Over the last twenty years, an exciting interplay has emerged between proof systems and algorithms. Some natural families of algorithms can be viewed as a generic translation from a proof that a solution exists into an algorithm for finding the solution itself. This connection has perhaps been the most consequential ... more >>>

Given an unpredictable Boolean function $f: \{0, 1\}^n \rightarrow \{0, 1\}$, the standard Yao's XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in [k]}f(x_i)$ given $x_1, ..., x_k \in \{0, 1\}^n$, whereas the Selective XOR lemma is a statement about the unpredictability of computing $\oplus_{i \in S}f(x_i)$ ... more >>>

We present a deterministic algorithm for reconstructing multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuits, i.e. multilinear depth-$4$ circuits with fan-in $k$ at the top $+$ gate. For any fixed $k$, given black-box access to a polynomial $f \in \mathbb{F}[x_{1},x_{2},\ldots ,x_{n}]$ computable by a multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuit of size $s$, the algorithm runs in time ... more >>>