Balancing Output Length and Query Bound in Hardness Preserving Constructions of Pseudorandom Functions

Abstract

We revisit hardness-preserving constructions of a pseudo-random function (PRF) from any length doubling pseudo-random generator (PRG) when there is a non-trivial upper bound $q$ on the number of queries that the adversary can make to the PRF. Very recently, Jain, Pietrzak, and Tentes (TCC 2012) gave a hardness-preserving construction of a PRF that makes only $O(\log q)$ calls to the underlying PRG when $q = 2^{n^\epsilon }$ and $\epsilon \ge \frac{1}{2}$. This dramatically improves upon the efficiency of the construction of Goldreich, Goldwasser, and Micali (FOCS 1984). However, they explicitly left open the question of whether such constructions exist when $\epsilon < \frac{1}{2}$. In this work, we give constructions of PRFs that make only $O(\log q)$ calls to the underlying PRG when $q = 2^{n^\epsilon }$, for $0<\epsilon <1$; our PRF outputs $O(n^{2\epsilon })$ bits (on every input), as opposed to the construction of Jain et al. that outputs $n$ bits. That is, our PRF is not length preserving; however it outputs more bits than the PRF of Jain et al. when $\epsilon >\frac{1}{2}$. We obtain our construction through the use of information-theoretic tools such as almost $\alpha$-wise independent hash functions coupled with a novel proof strategy.

Work done while Nishanth Chandran was at Microsoft Research, Redmond.

Work done while Sanjam Garg was visiting Microsoft Research, Redmond. Part of this research conducted while at the IBM Research, T.J. Watson funded by NSF Grant No.1017660.