Succinct Permanent Is NEXP-Hard with Many Hard Instances

Finding a problem that is both hard to solve and hard to solve on many instances is a long standing issue in theoretical computer science. In this work, we prove that the Succinct Permanent mod p is NEXP time hard in the worst case (via randomized polynomial time reduction). We find hard instances for any given heuristic, either by running the heuristic, applying the techniques in [5] to NEXP , or using auxiliary provers to gain a more efficient procedure for generating a hard instance. We provide a technique for building an exponential set (in the number of additional bits added to the found instance) of hard instances of the problem. ∗Partially supported by Lynne and William Frankel Center for Computer Science, the Israeli Ministry of Science (Russia Israel grant), and Rita Altura Trust Chair in Computer Science. Emails:{dolev,fandina}@cs.bgu.ac.il, [email protected]. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 86 (2012)