.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/number_theory/factorization/plot_01_pollard_rho.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_factorization_plot_01_pollard_rho.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_number_theory_factorization_plot_01_pollard_rho.py:


Pollard's rho factorization
===================================================

Pollard's rho method finds a prime factor :math:`p` of :math:`n` in
roughly :math:`\sqrt p` steps, by waiting for a birthday-paradox
collision in the sequence :math:`x \mapsto x^2 + 1 \bmod p`. This script
factors the Fermat number :math:`F_6 = 2^{64}+1` and plots the step
count against the smallest prime factor for a batch of semiprimes.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.number_theory import is_prime_miller_rabin, pollard_rho








.. GENERATED FROM PYTHON SOURCE LINES 19-21

The sixth Fermat number
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-25

.. code-block:: Python


    result = pollard_rho(2**64 + 1)
    print(f"2^64 + 1 = {result.factor} * {result.cofactor}  ({result.iterations} iterations)")





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

 .. code-block:: none

    2^64 + 1 = 274177 * 67280421310721  (808 iterations)




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Steps grow like the square root of the smallest factor
-----------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-58

.. code-block:: Python


    rng = np.random.default_rng(0)


    def random_prime(lo, hi):
        while True:
            n = int(rng.integers(lo, hi)) | 1
            if is_prime_miller_rabin(n):
                return n


    small_factors, steps = [], []
    for bits in range(8, 34, 2):
        for _ in range(6):
            p = random_prime(2 ** (bits - 1), 2**bits)
            q = random_prime(2**40, 2**41)
            r = pollard_rho(p * q)
            small_factors.append(min(r.factor, r.cofactor))
            steps.append(r.iterations)
    small_factors, steps = np.array(small_factors), np.array(steps)

    fig, ax = plt.subplots()
    ax.loglog(small_factors, steps, "o", alpha=0.6, label="Pollard rho")
    ref = np.logspace(np.log10(small_factors.min()), np.log10(small_factors.max()), 50)
    ax.loglog(ref, np.sqrt(np.pi * ref / 2), "--", color="firebrick", label="sqrt(pi p / 2)")
    ax.set_xlabel("smallest prime factor p")
    ax.set_ylabel("iterations")
    ax.set_title("Rho finds p in about sqrt(p) steps")
    ax.legend()
    plt.show()



.. image-sg:: /api/gallery/number_theory/factorization/images/sphx_glr_plot_01_pollard_rho_001.png
   :alt: Rho finds p in about sqrt(p) steps
   :srcset: /api/gallery/number_theory/factorization/images/sphx_glr_plot_01_pollard_rho_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.476 seconds)


.. _sphx_glr_download_api_gallery_number_theory_factorization_plot_01_pollard_rho.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_pollard_rho.ipynb <plot_01_pollard_rho.ipynb>`

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

      :download:`Download Python source code: plot_01_pollard_rho.py <plot_01_pollard_rho.py>`

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

      :download:`Download zipped: plot_01_pollard_rho.zip <plot_01_pollard_rho.zip>`


.. only:: html

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

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