Synthesizers and Their Application to the Parallel Construction of Pseudo-Random Functions

A pseudo-random function is a fundamental cryptographic primitive that is essential for encryption, identi cation and authentication. We present a new cryptographic primitive called pseudorandom synthesizer and show how to use it in order to get a parallel construction of a pseudo-random function. We show several NC 1 implementations of synthesizers based on concrete intractability assumptions as factoring and the Di e-Hellman assumption. This yields the rst parallel pseudorandom functions (based on standard intractability assumptions) and the only alternative to the original construction of Goldreich, Goldwasser and Micali. In addition, we show parallel constructions of synthesizers based on other primitives such as weak pseudo-random functions or trapdoor one-way permutations. The security of all our constructions is similar to the security of the underlying assumptions. The connection with problems in Computational Learning Theory is discussed. A preliminary version of this paper appeared at the Proc. 35th IEEE Symp. on Foundations of Computer Science (1995) pp. 170-181 y Incumbent of the Morris and Rose Goldman Career Development Chair, Dept. of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 76100, Israel. Research supported by BSF grant no. 94-00032 and a grant from the Israel Science Foundation administered by the Israeli Academy of Sciences. E-mail: [email protected]. z Dept. of Applied Mathematics and Computer Science, Weizmann Institute of Science, Rehovot 76100, Israel. Research supported by a Clore Scholars award and by a grant from the Israel Science Foundation administered by the Israeli Academy of Sciences. E-mail: [email protected].