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<div class="headertitle"><div class="title">extended_euclid_algorithm.cpp File Reference</div></div>
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<p>GCD using [extended Euclid's algorithm] (<a href="https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm">https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm</a>)
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;algorithm&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
</div><div class="textblock"><div class="dynheader">
Include dependency graph for extended_euclid_algorithm.cpp:</div>
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Functions</h2></td></tr>
<tr class="memitem:abe92d63a0ff9bda7e304df510d5dd217"><td class="memTemplParams" colspan="2">template&lt;typename T , typename T2 &gt; </td></tr>
<tr class="memitem:abe92d63a0ff9bda7e304df510d5dd217"><td class="memTemplItemLeft" align="right" valign="top">void&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#abe92d63a0ff9bda7e304df510d5dd217">update_step</a> (T *r, T *r0, const T2 quotient)</td></tr>
<tr class="separator:abe92d63a0ff9bda7e304df510d5dd217"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:a1792ac7c33aaf26b860ab55f5652ab25"><td class="memTemplParams" colspan="2">template&lt;typename T1 , typename T2 &gt; </td></tr>
<tr class="memitem:a1792ac7c33aaf26b860ab55f5652ab25"><td class="memTemplItemLeft" align="right" valign="top">void&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#a1792ac7c33aaf26b860ab55f5652ab25">extendedEuclid_1</a> (T1 A, T1 B, T1 *GCD, T2 *x, T2 *y)</td></tr>
<tr class="separator:a1792ac7c33aaf26b860ab55f5652ab25"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:acba15ca55b3e7dcb91f3c65d72ba052d"><td class="memTemplParams" colspan="2">template&lt;typename T , typename T2 &gt; </td></tr>
<tr class="memitem:acba15ca55b3e7dcb91f3c65d72ba052d"><td class="memTemplItemLeft" align="right" valign="top">void&#160;</td><td class="memTemplItemRight" valign="bottom"><a class="el" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#acba15ca55b3e7dcb91f3c65d72ba052d">extendedEuclid</a> (T A, T B, T *GCD, T2 *x, T2 *y)</td></tr>
<tr class="separator:acba15ca55b3e7dcb91f3c65d72ba052d"><td class="memSeparator" colspan="2">&#160;</td></tr>
<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#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>
<div class="textblock"><p>GCD using [extended Euclid's algorithm] (<a href="https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm">https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm</a>) </p>
<p>Finding coefficients of a and b ie x and y in Bézout's identity </p><p class="formulaDsp">
\[\text{gcd}(a, b) = a \times x + b \times y \]
</p>
<p> This is also used in finding Modular multiplicative inverse of a number. (A * B)M == 1 Here B is the MMI of A for given M, so extendedEuclid (A, M) gives B. </p>
</div><h2 class="groupheader">Function Documentation</h2>
<a id="acba15ca55b3e7dcb91f3c65d72ba052d" name="acba15ca55b3e7dcb91f3c65d72ba052d"></a>
<h2 class="memtitle"><span class="permalink"><a href="#acba15ca55b3e7dcb91f3c65d72ba052d">&#9670;&#160;</a></span>extendedEuclid()</h2>
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template&lt;typename T , typename T2 &gt; </div>
<table class="memname">
<tr>
<td class="memname">void extendedEuclid </td>
<td>(</td>
<td class="paramtype">T&#160;</td>
<td class="paramname"><em>A</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T&#160;</td>
<td class="paramname"><em>B</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T *&#160;</td>
<td class="paramname"><em>GCD</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T2 *&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T2 *&#160;</td>
<td class="paramname"><em>y</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Implementation using recursive algorithm</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">A</td><td>unsigned </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td>unsigned </td></tr>
<tr><td class="paramdir">[out]</td><td class="paramname">GCD</td><td>unsigned </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">x</td><td>signed </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">y</td><td>signed </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 70</span> {</div>
<div class="line"><span class="lineno"> 71</span> <span class="keywordflow">if</span> (B &gt; A)</div>
<div class="line"><span class="lineno"> 72</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a>(A, B); <span class="comment">// Ensure that A &gt;= B</span></div>
<div class="line"><span class="lineno"> 73</span> </div>
<div class="line"><span class="lineno"> 74</span> <span class="keywordflow">if</span> (B == 0) {</div>
<div class="line"><span class="lineno"> 75</span> *GCD = A;</div>
<div class="line"><span class="lineno"> 76</span> *x = 1;</div>
<div class="line"><span class="lineno"> 77</span> *y = 0;</div>
<div class="line"><span class="lineno"> 78</span> } <span class="keywordflow">else</span> {</div>
<div class="line"><span class="lineno"> 79</span> <a class="code hl_function" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#acba15ca55b3e7dcb91f3c65d72ba052d">extendedEuclid</a>(B, A % B, GCD, x, y);</div>
<div class="line"><span class="lineno"> 80</span> T2 temp = *x;</div>
<div class="line"><span class="lineno"> 81</span> *x = *y;</div>
<div class="line"><span class="lineno"> 82</span> *y = temp - (A / B) * (*y);</div>
<div class="line"><span class="lineno"> 83</span> }</div>
<div class="line"><span class="lineno"> 84</span>}</div>
<div class="ttc" id="aextended__euclid__algorithm_8cpp_html_acba15ca55b3e7dcb91f3c65d72ba052d"><div class="ttname"><a href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#acba15ca55b3e7dcb91f3c65d72ba052d">extendedEuclid</a></div><div class="ttdeci">void extendedEuclid(T A, T B, T *GCD, T2 *x, T2 *y)</div><div class="ttdef"><b>Definition:</b> extended_euclid_algorithm.cpp:70</div></div>
<div class="ttc" id="aswap_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a></div><div class="ttdeci">T swap(T... args)</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a1792ac7c33aaf26b860ab55f5652ab25">&#9670;&#160;</a></span>extendedEuclid_1()</h2>
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template&lt;typename T1 , typename T2 &gt; </div>
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<tr>
<td class="memname">void extendedEuclid_1 </td>
<td>(</td>
<td class="paramtype">T1&#160;</td>
<td class="paramname"><em>A</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T1&#160;</td>
<td class="paramname"><em>B</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T1 *&#160;</td>
<td class="paramname"><em>GCD</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T2 *&#160;</td>
<td class="paramname"><em>x</em>, </td>
</tr>
<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T2 *&#160;</td>
<td class="paramname"><em>y</em>&#160;</td>
</tr>
<tr>
<td></td>
<td>)</td>
<td></td><td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Implementation using iterative algorithm from <a href="https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Pseudocode" target="_blank">Wikipedia</a></p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in]</td><td class="paramname">A</td><td>unsigned </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">B</td><td>unsigned </td></tr>
<tr><td class="paramdir">[out]</td><td class="paramname">GCD</td><td>unsigned </td></tr>
<tr><td class="paramdir">[out]</td><td class="paramname">x</td><td>signed </td></tr>
<tr><td class="paramdir">[out]</td><td class="paramname">y</td><td>signed </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 41</span> {</div>
<div class="line"><span class="lineno"> 42</span> <span class="keywordflow">if</span> (B &gt; A)</div>
<div class="line"><span class="lineno"> 43</span> <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/algorithm/swap.html">std::swap</a>(A, B); <span class="comment">// Ensure that A &gt;= B</span></div>
<div class="line"><span class="lineno"> 44</span> </div>
<div class="line"><span class="lineno"> 45</span> T2 s = 0, s0 = 1;</div>
<div class="line"><span class="lineno"> 46</span> T2 t = 1, t0 = 0;</div>
<div class="line"><span class="lineno"> 47</span> T1 r = B, r0 = A;</div>
<div class="line"><span class="lineno"> 48</span> </div>
<div class="line"><span class="lineno"> 49</span> <span class="keywordflow">while</span> (r != 0) {</div>
<div class="line"><span class="lineno"> 50</span> T1 quotient = r0 / r;</div>
<div class="line"><span class="lineno"> 51</span> <a class="code hl_function" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#abe92d63a0ff9bda7e304df510d5dd217">update_step</a>(&amp;r, &amp;r0, quotient);</div>
<div class="line"><span class="lineno"> 52</span> <a class="code hl_function" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#abe92d63a0ff9bda7e304df510d5dd217">update_step</a>(&amp;s, &amp;s0, quotient);</div>
<div class="line"><span class="lineno"> 53</span> <a class="code hl_function" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#abe92d63a0ff9bda7e304df510d5dd217">update_step</a>(&amp;t, &amp;t0, quotient);</div>
<div class="line"><span class="lineno"> 54</span> }</div>
<div class="line"><span class="lineno"> 55</span> *GCD = r0;</div>
<div class="line"><span class="lineno"> 56</span> *x = s0;</div>
<div class="line"><span class="lineno"> 57</span> *y = t0;</div>
<div class="line"><span class="lineno"> 58</span>}</div>
<div class="ttc" id="aextended__euclid__algorithm_8cpp_html_abe92d63a0ff9bda7e304df510d5dd217"><div class="ttname"><a href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#abe92d63a0ff9bda7e304df510d5dd217">update_step</a></div><div class="ttdeci">void update_step(T *r, T *r0, const T2 quotient)</div><div class="ttdef"><b>Definition:</b> extended_euclid_algorithm.cpp:24</div></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&#160;</td>
<td class="paramname"></td><td>)</td>
<td></td>
</tr>
</table>
</div><div class="memdoc">
<p>Main function. </p>
<div class="fragment"><div class="line"><span class="lineno"> 87</span> {</div>
<div class="line"><span class="lineno"> 88</span> uint32_t a, b, <a class="code hl_function" href="../../d4/da0/gcd__iterative__euclidean_8cpp.html#ae48807fa2b7000afae599e67f327545e">gcd</a>;</div>
<div class="line"><span class="lineno"> 89</span> int32_t x, y;</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; b;</div>
<div class="line"><span class="lineno"> 91</span> <a class="code hl_function" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#acba15ca55b3e7dcb91f3c65d72ba052d">extendedEuclid</a>(a, b, &amp;<a class="code hl_function" href="../../d4/da0/gcd__iterative__euclidean_8cpp.html#ae48807fa2b7000afae599e67f327545e">gcd</a>, &amp;x, &amp;y);</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; <a class="code hl_function" href="../../d4/da0/gcd__iterative__euclidean_8cpp.html#ae48807fa2b7000afae599e67f327545e">gcd</a> &lt;&lt; <span class="stringliteral">&quot; &quot;</span> &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot; &quot;</span> &lt;&lt; y &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> <a class="code hl_function" href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#a1792ac7c33aaf26b860ab55f5652ab25">extendedEuclid_1</a>(a, b, &amp;<a class="code hl_function" href="../../d4/da0/gcd__iterative__euclidean_8cpp.html#ae48807fa2b7000afae599e67f327545e">gcd</a>, &amp;x, &amp;y);</div>
<div class="line"><span class="lineno"> 94</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="../../d4/da0/gcd__iterative__euclidean_8cpp.html#ae48807fa2b7000afae599e67f327545e">gcd</a> &lt;&lt; <span class="stringliteral">&quot; &quot;</span> &lt;&lt; x &lt;&lt; <span class="stringliteral">&quot; &quot;</span> &lt;&lt; y &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"> 95</span> <span class="keywordflow">return</span> 0;</div>
<div class="line"><span class="lineno"> 96</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>
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<div class="ttc" id="aextended__euclid__algorithm_8cpp_html_a1792ac7c33aaf26b860ab55f5652ab25"><div class="ttname"><a href="../../d9/d5d/extended__euclid__algorithm_8cpp.html#a1792ac7c33aaf26b860ab55f5652ab25">extendedEuclid_1</a></div><div class="ttdeci">void extendedEuclid_1(T1 A, T1 B, T1 *GCD, T2 *x, T2 *y)</div><div class="ttdef"><b>Definition:</b> extended_euclid_algorithm.cpp:41</div></div>
<div class="ttc" id="agcd__iterative__euclidean_8cpp_html_ae48807fa2b7000afae599e67f327545e"><div class="ttname"><a href="../../d4/da0/gcd__iterative__euclidean_8cpp.html#ae48807fa2b7000afae599e67f327545e">gcd</a></div><div class="ttdeci">int gcd(int num1, int num2)</div><div class="ttdef"><b>Definition:</b> gcd_iterative_euclidean.cpp:15</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#abe92d63a0ff9bda7e304df510d5dd217">&#9670;&#160;</a></span>update_step()</h2>
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<div class="memtemplate">
template&lt;typename T , typename T2 &gt; </div>
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<td class="memname">void update_step </td>
<td>(</td>
<td class="paramtype">T *&#160;</td>
<td class="paramname"><em>r</em>, </td>
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<tr>
<td class="paramkey"></td>
<td></td>
<td class="paramtype">T *&#160;</td>
<td class="paramname"><em>r0</em>, </td>
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<td class="paramkey"></td>
<td></td>
<td class="paramtype">const T2&#160;</td>
<td class="paramname"><em>quotient</em>&#160;</td>
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<td></td>
<td>)</td>
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<p>function to update the coefficients per iteration </p><p class="formulaDsp">
\[r_0,\,r = r,\, r_0 - \text{quotient}\times r\]
</p>
<dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramdir">[in,out]</td><td class="paramname">r</td><td>signed or unsigned </td></tr>
<tr><td class="paramdir">[in,out]</td><td class="paramname">r0</td><td>signed or unsigned </td></tr>
<tr><td class="paramdir">[in]</td><td class="paramname">quotient</td><td>unsigned </td></tr>
</table>
</dd>
</dl>
<div class="fragment"><div class="line"><span class="lineno"> 24</span> {</div>
<div class="line"><span class="lineno"> 25</span> T temp = *r;</div>
<div class="line"><span class="lineno"> 26</span> *r = *r0 - (quotient * temp);</div>
<div class="line"><span class="lineno"> 27</span> *r0 = temp;</div>
<div class="line"><span class="lineno"> 28</span>}</div>
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