Secure computation with constant communication overhead using multiplication embeddings

Abstract

Secure multi-party computation (MPC) allows mutually distrusting parties to compute securely over their private data. The hardness of MPC, essentially, lies in performing secure multiplications over suitable algebras. There are several cryptographic resources that help securely compute one multiplication over a large finite field, say GF[2 n ], with linear communication complexity. For example, the computational hardness assumption like noisy Reed-Solomon codewords are pseudorandom. However, it is not known if we can securely compute, say, a linear number of AND-gates from such resources, i.e., a linear number of multiplications over the base field GF[2]. Before our work, we could only perform o(n) secure AND-evaluations. Technically, we construct a perfectly secure protocol that realizes a linear number of multiplication gates over the base field using one multiplication gate over a degree-n extension field. This construction relies on the toolkit provided by algebraic function fields. Using this construction, we obtain the following results. We provide the first construction that computes a linear number of oblivious transfers with linear communication complexity from the computational hardness assumptions like noisy Reed-Solomon codewords are pseudorandom, or arithmetic-analogues of LPN-style assumptions. Next, we highlight the potential of our result for other applications to MPC by constructing the first correlation extractor that has 1 / 2 resilience and produces a linear number of oblivious transfers.