.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/proofs/snarks/plot_05_plonk.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_proofs_snarks_plot_05_plonk.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_proofs_snarks_plot_05_plonk.py:


PLONK: permutation arguments and a universal setup (2019)
=========================================================

Gabizon, Williamson and Ciobotaru wrote circuits as uniform gates,
:math:`q_L a + q_R b + q_O c + q_M ab + q_C = 0`, and enforced that wires
sharing a variable carry the same value with a *permutation argument*. With
random :math:`\beta, \gamma`, the grand product

.. math::

   \prod_{\text{positions}} \frac{w + \beta\,\mathrm{id} + \gamma}{w + \beta\,\sigma + \gamma}

equals 1 whp only if the values are constant on every cycle of the wiring
permutation :math:`\sigma`. The prover commits to the running product
:math:`z(X)` with KZG, and everything reduces to one quotient polynomial
checked at a random point. Its setup is *universal*: one powers-of-tau
string serves every circuit up to a size, with no ceremony per circuit.

.. GENERATED FROM PYTHON SOURCE LINES 22-27

.. code-block:: Python

    import matplotlib.pyplot as plt

    import blockchainkit as bk
    from blockchainkit.proofs.visualizers import plot_copy_constraints








.. GENERATED FROM PYTHON SOURCE LINES 28-30

One setup, two circuits
-----------------------

.. GENERATED FROM PYTHON SOURCE LINES 30-50

.. code-block:: Python


    srs = bk.proofs.trusted_setup(200, secret=24680)
    circuit = bk.proofs.PlonkCircuit()
    x = circuit.variable(3)
    circuit.assert_constant(circuit.add(circuit.mul(circuit.mul(x, x), x), x), 30)  # x**3 + x = 30
    key = bk.proofs.plonk_setup(circuit, srs)
    proof = bk.proofs.plonk_prove(key, srs, circuit.wire_values())
    assert bk.proofs.plonk_verify(key, srs, proof)

    power = bk.proofs.PlonkCircuit()
    v = power.variable(2)
    for _ in range(40):
        v = power.mul(v, power.variable(2))
    power.assert_constant(v, 2**41 % bk.proofs.FIELD_PRIME)
    power_key = bk.proofs.plonk_setup(power, srs)
    assert bk.proofs.plonk_verify(
        power_key, srs, bk.proofs.plonk_prove(power_key, srs, power.wire_values())
    )
    print(f"the same string served circuits of {key.n} and {power_key.n} rows")





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

 .. code-block:: none

    the same string served circuits of 4 and 64 rows




.. GENERATED FROM PYTHON SOURCE LINES 51-55

Breaking a copy constraint
--------------------------
Change the first gate to 2 * 2 = 4: the gate still holds, but x is now 2
there and 3 elsewhere. Only the permutation argument notices.

.. GENERATED FROM PYTHON SOURCE LINES 55-73

.. code-block:: Python


    rows = [list(row) for row in circuit.wire_values()]
    rows[0] = [2, 2, 4]
    honest = bk.proofs.permutation_product(key, circuit.wire_values(), beta=11, gamma=13)
    broken = bk.proofs.permutation_product(key, rows, beta=11, gamma=13)
    print("grand product, honest:", honest[-1], "| broken copy:", broken[-1])
    assert honest[-1] == 1 and broken[-1] != 1
    try:
        bk.proofs.plonk_prove(key, srs, rows)
    except ValueError as error:
        print("the prover cannot continue:", error)

    fig, ax = plt.subplots(figsize=(5, 4.5))
    plot_copy_constraints(key, ax=ax)
    fig.tight_layout()

    plt.show()




.. image-sg:: /api/gallery/proofs/snarks/images/sphx_glr_plot_05_plonk_001.png
   :alt: 4 variables on 12 wire positions
   :srcset: /api/gallery/proofs/snarks/images/sphx_glr_plot_05_plonk_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    grand product, honest: 1 | broken copy: 1729665758
    the prover cannot continue: a copy constraint is not satisfied




.. GENERATED FROM PYTHON SOURCE LINES 74-79

Exercise
--------
A PLONK proof here is 5 points, 14 field elements and 2 more points,
whatever the circuit. Measure ``len(proof.evaluations)`` for both circuits
above, and count how many gates the second circuit needed.


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

   **Total running time of the script:** (0 minutes 0.107 seconds)


.. _sphx_glr_download_api_gallery_proofs_snarks_plot_05_plonk.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/proofs/snarks/plot_05_plonk.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_05_plonk.ipynb <plot_05_plonk.ipynb>`

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

      :download:`Download Python source code: plot_05_plonk.py <plot_05_plonk.py>`

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

      :download:`Download zipped: plot_05_plonk.zip <plot_05_plonk.zip>`


.. only:: html

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

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