.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crypto/curves/plot_01_elliptic_curves.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_crypto_curves_plot_01_elliptic_curves.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_crypto_curves_plot_01_elliptic_curves.py:


Elliptic-curve cryptography: a finite group you can draw (Miller and Koblitz 1985)
==================================================================================

Replace modular exponentiation with repeated addition of curve points.
The public key Q=xG is easy to compute; recovering x is the discrete-log
problem. Our 19-element example is small enough to solve exhaustively.

What to look for
----------------

Follow repeated addition around a tiny set of points. Exhaustive search recovers the
private step count here because this example is deliberately small.

Read cells in order. An ``assert`` that produces no output has passed.
The final exercise asks you to change an input and explain the result.

The history behind this experiment: :doc:`/history/crypto_breakthroughs`.
See :doc:`/exercises/crypto` for a worked solution to the exercise.

.. GENERATED FROM PYTHON SOURCE LINES 23-36

.. code-block:: Python

    import matplotlib.pyplot as plt

    import blockchainkit as bk

    curve = bk.crypto.TOY_CURVE
    points = [(x, y) for x in range(curve.p) for y in range(curve.p) if curve.contains((x, y))]
    assert len(points) + 1 == curve.order  # Include infinity.
    assert bk.crypto.multiply(curve.order, curve.generator, curve) is None
    public = bk.crypto.public_key(7, curve)
    recovered = next(k for k in range(1, curve.order) if bk.crypto.public_key(k, curve) == public)
    assert recovered == 7
    print("Public key:", public, "; recovered tiny private key:", recovered)





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Public key: (0, 6) ; recovered tiny private key: 7




.. GENERATED FROM PYTHON SOURCE LINES 37-53

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(6, 5))
    ax.scatter(*zip(*points, strict=True), color="#2563eb", s=55)
    for k in (1, 2, 3, 7):
        x, y = bk.crypto.public_key(k, curve)
        ax.annotate(f"{k}G", (x, y), xytext=(6, 6), textcoords="offset points")
    ax.set(
        xlabel="x modulo 17",
        ylabel="y modulo 17",
        title="y² = x³ + 2x + 2 over F₁₇",
        xlim=(-1, 17),
        ylim=(-1, 17),
    )
    ax.set_aspect("equal")
    ax.grid(alpha=0.2)
    fig.tight_layout()




.. image-sg:: /api/gallery/crypto/curves/images/sphx_glr_plot_01_elliptic_curves_001.png
   :alt: y² = x³ + 2x + 2 over F₁₇
   :srcset: /api/gallery/crypto/curves/images/sphx_glr_plot_01_elliptic_curves_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 54-56

Larger parameters, same API
---------------------------

.. GENERATED FROM PYTHON SOURCE LINES 56-60

.. code-block:: Python

    large_public = bk.crypto.public_key(7)  # secp256k1 by default.
    assert bk.crypto.SECP256K1.contains(large_public)
    print("secp256k1 public point:", bk.crypto.encode_point(large_public).hex())





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    secp256k1 public point: 045cbdf0646e5db4eaa398f365f2ea7a0e3d419b7e0330e39ce92bddedcac4f9bc6aebca40ba255960a3178d6d861a54dba813d0b813fde7b5a5082628087264da




.. GENERATED FROM PYTHON SOURCE LINES 61-66

Exercise
--------
Verify G + (-G) = infinity and (a+b)G = aG+bG. Explain why the plot consists
of isolated points rather than a smooth curve. Inspect double-and-add and
identify where its control flow depends on private scalar bits.


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.159 seconds)


.. _sphx_glr_download_api_gallery_crypto_curves_plot_01_elliptic_curves.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/crypto/curves/plot_01_elliptic_curves.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_elliptic_curves.ipynb <plot_01_elliptic_curves.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_elliptic_curves.py <plot_01_elliptic_curves.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_elliptic_curves.zip <plot_01_elliptic_curves.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_