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<div class="headertitle"><div class="title">miller_rabin.cpp</div></div>
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<a href="../../d6/d42/miller__rabin_8cpp.html">Go to the documentation of this file.</a><div class="fragment"><div class="line"><a id="l00001" name="l00001"></a><span class="lineno"> 1</span></div>
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<div class="line"><a id="l00007" name="l00007"></a><span class="lineno"> 7</span> </div>
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<div class="line"><a id="l00008" name="l00008"></a><span class="lineno"> 8</span><span class="preprocessor">#include <cassert></span></div>
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<div class="line"><a id="l00009" name="l00009"></a><span class="lineno"> 9</span><span class="preprocessor">#include <iostream></span></div>
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<div class="line"><a id="l00010" name="l00010"></a><span class="lineno"> 10</span><span class="preprocessor">#include <random></span></div>
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<div class="line"><a id="l00011" name="l00011"></a><span class="lineno"> 11</span><span class="preprocessor">#include <vector></span></div>
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<div class="line"><a id="l00012" name="l00012"></a><span class="lineno"> 12</span></div>
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<div class="line"><a id="l00018" name="l00018"></a><span class="lineno"> 18</span><span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="foldopen" id="foldopen00019" data-start="{" data-end="}">
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<div class="line"><a id="l00019" name="l00019"></a><span class="lineno"><a class="line" href="../../d6/d42/miller__rabin_8cpp.html#a091662a787d5ad4866713021f580fddb"> 19</a></span>std::vector<T> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a091662a787d5ad4866713021f580fddb">reverse_binary</a>(T num) {</div>
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<div class="line"><a id="l00020" name="l00020"></a><span class="lineno"> 20</span> std::vector<T> result;</div>
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<div class="line"><a id="l00021" name="l00021"></a><span class="lineno"> 21</span> T temp = num;</div>
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<div class="line"><a id="l00022" name="l00022"></a><span class="lineno"> 22</span> <span class="keywordflow">while</span> (temp > 0) {</div>
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<div class="line"><a id="l00023" name="l00023"></a><span class="lineno"> 23</span> result.push_back(temp % 2);</div>
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<div class="line"><a id="l00024" name="l00024"></a><span class="lineno"> 24</span> temp = temp / 2;</div>
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<div class="line"><a id="l00025" name="l00025"></a><span class="lineno"> 25</span> }</div>
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<div class="line"><a id="l00026" name="l00026"></a><span class="lineno"> 26</span> <span class="keywordflow">return</span> result;</div>
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<div class="line"><a id="l00027" name="l00027"></a><span class="lineno"> 27</span>}</div>
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</div>
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<div class="line"><a id="l00028" name="l00028"></a><span class="lineno"> 28</span></div>
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<div class="line"><a id="l00042" name="l00042"></a><span class="lineno"> 42</span><span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="foldopen" id="foldopen00043" data-start="{" data-end="}">
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<div class="line"><a id="l00043" name="l00043"></a><span class="lineno"><a class="line" href="../../d6/d42/miller__rabin_8cpp.html#ad6c2c67ea416d0e80003a88623f98b29"> 43</a></span>T <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#ad6c2c67ea416d0e80003a88623f98b29">modular_exponentiation</a>(T base, <span class="keyword">const</span> std::vector<T> &rev_binary_exponent,</div>
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<div class="line"><a id="l00044" name="l00044"></a><span class="lineno"> 44</span> T mod) {</div>
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<div class="line"><a id="l00045" name="l00045"></a><span class="lineno"> 45</span> <span class="keywordflow">if</span> (mod == 1)</div>
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<div class="line"><a id="l00046" name="l00046"></a><span class="lineno"> 46</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a id="l00047" name="l00047"></a><span class="lineno"> 47</span> T b = 1;</div>
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<div class="line"><a id="l00048" name="l00048"></a><span class="lineno"> 48</span> <span class="keywordflow">if</span> (rev_binary_exponent.size() == 0)</div>
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<div class="line"><a id="l00049" name="l00049"></a><span class="lineno"> 49</span> <span class="keywordflow">return</span> b;</div>
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<div class="line"><a id="l00050" name="l00050"></a><span class="lineno"> 50</span> T A = base;</div>
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<div class="line"><a id="l00051" name="l00051"></a><span class="lineno"> 51</span> <span class="keywordflow">if</span> (rev_binary_exponent[0] == 1)</div>
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<div class="line"><a id="l00052" name="l00052"></a><span class="lineno"> 52</span> b = base;</div>
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<div class="line"><a id="l00053" name="l00053"></a><span class="lineno"> 53</span> </div>
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<div class="line"><a id="l00054" name="l00054"></a><span class="lineno"> 54</span> <span class="keywordflow">for</span> (<span class="keyword">typename</span> std::vector<T>::const_iterator it =</div>
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<div class="line"><a id="l00055" name="l00055"></a><span class="lineno"> 55</span> rev_binary_exponent.cbegin() + 1;</div>
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<div class="line"><a id="l00056" name="l00056"></a><span class="lineno"> 56</span> it != rev_binary_exponent.cend(); ++it) {</div>
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<div class="line"><a id="l00057" name="l00057"></a><span class="lineno"> 57</span> A = A * A % mod;</div>
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<div class="line"><a id="l00058" name="l00058"></a><span class="lineno"> 58</span> <span class="keywordflow">if</span> (*it == 1)</div>
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<div class="line"><a id="l00059" name="l00059"></a><span class="lineno"> 59</span> b = A * b % mod;</div>
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<div class="line"><a id="l00060" name="l00060"></a><span class="lineno"> 60</span> }</div>
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<div class="line"><a id="l00061" name="l00061"></a><span class="lineno"> 61</span> <span class="keywordflow">return</span> b;</div>
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<div class="line"><a id="l00062" name="l00062"></a><span class="lineno"> 62</span>}</div>
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</div>
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<div class="line"><a id="l00063" name="l00063"></a><span class="lineno"> 63</span></div>
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<div class="line"><a id="l00072" name="l00072"></a><span class="lineno"> 72</span><span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="foldopen" id="foldopen00073" data-start="{" data-end="}">
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<div class="line"><a id="l00073" name="l00073"></a><span class="lineno"><a class="line" href="../../d6/d42/miller__rabin_8cpp.html#a6f9c31c1047aa3191676d64571d4c506"> 73</a></span><span class="keywordtype">bool</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a6f9c31c1047aa3191676d64571d4c506">miller_test</a>(T d, T num) {</div>
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<div class="line"><a id="l00074" name="l00074"></a><span class="lineno"> 74</span> <span class="comment">// random number seed</span></div>
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<div class="line"><a id="l00075" name="l00075"></a><span class="lineno"> 75</span> std::random_device rd_seed;</div>
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<div class="line"><a id="l00076" name="l00076"></a><span class="lineno"> 76</span> <span class="comment">// random number generator</span></div>
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<div class="line"><a id="l00077" name="l00077"></a><span class="lineno"> 77</span> std::mt19937 gen(rd_seed());</div>
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<div class="line"><a id="l00078" name="l00078"></a><span class="lineno"> 78</span> <span class="comment">// Uniformly distributed range [2, num - 2] for random numbers</span></div>
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<div class="line"><a id="l00079" name="l00079"></a><span class="lineno"> 79</span> std::uniform_int_distribution<> distribution(2, num - 2);</div>
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<div class="line"><a id="l00080" name="l00080"></a><span class="lineno"> 80</span> <span class="comment">// Random number generated in the range [2, num -2].</span></div>
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<div class="line"><a id="l00081" name="l00081"></a><span class="lineno"> 81</span> T random = distribution(gen);</div>
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<div class="line"><a id="l00082" name="l00082"></a><span class="lineno"> 82</span> <span class="comment">// vector for reverse binary of the power</span></div>
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<div class="line"><a id="l00083" name="l00083"></a><span class="lineno"> 83</span> std::vector<T> <a class="code hl_function" href="../../df/def/power__for__huge__numbers_8cpp.html#ae249a2af508aa94266023ce8aa81426f">power</a> = <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a091662a787d5ad4866713021f580fddb">reverse_binary</a>(d);</div>
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<div class="line"><a id="l00084" name="l00084"></a><span class="lineno"> 84</span> <span class="comment">// x = random ^ d % num</span></div>
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<div class="line"><a id="l00085" name="l00085"></a><span class="lineno"> 85</span> T x = <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#ad6c2c67ea416d0e80003a88623f98b29">modular_exponentiation</a>(random, <a class="code hl_function" href="../../df/def/power__for__huge__numbers_8cpp.html#ae249a2af508aa94266023ce8aa81426f">power</a>, num);</div>
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<div class="line"><a id="l00086" name="l00086"></a><span class="lineno"> 86</span> <span class="comment">// miller conditions</span></div>
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<div class="line"><a id="l00087" name="l00087"></a><span class="lineno"> 87</span> <span class="keywordflow">if</span> (x == 1 || x == num - 1) {</div>
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<div class="line"><a id="l00088" name="l00088"></a><span class="lineno"> 88</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><a id="l00089" name="l00089"></a><span class="lineno"> 89</span> }</div>
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<div class="line"><a id="l00090" name="l00090"></a><span class="lineno"> 90</span> </div>
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<div class="line"><a id="l00091" name="l00091"></a><span class="lineno"> 91</span> <span class="keywordflow">while</span> (d != num - 1) {</div>
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<div class="line"><a id="l00092" name="l00092"></a><span class="lineno"> 92</span> x = (x * x) % num;</div>
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<div class="line"><a id="l00093" name="l00093"></a><span class="lineno"> 93</span> d *= 2;</div>
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<div class="line"><a id="l00094" name="l00094"></a><span class="lineno"> 94</span> <span class="keywordflow">if</span> (x == 1) {</div>
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<div class="line"><a id="l00095" name="l00095"></a><span class="lineno"> 95</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00096" name="l00096"></a><span class="lineno"> 96</span> }</div>
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<div class="line"><a id="l00097" name="l00097"></a><span class="lineno"> 97</span> <span class="keywordflow">if</span> (x == num - 1) {</div>
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<div class="line"><a id="l00098" name="l00098"></a><span class="lineno"> 98</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><a id="l00099" name="l00099"></a><span class="lineno"> 99</span> }</div>
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<div class="line"><a id="l00100" name="l00100"></a><span class="lineno"> 100</span> }</div>
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<div class="line"><a id="l00101" name="l00101"></a><span class="lineno"> 101</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00102" name="l00102"></a><span class="lineno"> 102</span>}</div>
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</div>
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<div class="line"><a id="l00103" name="l00103"></a><span class="lineno"> 103</span></div>
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<div class="line"><a id="l00124" name="l00124"></a><span class="lineno"> 124</span><span class="keyword">template</span> <<span class="keyword">typename</span> T></div>
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<div class="foldopen" id="foldopen00125" data-start="{" data-end="}">
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<div class="line"><a id="l00125" name="l00125"></a><span class="lineno"><a class="line" href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce"> 125</a></span><span class="keywordtype">bool</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce">miller_rabin_primality_test</a>(T num, T repeats) {</div>
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<div class="line"><a id="l00126" name="l00126"></a><span class="lineno"> 126</span> <span class="keywordflow">if</span> (num <= 4) {</div>
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<div class="line"><a id="l00127" name="l00127"></a><span class="lineno"> 127</span> <span class="comment">// If num == 2 or num == 3 then prime</span></div>
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<div class="line"><a id="l00128" name="l00128"></a><span class="lineno"> 128</span> <span class="keywordflow">if</span> (num == 2 || num == 3) {</div>
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<div class="line"><a id="l00129" name="l00129"></a><span class="lineno"> 129</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><a id="l00130" name="l00130"></a><span class="lineno"> 130</span> } <span class="keywordflow">else</span> {</div>
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<div class="line"><a id="l00131" name="l00131"></a><span class="lineno"> 131</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00132" name="l00132"></a><span class="lineno"> 132</span> }</div>
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<div class="line"><a id="l00133" name="l00133"></a><span class="lineno"> 133</span> }</div>
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<div class="line"><a id="l00134" name="l00134"></a><span class="lineno"> 134</span> <span class="comment">// If num is even then not prime</span></div>
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<div class="line"><a id="l00135" name="l00135"></a><span class="lineno"> 135</span> <span class="keywordflow">if</span> (num % 2 == 0) {</div>
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<div class="line"><a id="l00136" name="l00136"></a><span class="lineno"> 136</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00137" name="l00137"></a><span class="lineno"> 137</span> }</div>
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<div class="line"><a id="l00138" name="l00138"></a><span class="lineno"> 138</span> <span class="comment">// Finding d and r in num = 2^r * d + 1</span></div>
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<div class="line"><a id="l00139" name="l00139"></a><span class="lineno"> 139</span> T d = num - 1, r = 0;</div>
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<div class="line"><a id="l00140" name="l00140"></a><span class="lineno"> 140</span> <span class="keywordflow">while</span> (d % 2 == 0) {</div>
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<div class="line"><a id="l00141" name="l00141"></a><span class="lineno"> 141</span> d = d / 2;</div>
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<div class="line"><a id="l00142" name="l00142"></a><span class="lineno"> 142</span> r++;</div>
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<div class="line"><a id="l00143" name="l00143"></a><span class="lineno"> 143</span> }</div>
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<div class="line"><a id="l00144" name="l00144"></a><span class="lineno"> 144</span> </div>
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<div class="line"><a id="l00145" name="l00145"></a><span class="lineno"> 145</span> <span class="keywordflow">for</span> (T i = 0; i < repeats; ++i) {</div>
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<div class="line"><a id="l00146" name="l00146"></a><span class="lineno"> 146</span> <span class="keywordflow">if</span> (!<a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a6f9c31c1047aa3191676d64571d4c506">miller_test</a>(d, num)) {</div>
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<div class="line"><a id="l00147" name="l00147"></a><span class="lineno"> 147</span> <span class="keywordflow">return</span> <span class="keyword">false</span>;</div>
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<div class="line"><a id="l00148" name="l00148"></a><span class="lineno"> 148</span> }</div>
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<div class="line"><a id="l00149" name="l00149"></a><span class="lineno"> 149</span> }</div>
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<div class="line"><a id="l00150" name="l00150"></a><span class="lineno"> 150</span> <span class="keywordflow">return</span> <span class="keyword">true</span>;</div>
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<div class="line"><a id="l00151" name="l00151"></a><span class="lineno"> 151</span>}</div>
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</div>
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<div class="line"><a id="l00152" name="l00152"></a><span class="lineno"> 152</span></div>
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<div class="foldopen" id="foldopen00157" data-start="{" data-end="}">
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<div class="line"><a id="l00157" name="l00157"></a><span class="lineno"><a class="line" href="../../d6/d42/miller__rabin_8cpp.html#a88ec9ad42717780d6caaff9d3d6977f9"> 157</a></span><span class="keywordtype">void</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a88ec9ad42717780d6caaff9d3d6977f9">tests</a>() {</div>
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<div class="line"><a id="l00158" name="l00158"></a><span class="lineno"> 158</span> <span class="comment">// First test on 2</span></div>
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<div class="line"><a id="l00159" name="l00159"></a><span class="lineno"> 159</span> assert(((<span class="keywordtype">void</span>)<span class="stringliteral">"2 is prime but function says otherwise.\n"</span>,</div>
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<div class="line"><a id="l00160" name="l00160"></a><span class="lineno"> 160</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce">miller_rabin_primality_test</a>(2, 1) == <span class="keyword">true</span>));</div>
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<div class="line"><a id="l00161" name="l00161"></a><span class="lineno"> 161</span> std::cout << <span class="stringliteral">"First test passes."</span> << std::endl;</div>
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<div class="line"><a id="l00162" name="l00162"></a><span class="lineno"> 162</span> <span class="comment">// Second test on 5</span></div>
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<div class="line"><a id="l00163" name="l00163"></a><span class="lineno"> 163</span> assert(((<span class="keywordtype">void</span>)<span class="stringliteral">"5 should be prime but the function says otherwise.\n"</span>,</div>
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<div class="line"><a id="l00164" name="l00164"></a><span class="lineno"> 164</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce">miller_rabin_primality_test</a>(5, 3) == <span class="keyword">true</span>));</div>
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<div class="line"><a id="l00165" name="l00165"></a><span class="lineno"> 165</span> std::cout << <span class="stringliteral">"Second test passes."</span> << std::endl;</div>
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<div class="line"><a id="l00166" name="l00166"></a><span class="lineno"> 166</span> <span class="comment">// Third test on 23</span></div>
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<div class="line"><a id="l00167" name="l00167"></a><span class="lineno"> 167</span> assert(((<span class="keywordtype">void</span>)<span class="stringliteral">"23 should be prime but the function says otherwise.\n"</span>,</div>
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<div class="line"><a id="l00168" name="l00168"></a><span class="lineno"> 168</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce">miller_rabin_primality_test</a>(23, 3) == <span class="keyword">true</span>));</div>
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<div class="line"><a id="l00169" name="l00169"></a><span class="lineno"> 169</span> std::cout << <span class="stringliteral">"Third test passes."</span> << std::endl;</div>
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<div class="line"><a id="l00170" name="l00170"></a><span class="lineno"> 170</span> <span class="comment">// Fourth test on 16</span></div>
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<div class="line"><a id="l00171" name="l00171"></a><span class="lineno"> 171</span> assert(((<span class="keywordtype">void</span>)<span class="stringliteral">"16 is not a prime but the function says otherwise.\n"</span>,</div>
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<div class="line"><a id="l00172" name="l00172"></a><span class="lineno"> 172</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce">miller_rabin_primality_test</a>(16, 3) == <span class="keyword">false</span>));</div>
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<div class="line"><a id="l00173" name="l00173"></a><span class="lineno"> 173</span> std::cout << <span class="stringliteral">"Fourth test passes."</span> << std::endl;</div>
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<div class="line"><a id="l00174" name="l00174"></a><span class="lineno"> 174</span> <span class="comment">// Fifth test on 27</span></div>
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<div class="line"><a id="l00175" name="l00175"></a><span class="lineno"> 175</span> assert(((<span class="keywordtype">void</span>)<span class="stringliteral">"27 is not a prime but the function says otherwise.\n"</span>,</div>
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<div class="line"><a id="l00176" name="l00176"></a><span class="lineno"> 176</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce">miller_rabin_primality_test</a>(27, 3) == <span class="keyword">false</span>));</div>
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<div class="line"><a id="l00177" name="l00177"></a><span class="lineno"> 177</span> std::cout << <span class="stringliteral">"Fifth test passes."</span> << std::endl;</div>
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<div class="line"><a id="l00178" name="l00178"></a><span class="lineno"> 178</span>}</div>
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</div>
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<div class="line"><a id="l00179" name="l00179"></a><span class="lineno"> 179</span></div>
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<div class="foldopen" id="foldopen00183" data-start="{" data-end="}">
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<div class="line"><a id="l00183" name="l00183"></a><span class="lineno"><a class="line" href="../../d6/d42/miller__rabin_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4"> 183</a></span><span class="keywordtype">int</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a>() {</div>
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<div class="line"><a id="l00184" name="l00184"></a><span class="lineno"> 184</span> <a class="code hl_function" href="../../d6/d42/miller__rabin_8cpp.html#a88ec9ad42717780d6caaff9d3d6977f9">tests</a>();</div>
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<div class="line"><a id="l00185" name="l00185"></a><span class="lineno"> 185</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a id="l00186" name="l00186"></a><span class="lineno"> 186</span>}</div>
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</div>
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<div class="ttc" id="amiller__rabin_8cpp_html_a091662a787d5ad4866713021f580fddb"><div class="ttname"><a href="../../d6/d42/miller__rabin_8cpp.html#a091662a787d5ad4866713021f580fddb">reverse_binary</a></div><div class="ttdeci">std::vector< T > reverse_binary(T num)</div><div class="ttdef"><b>Definition</b> <a href="#l00019">miller_rabin.cpp:19</a></div></div>
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<div class="ttc" id="amiller__rabin_8cpp_html_a6f9c31c1047aa3191676d64571d4c506"><div class="ttname"><a href="../../d6/d42/miller__rabin_8cpp.html#a6f9c31c1047aa3191676d64571d4c506">miller_test</a></div><div class="ttdeci">bool miller_test(T d, T num)</div><div class="ttdef"><b>Definition</b> <a href="#l00073">miller_rabin.cpp:73</a></div></div>
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<div class="ttc" id="amiller__rabin_8cpp_html_a88ec9ad42717780d6caaff9d3d6977f9"><div class="ttname"><a href="../../d6/d42/miller__rabin_8cpp.html#a88ec9ad42717780d6caaff9d3d6977f9">tests</a></div><div class="ttdeci">void tests()</div><div class="ttdef"><b>Definition</b> <a href="#l00157">miller_rabin.cpp:157</a></div></div>
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<div class="ttc" id="amiller__rabin_8cpp_html_a901288288ef5ebe8e97414cc30797cce"><div class="ttname"><a href="../../d6/d42/miller__rabin_8cpp.html#a901288288ef5ebe8e97414cc30797cce">miller_rabin_primality_test</a></div><div class="ttdeci">bool miller_rabin_primality_test(T num, T repeats)</div><div class="ttdef"><b>Definition</b> <a href="#l00125">miller_rabin.cpp:125</a></div></div>
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<div class="ttc" id="amiller__rabin_8cpp_html_ad6c2c67ea416d0e80003a88623f98b29"><div class="ttname"><a href="../../d6/d42/miller__rabin_8cpp.html#ad6c2c67ea416d0e80003a88623f98b29">modular_exponentiation</a></div><div class="ttdeci">T modular_exponentiation(T base, const std::vector< T > &rev_binary_exponent, T mod)</div><div class="ttdef"><b>Definition</b> <a href="#l00043">miller_rabin.cpp:43</a></div></div>
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<div class="ttc" id="amiller__rabin_8cpp_html_ae66f6b31b5ad750f1fe042a706a4e3d4"><div class="ttname"><a href="../../d6/d42/miller__rabin_8cpp.html#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a></div><div class="ttdeci">int main()</div><div class="ttdef"><b>Definition</b> <a href="#l00183">miller_rabin.cpp:183</a></div></div>
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<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> <a href="../../df/def/power__for__huge__numbers_8cpp_source.html#l00056">power_for_huge_numbers.cpp:56</a></div></div>
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