Derandomizing Arthur-Merlin games using hitting sets

We prove that AM (and hence Graph Nonisomorphism) is in NP if for some ε>0, some language in NE∩ coNE requires nondeterministic circuits of size 2^en. This improves results of Arvind and Kobler (1997) and of Klivans and Van Melkebeek (1999) who have proven the same conclusion, but under stronger hardness assumptions, namely, either the existence of a language in NE∩ coNE which cannot be approximated by nondeterministic circuits of size less than 2^en or the existence of a language in NE∩ coNE which requires oracle circuits of size 2^en with oracle gates for SAT (satisfiability). The previous results on derandomizing AM were based on pseudorandom generators. In contrast, our approach is based on a strengthening of Andreev, Clementi and Rolim´s (1996) hitting set approach to derandomization. As a spin-off we show that this approach is strong enough to give an easy (if the existence of explicit dispersers can be assumed known) proof of the following implication: for some ε>0, if there is a language in E which requires nondeterministic circuits of size 2^en, then P=BPP. This differs from Impagliazzo and Wigderson´s (1995) theorem “only” by replacing deterministic circuits with nondeterministic ones