Electronic Colloquium on Computational Complexity, Report No. 22 (2006) On the Compressibility of N P Instances and Cryptographic Applications ‡

Abstract

We initiate the study of the compressibility of N P problems. We consider N P problems that have long instances but relatively short witnesses. The question is, can one efficiently compress an instance and store a shorter representation that maintains the information of whether the original input is in the language or not. We want the length of the compressed instance to be polynomial in the length of the witness rather than the length of original input. Such compression enables to succinctly store instances until a future setting will allow solving them, either via a technological or algorithmic breakthrough or simply until enough time has elapsed. We give a new classification of N P with respect to compression. This classification forms a stratification of N P that we call the VC hierarchy. The hierarchy is based on a new type of reduction called W-reduction and there are compression-complete problems for each class. Our motivation for studying this issue stem from the vast cryptographic implications compressibility has. For example, suppose that SAT is compressible, that is there exist a polynomial p(·, ·) so that given a formula consisting of m clauses over n variables it is possible to come up with an equivalent (w.r.t satisfiability) formula of size at most p(n, logm). Then if the reduction is what we call witness retrievabl