Synthesizers and their application to the parallel construction of pseudo-random functions

We present a new cryptographic primitive called pseudo-random synthesizer and show how to use it in order to get a parallel construction of a pseudo-random function. We show an NC¹ implementation of pseudo-random synthesizers based on the RSA or the Diffie-Hellman assumptions. This yields the first parallel (NC² ) pseudo-random function and the only alternative to the original construction of Goldreich, Gold-wasser and Micali (GGM). The security of our constructions is similar to the security of the underling assumptions. We discuss the connection with problems in computational learning theory