Succinct Permanent Is NEXP-Hard with Many Hard Instances

Abstract

The main motivation of this work is to study the average case hardness of the problems which belong to high complexity classes. In more detail, we are interested in provable hard problems which have a big set of hard instances. Moreover, we consider efficient generators of these hard instances of the problems. Our investigation has possible applications in cryptography. As a first step, we consider computational problems from the NEXP class.

We extend techniques presented in [7] in order to develop efficient generation of hard instances of exponentially hard problems. Particularly, for any given polynomial time (deterministic/probabilistic) heuristic claiming to solve NEXP hard problem our procedure finds instances on which the heuristic errs. Then we present techniques for generating hard instances for (super polynomial but) sub exponential time heuristics.

As a concrete example the Succinct Permanent $\bmod \; p$ problem is chosen. First, we prove the NEXP hardness of this problem (via randomized polynomial time reduction). Next, for any given polynomial time heuristic we construct hard instance. Finally, an efficient technique which expands one hard instance to exponential set (in the number of additional bits added to the found instance) of hard instances of the Succinct Permanent $\bmod \; p$ problem is provided.

Partially supported by the Israeli Ministry of Science (Russia Israel grant), Rita Altura Trust Chair in Computer Sciences, Lynne and William Frankel Center for Computer Sciences, Israel Science Foundation (grant number 428/11), Cabarnit Cyber Security MAGNET Consortium, Grant from the Technion’s Institute for Future Defense Technologies Research named for the Medvedi, Shwartzman and Gensler families, and the Israeli Internet Association.