Information-theoretically secure oblivious polynomial evaluation in the commodity-based model

Oblivious polynomial evaluation (OPE) consists of a two-party protocol where a sender inputs a polynomial \(p(x)\) and a receiver inputs a single value \(x_{0}\). At the end of the protocol, the sender learns nothing and the receiver learns \(p(x_{0})\). This paper deals with the problem of oblivious polynomial evaluation under an information-theoretic perspective, which is based on the definitions of unconditional security developed by Crépeau et al. (Information-theoretic conditions for two-party secure function evaluation. EUROCRYPT 2006, LNCS 4004. Springer, Berlin, Heidelberg, pp 538–554, 2006). In this paper, we propose an information-theoretic model for oblivious polynomial evaluation relying on pre-distributed data and prove very general lower bounds on the size of the pre-distributed data, as well as the size of the communications in any protocol. It is demonstrated that these bounds are tight by obtaining a round-optimal OPE protocol, which meets the lower bounds simultaneously. We present a natural generalization to OPE called oblivious linear functional evaluation.