Abstract
A circuit C compresses a function f:{0,1}^n > {0,1}^m if given an input x in {0,1}^n the circuit C can shrink x to a shorter lbit string x' such that later, a computationallyunbounded solver D will be able to compute f(x) based on x'. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size s=n^c. Motivated by cryptographic applications, we focus on averagecase (l,epsilon) incompressibility, which guarantees that on a random input x in {0,1}^n, for every size s circuit C:{0,1}^n > {0,1}^l and any unbounded solver D, the success probability Pr_x[D(C(x))=f(x)] is upperbounded by 2^(m)+epsilon. While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work we present the following results:
1. Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function f:{0,1}^n > {0,1} which is incompressible by size n^c circuits with communication l=(1o(1)) * n and error epsilon=n^(c). Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14).
2. We show that it is possible to achieve negligible error parameter epsilon=n^(omega(1)) for nonboolean functions. Specifically, assuming that E is hard for exponential size Sigma_3circuits, we construct a nonboolean function f:{0,1}^n > {0,1}^m which is incompressible by size n^c circuits with l=Omega(n) and extremely small epsilon=n^(c) * 2^(m). Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest.
3. We show that the task of constructing an incompressible boolean function f:{0,1}^n > {0,1} with negligible error parameter epsilon cannot be achieved by "existing proof techniques". Namely, nondeterministic reductions (or even Sigma_i reductions) cannot get epsilon=n^(omega(1)) for boolean incompressible functions. Our results also apply to constructions of standard NisanWigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospective, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (SIAM J. Comp., 2010).
BibTeX  Entry
@InProceedings{applebaum_et_al:LIPIcs:2015:5056,
author = {Benny Applebaum and Sergei Artemenko and Ronen Shaltiel and Guang Yang},
title = {{Incompressible Functions, RelativeError Extractors, and the Power of Nondeterministic Reductions (Extended Abstract)}},
booktitle = {30th Conference on Computational Complexity (CCC 2015)},
pages = {582600},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897811},
ISSN = {18688969},
year = {2015},
volume = {33},
editor = {David Zuckerman},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5056},
URN = {urn:nbn:de:0030drops50567},
doi = {10.4230/LIPIcs.CCC.2015.582},
annote = {Keywords: compression, pseudorandomness, extractors, nondeterministic reductions}
}
Keywords: 

compression, pseudorandomness, extractors, nondeterministic reductions 
Collection: 

30th Conference on Computational Complexity (CCC 2015) 
Issue Date: 

2015 
Date of publication: 

06.06.2015 