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<h1 id="elliptic-curves">Elliptic Curves</h1>
<h2 id="points">Points</h2>
<div class="doc doc-object doc-module">
<a id="Cryptotools.Groups.point"></a>
<div class="doc doc-contents first">
<div class="doc doc-children">
<div class="doc doc-object doc-class">
<h2 id="Cryptotools.Groups.point.Point" class="doc doc-heading">
<code>Point</code>
</h2>
<div class="doc doc-contents ">
<p>This simple class represent the Point at the coordinate x and y in a plan</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">Attributes:</th>
<td class="field-body">
<ul class="first simple">
<li>
<b><code>x</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>Position at the x</p>
</div>
</li>
<li>
<b><code>y</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>Position at the y</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<details class="quote">
<summary>Source code in <code>Cryptotools/Groups/point.py</code></summary>
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<span class="normal">34</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">class</span><span class="w"> </span><span class="nc">Point</span><span class="p">:</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This simple class represent the Point at the coordinate x and y in a plan</span>
<span class="sd"> Attributes:</span>
<span class="sd"> x (Integer): Position at the x</span>
<span class="sd"> y (Integer): Position at the y</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_x</span> <span class="o">=</span> <span class="n">x</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_y</span> <span class="o">=</span> <span class="n">y</span>
<span class="nd">@property</span>
<span class="k">def</span><span class="w"> </span><span class="nf">x</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_x</span>
<span class="nd">@property</span>
<span class="k">def</span><span class="w"> </span><span class="nf">y</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y</span>
<span class="nd">@x</span><span class="o">.</span><span class="n">setter</span>
<span class="k">def</span><span class="w"> </span><span class="nf">x</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_x</span> <span class="o">=</span> <span class="n">x</span>
<span class="nd">@y</span><span class="o">.</span><span class="n">setter</span>
<span class="k">def</span><span class="w"> </span><span class="nf">y</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_y</span> <span class="o">=</span> <span class="n">y</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_x</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_x</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">x</span> <span class="ow">and</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">y</span>
</code></pre></div></td></tr></table></div>
</details>
<div class="doc doc-children">
</div>
</div>
</div>
</div>
</div>
</div><h2 id="curves">Curves</h2>
<div class="doc doc-object doc-module">
<a id="Cryptotools.Groups.curve"></a>
<div class="doc doc-contents first">
<div class="doc doc-children">
<div class="doc doc-object doc-class">
<h2 id="Cryptotools.Groups.curve.Curve" class="doc doc-heading">
<code>Curve</code>
</h2>
<div class="doc doc-contents ">
<details class="quote">
<summary>Source code in <code>Cryptotools/Groups/curve.py</code></summary>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal"> 8</span>
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<span class="c1"># Curve</span>
<span class="n">WEIERSTRASS</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">MONTGOMERY</span> <span class="o">=</span> <span class="mi">1</span>
<span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">t</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_a</span> <span class="o">=</span> <span class="n">a</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_b</span> <span class="o">=</span> <span class="n">b</span>
<span class="k">if</span> <span class="n">t</span> <span class="ow">not</span> <span class="ow">in</span> <span class="p">(</span><span class="n">Curve</span><span class="o">.</span><span class="n">WEIERSTRASS</span><span class="p">,</span> <span class="n">Curve</span><span class="o">.</span><span class="n">MONTGOMERY</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">Exception</span><span class="p">(</span><span class="sa">f</span><span class="s2">&quot;The type of the curve is not recognized&quot;</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_type</span> <span class="o">=</span> <span class="n">t</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_xtmp</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">500</span><span class="p">)</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_x</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_y</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_yn</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_points</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_pointsSym</span> <span class="o">=</span> <span class="nb">list</span><span class="p">()</span>
<span class="k">def</span><span class="w"> </span><span class="nf">f</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_type</span> <span class="o">==</span> <span class="n">Curve</span><span class="o">.</span><span class="n">WEIERSTRASS</span><span class="p">:</span>
<span class="n">y</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">_a</span> <span class="o">*</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_b</span>
<span class="k">if</span> <span class="bp">self</span><span class="o">.</span><span class="n">_type</span> <span class="o">==</span> <span class="n">Curve</span><span class="o">.</span><span class="n">MONTGOMERY</span><span class="p">:</span>
<span class="n">y</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="mi">3</span> <span class="o">*</span> <span class="nb">pow</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="n">x</span>
<span class="k">if</span> <span class="n">y</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">sqrt</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="k">return</span> <span class="kc">None</span>
<span class="k">def</span><span class="w"> </span><span class="nf">generatePoints</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">for</span> <span class="n">x</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_xtmp</span><span class="p">:</span>
<span class="n">y</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">f</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="k">if</span> <span class="n">y</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="k">continue</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_x</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_y</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_yn</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="o">-</span><span class="n">y</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_points</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span>
<span class="n">x</span><span class="p">,</span>
<span class="n">y</span>
<span class="p">))</span>
<span class="bp">self</span><span class="o">.</span><span class="n">_pointsSym</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">Point</span><span class="p">(</span>
<span class="n">x</span><span class="p">,</span>
<span class="o">-</span><span class="n">y</span>
<span class="p">))</span>
<span class="nd">@property</span>
<span class="k">def</span><span class="w"> </span><span class="nf">x</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_x</span>
<span class="nd">@property</span>
<span class="k">def</span><span class="w"> </span><span class="nf">y</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_y</span>
<span class="nd">@property</span>
<span class="k">def</span><span class="w"> </span><span class="nf">yn</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_yn</span>
<span class="k">def</span><span class="w"> </span><span class="nf">getPoints</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_points</span>
<span class="k">def</span><span class="w"> </span><span class="nf">getPointsSym</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pointsSym</span>
<span class="k">def</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Point</span><span class="p">:</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function operathe addition operation on two points P and Q</span>
<span class="sd"> Args:</span>
<span class="sd"> P (Object): The first Point on the curve</span>
<span class="sd"> Q (Object): The second Point on the curve</span>
<span class="sd"> Returns:</span>
<span class="sd"> Return the Point object R</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1">## Check if P or Q are infinity</span>
<span class="k">if</span> <span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="k">elif</span> <span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
<span class="k">elif</span> <span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
<span class="c1"># point doubling</span>
<span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
<span class="c1"># Infinity</span>
<span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span> <span class="o">!=</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span> <span class="ow">or</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="c1"># Point doubling</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">inv</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="c1"># It&#39;s working with the inverse modular, WHY ???</span>
<span class="n">m</span> <span class="o">=</span> <span class="p">((</span><span class="mi">3</span> <span class="o">*</span> <span class="nb">pow</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_a</span><span class="p">)</span> <span class="o">*</span> <span class="n">inv</span>
<span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">inv</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">*</span> <span class="n">inv</span>
<span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
<span class="c1"># May call this Exception: base is not invertible for the given modulus</span>
<span class="c1"># I return an Infinity point until I fixed that</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">xr</span> <span class="o">=</span> <span class="p">(</span><span class="nb">pow</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
<span class="n">yr</span> <span class="o">=</span> <span class="p">(</span><span class="n">m</span> <span class="o">*</span> <span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">xr</span><span class="p">))</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">xr</span><span class="p">,</span> <span class="n">yr</span><span class="p">)</span>
<span class="k">def</span><span class="w"> </span><span class="nf">scalar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Point</span><span class="p">:</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add.</span>
<span class="sd"> Args:</span>
<span class="sd"> P (point): the Point to multiplication</span>
<span class="sd"> n (Integer): multiplicate n time P</span>
<span class="sd"> Returns:</span>
<span class="sd"> Return the result of the Scalar multiplication</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">n</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span>
<span class="n">binary</span> <span class="o">=</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="c1"># We need to reverse the binary</span>
<span class="n">nP</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">Rtmp</span> <span class="o">=</span> <span class="n">P</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="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">nP</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">nP</span><span class="p">,</span> <span class="n">Rtmp</span><span class="p">)</span>
<span class="n">Rtmp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Rtmp</span><span class="p">,</span> <span class="n">Rtmp</span><span class="p">)</span> <span class="c1"># Double P</span>
<span class="k">return</span> <span class="n">nP</span>
<span class="k">def</span><span class="w"> </span><span class="nf">find_reverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">P</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function return the reverse of the Point P</span>
<span class="sd"> Args:</span>
<span class="sd"> P (Point): Point object to find </span>
<span class="sd"> Returns:</span>
<span class="sd"> Return the object Pr, which is the reverse point of P</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">Pr</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pointsSym</span><span class="p">:</span>
<span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">x</span> <span class="ow">and</span> <span class="o">-</span><span class="n">P</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
<span class="n">Pr</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
<span class="k">break</span>
<span class="k">return</span> <span class="n">Pr</span>
</code></pre></div></td></tr></table></div>
</details>
<div class="doc doc-children">
<div class="doc doc-object doc-function">
<h3 id="Cryptotools.Groups.curve.Curve.add" class="doc doc-heading">
<code class="highlight language-python"><span class="n">add</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span></code>
</h3>
<div class="doc doc-contents ">
<p>This function operathe addition operation on two points P and Q</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>P</code></b>
(<code><span title="Object">Object</span></code>)
–
<div class="doc-md-description">
<p>The first Point on the curve</p>
</div>
</li>
<li>
<b><code>Q</code></b>
(<code><span title="Object">Object</span></code>)
–
<div class="doc-md-description">
<p>The second Point on the curve</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="Cryptotools.Groups.elliptic.Point">Point</span></code>
–
<div class="doc-md-description">
<p>Return the Point object R</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<details class="mkdocstrings-source">
<summary>Source code in <code>Cryptotools/Groups/curve.py</code></summary>
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<span class="normal">119</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span><span class="w"> </span><span class="nf">add</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">Q</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Point</span><span class="p">:</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function operathe addition operation on two points P and Q</span>
<span class="sd"> Args:</span>
<span class="sd"> P (Object): The first Point on the curve</span>
<span class="sd"> Q (Object): The second Point on the curve</span>
<span class="sd"> Returns:</span>
<span class="sd"> Return the Point object R</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="c1">## Check if P or Q are infinity</span>
<span class="k">if</span> <span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="ow">and</span> <span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="k">elif</span> <span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
<span class="k">elif</span> <span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">==</span> <span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">):</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
<span class="c1"># point doubling</span>
<span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
<span class="c1"># Infinity</span>
<span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span> <span class="o">!=</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span> <span class="ow">or</span> <span class="n">Q</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="c1"># Point doubling</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">inv</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="c1"># It&#39;s working with the inverse modular, WHY ???</span>
<span class="n">m</span> <span class="o">=</span> <span class="p">((</span><span class="mi">3</span> <span class="o">*</span> <span class="nb">pow</span><span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">_a</span><span class="p">)</span> <span class="o">*</span> <span class="n">inv</span>
<span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">inv</span> <span class="o">=</span> <span class="nb">pow</span><span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="p">(</span><span class="n">Q</span><span class="o">.</span><span class="n">y</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span><span class="p">)</span> <span class="o">*</span> <span class="n">inv</span>
<span class="k">except</span> <span class="ne">ValueError</span><span class="p">:</span>
<span class="c1"># May call this Exception: base is not invertible for the given modulus</span>
<span class="c1"># I return an Infinity point until I fixed that</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">xr</span> <span class="o">=</span> <span class="p">(</span><span class="nb">pow</span><span class="p">(</span><span class="n">m</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">Q</span><span class="o">.</span><span class="n">x</span><span class="p">)</span>
<span class="n">yr</span> <span class="o">=</span> <span class="p">(</span><span class="n">m</span> <span class="o">*</span> <span class="p">(</span><span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">-</span> <span class="n">xr</span><span class="p">))</span> <span class="o">-</span> <span class="n">P</span><span class="o">.</span><span class="n">y</span>
<span class="k">return</span> <span class="n">Point</span><span class="p">(</span><span class="n">xr</span><span class="p">,</span> <span class="n">yr</span><span class="p">)</span>
</code></pre></div></td></tr></table></div>
</details>
</div>
</div>
<div class="doc doc-object doc-function">
<h3 id="Cryptotools.Groups.curve.Curve.find_reverse" class="doc doc-heading">
<code class="highlight language-python"><span class="n">find_reverse</span><span class="p">(</span><span class="n">P</span><span class="p">)</span></code>
</h3>
<div class="doc doc-contents ">
<p>This function return the reverse of the Point P
Args:
P (Point): Point object to find </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">Returns:</th>
<td class="field-body">
<ul class="first simple">
<li>
–
<div class="doc-md-description">
<p>Return the object Pr, which is the reverse point of P</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<details class="mkdocstrings-source">
<summary>Source code in <code>Cryptotools/Groups/curve.py</code></summary>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">145</span>
<span class="normal">146</span>
<span class="normal">147</span>
<span class="normal">148</span>
<span class="normal">149</span>
<span class="normal">150</span>
<span class="normal">151</span>
<span class="normal">152</span>
<span class="normal">153</span>
<span class="normal">154</span>
<span class="normal">155</span>
<span class="normal">156</span>
<span class="normal">157</span>
<span class="normal">158</span>
<span class="normal">159</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span><span class="w"> </span><span class="nf">find_reverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">P</span><span class="p">):</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function return the reverse of the Point P</span>
<span class="sd"> Args:</span>
<span class="sd"> P (Point): Point object to find </span>
<span class="sd"> Returns:</span>
<span class="sd"> Return the object Pr, which is the reverse point of P</span>
<span class="sd"> &quot;&quot;&quot;</span>
<span class="n">Pr</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">p</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">_pointsSym</span><span class="p">:</span>
<span class="k">if</span> <span class="n">P</span><span class="o">.</span><span class="n">x</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">x</span> <span class="ow">and</span> <span class="o">-</span><span class="n">P</span><span class="o">.</span><span class="n">y</span> <span class="o">==</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
<span class="n">Pr</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">y</span><span class="p">)</span>
<span class="k">break</span>
<span class="k">return</span> <span class="n">Pr</span>
</code></pre></div></td></tr></table></div>
</details>
</div>
</div>
<div class="doc doc-object doc-function">
<h3 id="Cryptotools.Groups.curve.Curve.scalar" class="doc doc-heading">
<code class="highlight language-python"><span class="n">scalar</span><span class="p">(</span><span class="n">P</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span></code>
</h3>
<div class="doc doc-contents ">
<p>This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add.</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>P</code></b>
(<code><a class="autorefs autorefs-internal" title="Cryptotools.Groups.point" href="#Cryptotools.Groups.point">point</a></code>)
–
<div class="doc-md-description">
<p>the Point to multiplication</p>
</div>
</li>
<li>
<b><code>n</code></b>
(<code><span title="Integer">Integer</span></code>)
–
<div class="doc-md-description">
<p>multiplicate n time P</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="Cryptotools.Groups.elliptic.Point">Point</span></code>
–
<div class="doc-md-description">
<p>Return the result of the Scalar multiplication</p>
</div>
</li>
</ul>
</td>
</tr>
</tbody>
</table>
<details class="mkdocstrings-source">
<summary>Source code in <code>Cryptotools/Groups/curve.py</code></summary>
<div class="highlight"><table class="highlighttable"><tr><td class="linenos"><div class="linenodiv"><pre><span></span><span class="normal">121</span>
<span class="normal">122</span>
<span class="normal">123</span>
<span class="normal">124</span>
<span class="normal">125</span>
<span class="normal">126</span>
<span class="normal">127</span>
<span class="normal">128</span>
<span class="normal">129</span>
<span class="normal">130</span>
<span class="normal">131</span>
<span class="normal">132</span>
<span class="normal">133</span>
<span class="normal">134</span>
<span class="normal">135</span>
<span class="normal">136</span>
<span class="normal">137</span>
<span class="normal">138</span>
<span class="normal">139</span>
<span class="normal">140</span>
<span class="normal">141</span>
<span class="normal">142</span>
<span class="normal">143</span></pre></div></td><td class="code"><div><pre><span></span><code><span class="k">def</span><span class="w"> </span><span class="nf">scalar</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">P</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Point</span><span class="p">:</span>
<span class="w"> </span><span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> This function compute a Scalar Multiplication of P, n time. This algorithm is also known as Double and Add.</span>
<span class="sd"> Args:</span>
<span class="sd"> P (point): the Point to multiplication</span>
<span class="sd"> n (Integer): multiplicate n time P</span>
<span class="sd"> Returns:</span>
<span class="sd"> Return the result of the Scalar multiplication</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">n</span><span class="p">)[</span><span class="mi">2</span><span class="p">:]</span>
<span class="n">binary</span> <span class="o">=</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="c1"># We need to reverse the binary</span>
<span class="n">nP</span> <span class="o">=</span> <span class="n">Point</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">Rtmp</span> <span class="o">=</span> <span class="n">P</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="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">nP</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">nP</span><span class="p">,</span> <span class="n">Rtmp</span><span class="p">)</span>
<span class="n">Rtmp</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">add</span><span class="p">(</span><span class="n">Rtmp</span><span class="p">,</span> <span class="n">Rtmp</span><span class="p">)</span> <span class="c1"># Double P</span>
<span class="k">return</span> <span class="n">nP</span>
</code></pre></div></td></tr></table></div>
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