Paper 2012/616 Original: eprint.iacr.org/2012/616 ↗

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 $\epsilon \geq \frac{1}{2}$. 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 {\em et al.} that outputs $n$ bits. That is, our PRF is not length preserving; however it outputs more bits than the PRF of Jain {\em et al.} when $\epsilon>\frac{1}{2}$. We obtain our construction through the use of information-theoretic tools such as {\em almost} $\alpha$-wise independent hash functions coupled with a novel proof strategy.