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</li>
<li class="toctree-l1"><a class="reference internal" href="/group-theory/">Group theory</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="/curves/">Curves</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="/rsa/">RSA</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="/utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="/examples-rsa-keys/">Generating RSA Keys</a>
<li class="toctree-l1"><a class="reference internal" href="/example-rsa-keys/">Generating RSA Keys</a>
</li>
<li class="toctree-l1"><a class="" href="/example-curves.md">Generating Curves</a>
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<script>
// Current page data
var mkdocs_page_name = "Generating RSA Keys";
var mkdocs_page_input_path = "examples-rsa-keys.md";
var mkdocs_page_input_path = "example-rsa-keys.md";
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</li>
<li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../curves/">Curves</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul class="current">
<li class="toctree-l1 current"><a class="reference internal current" href="#">Generating RSA Keys</a>
<ul class="current">
</ul>
</li>
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
</li>
</ul>
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@@ -116,7 +125,7 @@ print(f&quot;Plaintext: {plaintext}&quot;)
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</li>
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<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
</li>
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<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
</li>
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
</li>
</ul>
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@@ -3015,7 +3024,7 @@ In Group Theory, an identity element is an element in the group which do not cha
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</li>
<li class="toctree-l1"><a class="reference internal" href="group-theory/">Group theory</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="curves/">Curves</a>
</li>
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<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="rsa/">RSA</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="examples-rsa-keys/">Generating RSA Keys</a>
<li class="toctree-l1"><a class="reference internal" href="example-rsa-keys/">Generating RSA Keys</a>
</li>
<li class="toctree-l1"><a class="" href="example-curves.md">Generating Curves</a>
</li>
</ul>
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@@ -91,14 +100,20 @@
<li>Low-Level Cryptographic<ul>
<li><a href="/number-theory">Number Theory</a></li>
<li><a href="/group-theory">Group Theory</a></li>
<li><a href="/curves">Curves</a></li>
</ul>
</li>
<li>Public Keys:<ul>
<li><a href="/rsa">RSA</a></li>
</ul>
</li>
<li>Utils<ul>
<li><a href="/utils">Utils</a></li>
</ul>
</li>
<li>Examples:<ul>
<li><a href="/examples-rsa-keys">Generating RSA keys</a></li>
<li><a href="/example-rsa-keys">Generating RSA keys</a></li>
<li><a href="/example-curves">Generate curves</a></li>
</ul>
</li>
</ul>
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</li>
<li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../curves/">Curves</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
</li>
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
</li>
</ul>
</div>
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</li>
<li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../curves/">Curves</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
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</li>
<li class="toctree-l1"><a class="reference internal" href="../group-theory/">Group theory</a>
</li>
<li class="toctree-l1"><a class="reference internal" href="../curves/">Curves</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../rsa/">RSA</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
</li>
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
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</ul>
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</ul>
<p class="caption"><span class="caption-text">Public Keys</span></p>
<ul class="current">
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</ul>
</li>
</ul>
<p class="caption"><span class="caption-text">Utils</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../utils/">Utils</a>
</li>
</ul>
<p class="caption"><span class="caption-text">Examples</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../examples-rsa-keys/">Generating RSA Keys</a>
<li class="toctree-l1"><a class="reference internal" href="../example-rsa-keys/">Generating RSA Keys</a>
</li>
<li class="toctree-l1"><a class="" href="../example-curves.md">Generating Curves</a>
</li>
</ul>
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@@ -1041,8 +1050,8 @@ The key is a tuple of the key and the modulus n</p>
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<p>CryptoTools provides several utils function wich can be used for cryptography purposes</p>
<div class="doc doc-object doc-module">
<a id="Cryptotools.Utils.utils"></a>
<div class="doc doc-contents first">
<div class="doc doc-children">
<div class="doc doc-object doc-function">
<h2 id="Cryptotools.Utils.utils.bin_expo" class="doc doc-heading">
<code class="highlight language-python"><span class="n">bin_expo</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span></code>
</h2>
<div class="doc doc-contents ">
<p>This function perform an binary exponentiation, also known as Exponentiation squaring.
A binary exponentiation is the process for computing an integer power of a number, such as a ** n.
For doing that, the first step is to convert the exponent into a binary representation
And for each 1 bit, we compute the exponent.</p>
<table class="field-list">
<colgroup>
<col class="field-name" />
<col class="field-body" />
</colgroup>
<tbody valign="top">
<tr class="field">
<th class="field-name">Parameters:</th>
<td class="field-body">
<ul class="first simple">
<li>
<b><code>n</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>it's the base</p>
</div>
</li>
<li>
<b><code>e</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>it's the exponent</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<table class="field-list">
<colgroup>
<col class="field-name" />
<col class="field-body" />
</colgroup>
<tbody valign="top">
<tr class="field">
<th class="field-name">Returns:</th>
<td class="field-body">
<ul class="first simple">
<li>
<code><span title="int">int</span></code>
–
<div class="doc-md-description">
<p>Return the result of the exponentation of n ** e</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<details class="quote">
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">35</span>
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<span class="normal">62</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span> <span class="nf">bin_expo</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">int</span><span class="p">:</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function perform an binary exponentiation, also known as Exponentiation squaring.</span>
<span class="sd"> A binary exponentiation is the process for computing an integer power of a number, such as a ** n.</span>
<span class="sd"> For doing that, the first step is to convert the exponent into a binary representation</span>
<span class="sd"> And for each 1 bit, we compute the exponent.</span>
<span class="sd"> Args:</span>
<span class="sd"> n (Integer): it&#39;s the base</span>
<span class="sd"> e (Integer): it&#39;s the exponent</span>
<span class="sd"> Returns:</span>
<span class="sd"> Return the result of the exponentation of n ** e</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">binary</span> <span class="o">=</span> <span class="nb">bin</span><span class="p">(</span><span class="n">e</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span> <span class="c1"># Remove the prefix 0b</span>
<span class="n">r</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">exp</span> <span class="o">=</span> <span class="mi">1</span>
<span class="c1"># We need to reverse, and to start from the right to left</span>
<span class="c1"># Otherwise, we do on left to the right</span>
<span class="c1"># It&#39;s dirty, maybe we can find another way to do that</span>
<span class="k">for</span> <span class="n">b</span> <span class="ow">in</span> <span class="n">binary</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]:</span>
<span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="s1">&#39;1&#39;</span><span class="p">:</span>
<span class="n">r</span> <span class="o">*=</span> <span class="n">n</span> <span class="o">**</span> <span class="n">exp</span>
<span class="nb">print</span><span class="p">(</span><span class="n">n</span> <span class="o">**</span> <span class="n">exp</span><span class="p">,</span> <span class="n">exp</span><span class="p">)</span>
<span class="n">exp</span> <span class="o">*=</span> <span class="mi">2</span>
<span class="k">return</span> <span class="n">r</span>
</code></pre></div></td></tr></table></div>
</details>
</div>
</div>
<div class="doc doc-object doc-function">
<h2 id="Cryptotools.Utils.utils.egcd" class="doc doc-heading">
<code class="highlight language-python"><span class="n">egcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span></code>
</h2>
<div class="doc doc-contents ">
<p>This function compute the Extended Euclidean algorithm
https://user.eng.umd.edu/~danadach/Cryptography_20/ExtEuclAlg.pdf
https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm</p>
<table class="field-list">
<colgroup>
<col class="field-name" />
<col class="field-body" />
</colgroup>
<tbody valign="top">
<tr class="field">
<th class="field-name">Parameters:</th>
<td class="field-body">
<ul class="first simple">
<li>
<b><code>a</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>the number a</p>
</div>
</li>
<li>
<b><code>b</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>the number b</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<details class="quote">
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">18</span>
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<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function compute the Extended Euclidean algorithm</span>
<span class="sd"> https://user.eng.umd.edu/~danadach/Cryptography_20/ExtEuclAlg.pdf</span>
<span class="sd"> https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm</span>
<span class="sd"> Args:</span>
<span class="sd"> a (Integer): the number a</span>
<span class="sd"> b (Integer): the number b</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">if</span> <span class="n">a</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">b</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span>
<span class="n">gcd</span><span class="p">,</span> <span class="n">x1</span><span class="p">,</span> <span class="n">y1</span> <span class="o">=</span> <span class="n">egcd</span><span class="p">(</span><span class="n">b</span> <span class="o">%</span> <span class="n">a</span><span class="p">,</span> <span class="n">a</span><span class="p">)</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">y1</span> <span class="o">-</span> <span class="p">(</span><span class="n">b</span> <span class="o">//</span> <span class="n">a</span><span class="p">)</span> <span class="o">*</span> <span class="n">x1</span>
<span class="n">y</span> <span class="o">=</span> <span class="n">x1</span>
<span class="k">return</span> <span class="n">gcd</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
</code></pre></div></td></tr></table></div>
</details>
</div>
</div>
<div class="doc doc-object doc-function">
<h2 id="Cryptotools.Utils.utils.exponent_squaring" class="doc doc-heading">
<code class="highlight language-python"><span class="n">exponent_squaring</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">)</span></code>
</h2>
<div class="doc doc-contents ">
<p>This function perform an exponentiation squaring, which compute an integer power of a number based on the following algorithm:
n ** e = {
1 # if n is 0
(n ** (e / 2)) ** 2 # if n is even
((n ** (e - 1 / 2)) ** 2) * n # if n is odd
}</p>
<table class="field-list">
<colgroup>
<col class="field-name" />
<col class="field-body" />
</colgroup>
<tbody valign="top">
<tr class="field">
<th class="field-name">Parameters:</th>
<td class="field-body">
<ul class="first simple">
<li>
<b><code>n</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>n is the base of n ** e</p>
</div>
</li>
<li>
<b><code>e</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>e is the exponent</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<details class="quote">
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">64</span>
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<span class="normal">84</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span> <span class="nf">exponent_squaring</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">e</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function perform an exponentiation squaring, which compute an integer power of a number based on the following algorithm:</span>
<span class="sd"> n ** e = {</span>
<span class="sd"> 1 # if n is 0</span>
<span class="sd"> (n ** (e / 2)) ** 2 # if n is even</span>
<span class="sd"> ((n ** (e - 1 / 2)) ** 2) * n # if n is odd</span>
<span class="sd"> }</span>
<span class="sd"> Args:</span>
<span class="sd"> n (Integer): n is the base of n ** e</span>
<span class="sd"> e (Integer): e is the exponent</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">if</span> <span class="n">e</span> <span class="o">&lt;</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">exponent_squaring</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="n">n</span><span class="p">,</span> <span class="o">-</span><span class="n">e</span><span class="p">)</span>
<span class="k">elif</span> <span class="n">e</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">1</span>
<span class="k">elif</span> <span class="n">e</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span> <span class="c1"># n is odd</span>
<span class="k">return</span> <span class="n">exponent_squaring</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">e</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">n</span>
<span class="k">elif</span> <span class="n">e</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="c1"># n is even</span>
<span class="k">return</span> <span class="n">exponent_squaring</span><span class="p">(</span><span class="n">n</span> <span class="o">*</span> <span class="n">n</span><span class="p">,</span> <span class="n">e</span> <span class="o">/</span> <span class="mi">2</span><span class="p">)</span>
</code></pre></div></td></tr></table></div>
</details>
</div>
</div>
<div class="doc doc-object doc-function">
<h2 id="Cryptotools.Utils.utils.gcd" class="doc doc-heading">
<code class="highlight language-python"><span class="n">gcd</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span></code>
</h2>
<div class="doc doc-contents ">
<p>This function calculate the GCD (Greatest Common Divisor of the number a of b
Args:
a (Integer): the number a
b (integer): the number b</p>
<details class="return" open>
<summary>Return</summary>
<p>the GCD</p>
</details>
<details class="quote">
<summary>Source code in <code>Cryptotools/Utils/utils.py</code></summary>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"> 3</span>
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<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function calculate the GCD (Greatest Common Divisor of the number a of b</span>
<span class="sd"> Args:</span>
<span class="sd"> a (Integer): the number a</span>
<span class="sd"> b (integer): the number b</span>
<span class="sd"> Return:</span>
<span class="sd"> the GCD </span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">if</span> <span class="n">b</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">a</span>
<span class="k">return</span> <span class="n">gcd</span><span class="p">(</span><span class="n">b</span><span class="p">,</span> <span class="n">a</span><span class="o">%</span><span class="n">b</span><span class="p">)</span>
</code></pre></div></td></tr></table></div>
</details>
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