Group Structure in Correlations and Its Applications in Cryptography

Abstract

Correlated random variables are a key tool in cryptographic applications like secure multi-party computation. We investigate the power of a class of correlations that we term group correlations: A group correlation is a uniform distribution over pairs $(x,y) \in G^2$ such that $x+y\in S$, where $G$ is a (possibly non-abelian) group and $S$ is a subset of $G$. We also introduce bi-affine correlations and show how they relate to group correlations. We present several structural results, new protocols, and applications of these correlations. The new applications include a completeness result for black-box group computation, perfectly secure protocols for evaluating a broad class of black box ``mixed-groups\u27\u27 circuits with bi-affine homomorphism, and new information-theoretic results. Finally, we uncover a striking structure underlying OLE: In particular, we show that OLE over $\mathrm{GF}(2^n)$, is isomorphic to a group correlation over $\mathbb{Z}_4^n$