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Group Structure in Correlations and its Applications in Cryptography

Correlated random variables are a key tool in cryptographic applications like secure multiparty 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) ∈ G such that x + y ∈ S, where G is a (possibly non-abelian) group and S is a subset of G. We also introduce bi-a ne 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 circuits with bi-a ne homomorphisms, and new information-theoretic results. Finally, we uncover a striking structure underlying OLE: In particular, we show that OLE over F2n , is isomorphic to a group correlation over Z4 .