Hardness of Sparse Sets and Minimal Circuit Size Problem
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We develop a polynomial method on finite fields to amplify the hardness of spare sets in nondeterministic time complexity classes on a randomized streaming model. One of our results shows that if there exists a 2^n^o(1)-sparse set in NTIME(2^n^o(1)) that does not have any randomized streaming algorithm with n^o(1) updating time, and n^o(1) space, then NEXP≠BPP, where a f(n)-sparse set is a language that has at most f(n) strings of length n. We also show that if MCSP is ZPP-hard under polynomial time truth-table reductions, then EXP≠ZPP.
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