**Hardness Preserving Constructions of Pseudorandom Functions, Revisited**

*Nishanth Chandran and Sanjam Garg*

**Abstract: **We revisit hardness-preserving constructions of a PRF from any length doubling 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 GGM construction. However, they explicitly left open the question of whether such constructions exist when $\epsilon < \frac{1}{2}$. In this work, we make progress towards answering this question. In particular we give constructions of PRFs that make only $O(\log q)$ calls to the underlying PRG even when $q = 2^{n^\epsilon}$, for $0<\epsilon<\frac{1}{2}$. Our constructions present a tradeoff between the output length of the PRF and the level of hardness preserved. We obtain our construction through the use of {\em almost} $\alpha$-wise independent hash functions coupled with a novel proof strategy.

**Category / Keywords: **foundations /

**Date: **received 31 Oct 2012

**Contact author: **nishanth at cs ucla edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20121101:172718 (All versions of this report)

**Short URL: **ia.cr/2012/616

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