Derandomizing Arthur-Merlin Games using Hitting Sets

We prove that AM (and hence Graph Nonisomorphism) is in NPif for some epsilon > 0, some language in NE intersection coNE requires nondeterministiccircuits of size 2^(epsilon n). This improves recent results of Arvindand K¨obler and of Klivans and Van Melkebeek who proved the sameconclusion, but under stronger hardness assumptions, namely, eitherthe existence of a language in NE intersection coNE which cannot be approximatedby nondeterministic circuits of size less than 2^(epsilon n) or the existenceof a language in NE intersection coNE which requires oracle circuits of size 2^(epsilon n)with oracle gates for SAT (satisfiability).The previous results on derandomizing AM were based on pseudorandomgenerators. In contrast, our approach is based on a strengtheningof Andreev, Clementi and Rolim's hitting set approach to derandomization.As a spin-off, we show that this approach is strong enoughto give an easy (if the existence of explicit dispersers can be assumedknown) proof of the following implication: For some epsilon > 0, if there isa language in E which requires nondeterministic circuits of size 2^(epsilon n),then P=BPP. This differs from Impagliazzo and Wigderson's theorem"only" by replacing deterministic circuits with nondeterministicones