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

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 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.