:mod:`qap`
==========

.. py:module:: qap


Module Contents
---------------

Classes
~~~~~~~

.. autoapisummary::

   qap.QAP



.. py:class:: QAP(A=Iterable, B=Iterable, C=Iterable)

   Given $n$ equations we pick arbitrary distinct $r_1,\cdots,r_n \in \mathbb{F}$ and define
   $$
   t(x) = \prod_{q=1}^n (x - r_q)
   $$
   Since $t(X)$ is the lowest degreee monomial with $t(r_q) = 0 $ in each point, we can refomulate as:
   $$
   \sum_{i=0}^m a_i u_i(X) \circ \sum_{i=0}^m a_i v_i(X) = \sum_{i=0}^m a_i w_i(X) \mod t(X)
   $$
   we will be working with quadratic arithmetic programsRthat have thefollowing description

   $$
   R = (\mathbb{F}, aux, l, \{u_i(X), v_i(X), w_i(X)\}_{i=0}^n, t(X))
   $$

   .. method:: O(self, ws)


   .. method:: H(self, ws)


   .. method:: qap(self)
      :property:


   .. method:: proof(self, x: FiniteField, s: Iterable[FiniteField], start=0, end=None)

      c: Callange
      s: witness vertex


   .. method:: verify(s, A, B, C, Z, H)