**Saving Private Randomness in One-Way Functions and Pseudorandom Generators**

*Nenad Dedic and Danny Harnik and Leonid Reyzin*

**Abstract: **Can a one-way function f on n input bits be used
with fewer than $n$ bits while retaining comparable hardness of
inversion? We show that the answer to this fundamental question is
negative, if one is limited black-box reductions.

Instead, we ask whether one can save on secret random bits at the expense of more public random bits. Using a shorter secret input is highly desirable, not only because it saves resources, but also because it can yield tighter reductions from higher-level primitives to one-way functions. Our first main result shows that if the number of output elements of f is at most $2^k$, then a simple construction using pairwise-independent hash functions results in a new one-way function that uses only k secret bits. We also demonstrate that it is not the knowledge of security of f, but rather of its structure, that enables the savings: a black-box reduction cannot, for a general f, reduce the secret-input length, even given the knowledge that security of f is only $2^{-k}$; nor can a black-box reduction use fewer than k secret input bits when f has $2^k$ distinct outputs.

Our second main result is an application of the public-randomness approach: we show a construction of a pseudorandom generator based on any regular one-way function with output range of known size $2^k$. The construction requires a seed of only 2n+O(k\log k) bits (as opposed to O(n \log n) in previous constructions); the savings come from the reusability of public randomness. The secret part of the seed is of length only k (as opposed to n in previous constructions), less than the length of the one-way function input.

**Category / Keywords: **foundations / pseudorandomness, one-way function, randomized iterate, pseudorandom generator, regular one-way function

**Publication Info: **This is the full version of TCC 2008 paper.

**Date: **received 7 Dec 2007, last revised 10 Dec 2007

**Contact author: **nenad dedic at gmail com

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Note: **PDF rendering problems corrected. Affiliations updated.

**Version: **20071210:163944 (All versions of this report)

**Short URL: **ia.cr/2007/458

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