Smooth Projective Hashing and Two-Message Oblivious Transfer

Abstract

We present a general framework for constructing two-message oblivious transfer protocols using a modification of Cramer and Shoup’s notion of smooth projective hashing (2002). Our framework is actually an abstraction of the two-message oblivious transfer protocols of Naor and Pinkas (2001) and Aiello et al. (2001), whose security is based on the Decisional Diffie Hellman Assumption. In particular, we give two new oblivious transfer protocols. The security of one is based on the N’th-Residuosity Assumption, and the security of the other is based on both the Quadratic Residuosity Assumption and the Extended Riemann Hypothesis. Our security guarantees are not simulation based, and are similar to those of previous constructions.

When using smooth projective hashing in this context, we must deal with maliciously chosen smooth projective hash families. This raises new technical difficulties, and in particular it is here that the Extended Riemann Hypothesis comes into play.