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<p>An algorithm to divide two numbers under modulo p <a href="https://www.geeksforgeeks.org/modular-division" target="_blank">Modular Division</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_division.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 assert <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="../../d1/d64/namespacemodular__division.html">modular_division</a></td></tr>
<tr class="memdesc:d1/d64/namespacemodular__division"><td class="mdescLeft">&#160;</td><td class="mdescRight">Functions for <a href="https://www.geeksforgeeks.org/modular-division" target="_blank">Modular Division</a> implementation. <br /></td></tr>
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Functions</h2></td></tr>
<tr class="memitem:a66cdf93153cbd1408bd74ac68961d179" id="r_a66cdf93153cbd1408bd74ac68961d179"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a66cdf93153cbd1408bd74ac68961d179">math::modular_division::power</a> (uint64_t a, uint64_t b, uint64_t c)</td></tr>
<tr class="memdesc:a66cdf93153cbd1408bd74ac68961d179"><td class="mdescLeft">&#160;</td><td class="mdescRight">This function calculates a raised to exponent b under modulo c using modular exponentiation. <br /></td></tr>
<tr class="separator:a66cdf93153cbd1408bd74ac68961d179"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a905e368ae121beb7e7ea35349ddcdac7" id="r_a905e368ae121beb7e7ea35349ddcdac7"><td class="memItemLeft" align="right" valign="top">uint64_t&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a> (uint64_t a, uint64_t b, uint64_t p)</td></tr>
<tr class="memdesc:a905e368ae121beb7e7ea35349ddcdac7"><td class="mdescLeft">&#160;</td><td class="mdescRight">This function calculates modular division. <br /></td></tr>
<tr class="separator:a905e368ae121beb7e7ea35349ddcdac7"><td class="memSeparator" colspan="2">&#160;</td></tr>
<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="separator:aa8dca7b867074164d5f45b0f3851269d"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97" id="r_a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
<tr class="memdesc:a0ddf1224851353fc92bfbff6f499fa97"><td class="mdescLeft">&#160;</td><td class="mdescRight">Main function. <br /></td></tr>
<tr class="separator:a0ddf1224851353fc92bfbff6f499fa97"><td class="memSeparator" colspan="2">&#160;</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>An algorithm to divide two numbers under modulo p <a href="https://www.geeksforgeeks.org/modular-division" target="_blank">Modular Division</a> </p>
<p>To calculate division of two numbers under modulo p Modulo operator is not distributive under division, therefore we first have to calculate the inverse of divisor using <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem" target="_blank">Fermat's little theorem</a> Now, we can multiply the dividend with the inverse of divisor and modulo is distributive over multiplication operation. Let, We have 3 numbers a, b, p To compute (a/b)p (a/b)p ≡ (a*(inverse(b)))p ≡ ((ap)*inverse(b)p)p NOTE: For the existence of inverse of 'b', 'b' and 'p' must be coprime For simplicity we take p as prime Time Complexity: O(log(b)) Example: ( 24 / 3 ) % 5 =&gt; 8 % 5 = 3 &mdash; (i) Now the inverse of 3 is 2 (24 * 2) % 5 = (24 % 5) * (2 % 5) = (4 * 2) % 5 = 3 &mdash; (ii) (i) and (ii) are equal hence the answer is correct. </p><dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html" title="C++ Program to find the modular inverse using Fermat&#39;s Little Theorem">modular_inverse_fermat_little_theorem.cpp</a>, <a class="el" href="../../d0/d6d/modular__exponentiation_8cpp.html" title="C++ Program for Modular Exponentiation Iteratively.">modular_exponentiation.cpp</a> </dd></dl>
<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/shubhamamsa" target="_blank">Shubham Yadav</a> </dd></dl>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="a0ddf1224851353fc92bfbff6f499fa97" name="a0ddf1224851353fc92bfbff6f499fa97"></a>
<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">&#9670;&#160;</a></span>main()</h2>
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<td class="memname">int main </td>
<td>(</td>
<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>argc</em></span>, </td>
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<td class="paramkey"></td>
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<p>Main function. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">argc</td><td>commandline argument count (ignored) </td></tr>
<tr><td class="paramname">argv</td><td>commandline array of arguments (ignored) </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 113</span> {</div>
<div class="line"><span class="lineno"> 114</span> <a class="code hl_function" href="#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// execute the tests</span></div>
<div class="line"><span class="lineno"> 115</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 116</span>}</div>
<div class="ttc" id="amodular__division_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdef"><b>Definition</b> modular_division.cpp:89</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a905e368ae121beb7e7ea35349ddcdac7">&#9670;&#160;</a></span>mod_division()</h2>
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<td class="memname">uint64_t math::modular_division::mod_division </td>
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<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
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<p>This function calculates modular division. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">a</td><td>integer dividend </td></tr>
<tr><td class="paramname">b</td><td>integer divisor </td></tr>
<tr><td class="paramname">p</td><td>integer modulo </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>a/b modulo c </dd></dl>
<p>Calculate the inverse of b</p>
<p>Calculate the final result</p>
<div class="fragment"><div class="line"><span class="lineno"> 75</span> {</div>
<div class="line"><span class="lineno"> 76</span> uint64_t inverse = <a class="code hl_function" href="../../df/def/power__for__huge__numbers_8cpp.html#ae249a2af508aa94266023ce8aa81426f">power</a>(b, p - 2, p) % p; <span class="comment">/// Calculate the inverse of b</span></div>
<div class="line"><span class="lineno"> 77</span> uint64_t <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> =</div>
<div class="line"><span class="lineno"> 78</span> ((a % p) * (inverse % p)) % p; <span class="comment">/// Calculate the final result</span></div>
<div class="line"><span class="lineno"> 79</span> <span class="keywordflow">return</span> <a class="code hl_function" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>;</div>
<div class="line"><span class="lineno"> 80</span>}</div>
<div class="ttc" id="afibonacci__sum_8cpp_html_aadb40ac4c74a7efc0680b83eeee138aa"><div class="ttname"><a href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">math::fibonacci_sum::result</a></div><div class="ttdeci">uint64_t result(uint64_t n)</div><div class="ttdef"><b>Definition</b> fibonacci_sum.cpp:77</div></div>
<div class="ttc" id="apower__for__huge__numbers_8cpp_html_ae249a2af508aa94266023ce8aa81426f"><div class="ttname"><a href="../../df/def/power__for__huge__numbers_8cpp.html#ae249a2af508aa94266023ce8aa81426f">power</a></div><div class="ttdeci">void power(int x, int n)</div><div class="ttdef"><b>Definition</b> power_for_huge_numbers.cpp:56</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a66cdf93153cbd1408bd74ac68961d179">&#9670;&#160;</a></span>power()</h2>
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<td class="memname">uint64_t math::modular_division::power </td>
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<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>a</em></span>, </td>
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<td class="paramkey"></td>
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<td class="paramtype">uint64_t</td> <td class="paramname"><span class="paramname"><em>b</em></span>, </td>
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<td></td>
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<p>This function calculates a raised to exponent b under modulo c using modular exponentiation. </p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">a</td><td>integer base </td></tr>
<tr><td class="paramname">b</td><td>unsigned integer exponent </td></tr>
<tr><td class="paramname">c</td><td>integer modulo </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>a raised to power b modulo c </dd></dl>
<p>Initialize the answer to be returned</p>
<p>Update a if it is more than or equal to c</p>
<p>In case a is divisible by c;</p>
<p>If b is odd, multiply a with answer</p>
<p>b must be even now</p>
<p>b = b/2</p>
<div class="fragment"><div class="line"><span class="lineno"> 50</span> {</div>
<div class="line"><span class="lineno"> 51</span> uint64_t ans = 1; <span class="comment">/// Initialize the answer to be returned</span></div>
<div class="line"><span class="lineno"> 52</span> a = a % c; <span class="comment">/// Update a if it is more than or equal to c</span></div>
<div class="line"><span class="lineno"> 53</span> <span class="keywordflow">if</span> (a == 0) {</div>
<div class="line"><span class="lineno"> 54</span> <span class="keywordflow">return</span> 0; <span class="comment">/// In case a is divisible by c;</span></div>
<div class="line"><span class="lineno"> 55</span> }</div>
<div class="line"><span class="lineno"> 56</span> <span class="keywordflow">while</span> (b &gt; 0) {<span class="comment"></span></div>
<div class="line"><span class="lineno"> 57</span><span class="comment"> /// If b is odd, multiply a with answer</span></div>
<div class="line"><span class="lineno"> 58</span><span class="comment"></span> <span class="keywordflow">if</span> (b &amp; 1) {</div>
<div class="line"><span class="lineno"> 59</span> ans = ((ans % c) * (a % c)) % c;</div>
<div class="line"><span class="lineno"> 60</span> }<span class="comment"></span></div>
<div class="line"><span class="lineno"> 61</span><span class="comment"> /// b must be even now</span></div>
<div class="line"><span class="lineno"> 62</span><span class="comment"></span> b = b &gt;&gt; 1; <span class="comment">/// b = b/2</span></div>
<div class="line"><span class="lineno"> 63</span> a = ((a % c) * (a % c)) % c;</div>
<div class="line"><span class="lineno"> 64</span> }</div>
<div class="line"><span class="lineno"> 65</span> <span class="keywordflow">return</span> ans;</div>
<div class="line"><span class="lineno"> 66</span>}</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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<td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
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<p>Function for testing power function. test cases and assert statement. </p><dl class="section return"><dt>Returns</dt><dd><code>void</code> </dd></dl>
<div class="fragment"><div class="line"><span class="lineno"> 89</span> {</div>
<div class="line"><span class="lineno"> 90</span> uint64_t test_case_1 = <a class="code hl_function" href="#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(8, 2, 2);</div>
<div class="line"><span class="lineno"> 91</span> assert(test_case_1 == 0);</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;Test 1 Passed!&quot;</span> &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"> 93</span> uint64_t test_case_2 = <a class="code hl_function" href="#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(15, 3, 7);</div>
<div class="line"><span class="lineno"> 94</span> assert(test_case_2 == 5);</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;Test 2 Passed!&quot;</span> &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"> 96</span> uint64_t test_case_3 = <a class="code hl_function" href="#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(10, 5, 2);</div>
<div class="line"><span class="lineno"> 97</span> assert(test_case_3 == 0);</div>
<div class="line"><span class="lineno"> 98</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;Test 3 Passed!&quot;</span> &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"> 99</span> uint64_t test_case_4 = <a class="code hl_function" href="#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(81, 3, 5);</div>
<div class="line"><span class="lineno"> 100</span> assert(test_case_4 == 2);</div>
<div class="line"><span class="lineno"> 101</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;Test 4 Passed!&quot;</span> &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"> 102</span> uint64_t test_case_5 = <a class="code hl_function" href="#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(12848, 73, 29);</div>
<div class="line"><span class="lineno"> 103</span> assert(test_case_5 == 2);</div>
<div class="line"><span class="lineno"> 104</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;Test 5 Passed!&quot;</span> &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"> 105</span>}</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__division_8cpp_html_a905e368ae121beb7e7ea35349ddcdac7"><div class="ttname"><a href="#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a></div><div class="ttdeci">uint64_t mod_division(uint64_t a, uint64_t b, uint64_t p)</div><div class="ttdoc">This function calculates modular division.</div><div class="ttdef"><b>Definition</b> modular_division.cpp:75</div></div>
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