In this work we prove a high dimensional analogue of the beloved Goldreich-Levin theorem (STOC 1989). We consider the following algorithmic problem: given oracle access to a function $f:\mathbb{Z}_q^m\rightarrow\mathbb{Z}_q^n$ such that ${\rm Prob}_{{\bf x}\sim\mathbb{Z}_q^m}\bigl[f({\bf x})={\bf Ax}\bigr]\geq\varepsilon$ for some ${\bf A}\in\mathbb{Z}_q^{n\times m}$ and $\varepsilon>0$, recover ${\bf A}$ (or a list of ... more >>>

Threshold cryptography is typically based on the idea of secret-sharing a private-key $s\in F$ ``in the exponent'' of some cryptographic group $G$, or more generally, encoding $s$ in some linearly homomorphic domain. In each invocation of the threshold system (e.g., for signing or decrypting) an ``encoding'' of the secret is ... more >>>

The random $\Delta$-CNF model is one of the most important distribution over $\Delta\text{-}\mathrm{SAT}$ instances. It is closely connected to various areas of computer science, statistical physics, and is a benchmark for satisfiability algorithms. Fleming, Pankratov, Pitassi, and Robere and independently Hrubes and Pudlak showed that when $\Delta = \Theta(\log n)$, ... more >>>