Efficient reconstruction of depth three circuits with top fan-in two

Abstract

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)