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 (Advances in Cryptology-EUROCRYPT'02, Lecture Notes in Computer Science, vol. 2332, pp. 45-64, Springer, Berlin, 2002). This framework is an abstraction of the two-message oblivious transfer protocols of Naor and Pinkas (SODA'01, pp. 448-457, ACM, New York, 2001) and Aiello et al. (Advances in Cryptology-EUROCRYPT'01, Lecture Notes in Computer Science, vol. 2045, pp. 119-135, Springer, Berlin, 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 Quadratic Residuosity Assumption, and the security of the other is based on the Nth Residuosity Assumption. Compared to other applications of smooth projective hashing, in our context we must deal also with maliciously chosen parameters, which raises new technical difficulties. We also improve on prior constructions of factoring-based smooth universal hashing, in that our constructions do not require that the underlying RSA modulus is a product of safe primes. (This holds for the schemes based on the Quadratic Residuosity Assumption as well as the ones based on the Nth Residuosity Assumption.) In fact, we observe that the safe-prime requirement is unnecessary for many prior constructions. In particular, the factoring-based CCA secure encryption schemes due to Cramer-Shoup, Gennaro-Lindell, and Camenisch-Shoup remain secure even if the underlying RSA modulus is not a product of safe primes. © 2010 International Association for Cryptologic Research.