## Cryptology ePrint Archive: Report 2007/458

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.

Short URL: ia.cr/2007/458

[ Cryptology ePrint archive ]