.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/crypto/public_keys/plot_02_diffie_hellman.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_public_keys_plot_02_diffie_hellman.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_public_keys_plot_02_diffie_hellman.py:


Diffie-Hellman: agreeing on a secret over a public channel (1976)
=================================================================

Diffie and Hellman showed that two people can reach the same secret by
exchanging only public values: each raises the other's public element to
their own private exponent, and both arrive at g**(a*b). An eavesdropper sees
g**a and g**b but would need a discrete logarithm to continue.

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

Both parties compute the same shared element without ever sending it. Then
a man in the middle substitutes his own public value: the arithmetic still
works, but Alice now shares a secret with Mallory. Agreement on a secret is
not authentication of the other person.

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

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Agree on a shared group element
-------------------------------
Neither party sends the shared value. Both reach g**(a*b) modulo p.

.. GENERATED FROM PYTHON SOURCE LINES 26-38

.. code-block:: Python

    import matplotlib.pyplot as plt

    import blockchainkit as bk

    group = bk.crypto.DHGroup()  # p = 23, q = 11, g = 2: tiny, so every value is visible.
    alice_secret, bob_secret = 3, 7
    alice_public = group.public(alice_secret)
    bob_public = group.public(bob_secret)
    shared = group.shared(bob_public, alice_secret)
    assert shared == group.shared(alice_public, bob_secret)
    print(f"Public values: {alice_public}, {bob_public}; shared element: {shared}")





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

 .. code-block:: none

    Public values: 8, 13; shared element: 12




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Authentication is a separate problem
------------------------------------
Mallory substitutes her own public value. Alice now shares a secret with
Mallory, not Bob. Real protocols authenticate the exchange and use a KDF.

.. GENERATED FROM PYTHON SOURCE LINES 43-49

.. code-block:: Python

    mallory_secret = 4
    alice_mallory = group.shared(group.public(mallory_secret), alice_secret)
    assert alice_mallory == group.shared(alice_public, mallory_secret)
    assert alice_mallory != shared
    print("After key substitution, Alice's shared element is", alice_mallory)





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

 .. code-block:: none

    After key substitution, Alice's shared element is 2




.. GENERATED FROM PYTHON SOURCE LINES 50-52

The same exchange in a group too large to search by hand
--------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 52-56

.. code-block:: Python

    big = bk.crypto.TEACHING_GROUP
    a, b = 0x1234_5678_9ABC, 0x0FED_CBA9_8765
    assert big.shared(big.public(b), a) == big.shared(big.public(a), b)








.. GENERATED FROM PYTHON SOURCE LINES 57-59

Exponentiation scrambles the subgroup
-------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 59-67

.. code-block:: Python

    fig, ax = plt.subplots(figsize=(7, 3.8))
    exponents = list(range(1, group.q))
    ax.scatter(exponents, [group.public(x) for x in exponents], color="#2563eb")
    ax.set(
        xlabel="private exponent x", ylabel="public element g**x mod p", title="DH in a tiny subgroup"
    )
    fig.tight_layout()




.. image-sg:: /api/gallery/crypto/public_keys/images/sphx_glr_plot_02_diffie_hellman_001.png
   :alt: DH in a tiny subgroup
   :srcset: /api/gallery/crypto/public_keys/images/sphx_glr_plot_02_diffie_hellman_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 68-72

Exercise
--------
Exhaustively recover Alice's exponent from her public value. Why does this
experiment say nothing about the cost for a carefully chosen large group?


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

   **Total running time of the script:** (0 minutes 0.135 seconds)


.. _sphx_glr_download_api_gallery_crypto_public_keys_plot_02_diffie_hellman.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/public_keys/plot_02_diffie_hellman.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_diffie_hellman.ipynb <plot_02_diffie_hellman.ipynb>`

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

      :download:`Download Python source code: plot_02_diffie_hellman.py <plot_02_diffie_hellman.py>`

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

      :download:`Download zipped: plot_02_diffie_hellman.zip <plot_02_diffie_hellman.zip>`


.. only:: html

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

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