  We develop efficient randomized algorithms to solve the black-box
reconstruction problem for polynomials over finite fields, computable by depth
three arithmetic circuits with alternating addition/multiplication gates, such
that output gate is an addition gate with in-degree two. These circuits compute
polynomials of form $G\times(T_1 + T_2)$, where $G,T_1,T_2$ are product of
affine forms, and polynomials $T_1,T_2$ have no common factors. Rank of such a
circuit is defined as dimension of vector space spanned by all affine factors
of $T_1$ and $T_2$. For any polynomial $f$ computable by such a circuit,
$rank(f)$ is defined to be the minimum rank of any such circuit computing it.
Our work develops randomized reconstruction algorithms which take as input
black-box access to a polynomial $f$ (over finite field $\mathbb{F}$),
computable by such a circuit. Here are the results.
  1 [Low rank]: When $5\leq rank(f) = O(\log^3 d)$, it runs in time
$(nd^{\log^3d}\log |\mathbb{F}|)^{O(1)}$, and, with high probability, outputs a
depth three circuit computing $f$, with top addition gate having in-degree
$\leq d^{rank(f)}$.
  2 [High rank]: When $rank(f) = \Omega(\log^3 d)$, it runs in time $(nd\log
|\mathbb{F}|)^{O(1)}$, and, with high probability, outputs a depth three
circuit computing $f$, with top addition gate having in-degree two.
  Ours is the first blackbox reconstruction algorithm for this circuit class,
that runs in time polynomial in $\log |\mathbb{F}|$. This problem has been
mentioned as an open problem in [GKL12] (STOC 2012)
