.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/number_theory/modular_arithmetic/plot_02_toy_rsa.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_number_theory_modular_arithmetic_plot_02_toy_rsa.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_number_theory_modular_arithmetic_plot_02_toy_rsa.py:


A toy RSA encryption/decryption round trip
=================================================

RSA's key generation, encryption, and decryption are exactly
:func:`~mathematicskit.number_theory.systems.modular_arithmetic.mod_inverse`
(to find the private exponent) and
:func:`~mathematicskit.number_theory.systems.modular_arithmetic.fast_mod_pow`
(to encrypt/decrypt) -- demonstrated here with small (textbook-toy,
NOT cryptographically secure) primes.

.. GENERATED FROM PYTHON SOURCE LINES 14-16

.. code-block:: Python

    from mathematicskit.number_theory import euler_totient, fast_mod_pow, mod_inverse








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Key generation
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-29

.. code-block:: Python


    p, q = 61, 53  # small toy primes (real RSA uses ~1024-bit primes)
    n = p * q
    phi_n = euler_totient(n)
    e = 17  # public exponent, coprime to phi(n)
    d = mod_inverse(e, phi_n)  # private exponent

    print(f"n = {n}, phi(n) = {phi_n}")
    print(f"public key: (e={e}, n={n}); private key: (d={d}, n={n})")





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

 .. code-block:: none

    n = 3233, phi(n) = 3120
    public key: (e=17, n=3233); private key: (d=2753, n=3233)




.. GENERATED FROM PYTHON SOURCE LINES 30-32

Encrypt and decrypt a message
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-39

.. code-block:: Python


    message = 65
    ciphertext = fast_mod_pow(message, e, n)
    decrypted = fast_mod_pow(ciphertext, d, n)

    print(f"message={message} -> ciphertext={ciphertext} -> decrypted={decrypted}")
    assert decrypted == message




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

 .. code-block:: none

    message=65 -> ciphertext=2790 -> decrypted=65





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

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_number_theory_modular_arithmetic_plot_02_toy_rsa.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_toy_rsa.ipynb <plot_02_toy_rsa.ipynb>`

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

      :download:`Download Python source code: plot_02_toy_rsa.py <plot_02_toy_rsa.py>`

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

      :download:`Download zipped: plot_02_toy_rsa.zip <plot_02_toy_rsa.zip>`


.. only:: html

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

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