {</div>
<div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>&#160; <span class="keywordflow">for</span> (int64_t i = 2; i * i &lt;= m; i++) {</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; <span class="keywordflow">if</span> (m % i == 0) {</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160; <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; }</div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160; }</div>
<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>&#160; }</div>
<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>&#160; <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
<div class="line"><a name="l00079"></a><span class="lineno"> 79</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&nbsp;</a></span>main()</h2>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; {</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; int64_t <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, m;</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; <span class="comment">// Take input of a and m.</span></div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160; <a class="codeRef" 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"><a name="l00088"></a><span class="lineno"> 88</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>&#160; <a class="codeRef" 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"><a name="l00090"></a><span class="lineno"> 90</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a> &gt;&gt; m;</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>&#160; <span class="keywordflow">if</span> (<a class="code" href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html#a09660096b134753128952246f4f4e4bd">isPrime</a>(m)) {</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>&#160; <a class="codeRef" 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"><a name="l00093"></a><span class="lineno"> 93</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="code" href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html#a4c6eefd1254eab3e8d34bf02c205e0f4">binExpo</a>(<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a>, m - 2, m) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>&#160; <a class="codeRef" 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"><a name="l00096"></a><span class="lineno"> 96</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>&#160; }</div>
<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>&#160;}</div>
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<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="amatrix__exponentiation_8cpp_html_a9ecd98ce0064e2f7f97bb6a92f58ee72"><div class="ttname"><a href="../../d7/d35/matrix__exponentiation_8cpp.html#a9ecd98ce0064e2f7f97bb6a92f58ee72">a</a></div><div class="ttdeci">vector&lt; ll &gt; a</div><div class="ttdef"><b>Definition:</b> matrix_exponentiation.cpp:53</div></div>
<div class="ttc" id="amodular__inverse__fermat__little__theorem_8cpp_html_a4c6eefd1254eab3e8d34bf02c205e0f4"><div class="ttname"><a href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html#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>
<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>
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<div class="ttc" id="amodular__inverse__fermat__little__theorem_8cpp_html_a09660096b134753128952246f4f4e4bd"><div class="ttname"><a href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html#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>
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