Hardness of Sparse Sets and Minimal Circuit Size Problem

Abstract

We study the magnification of hardness of sparse 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 $\mathrm{NDTIME}(2^{n^{o(1)}})$ that does not have any randomized streaming algorithm with $n^{o(1)}$ updating time, and $n^{o(1)}$ space, then $\mathrm{NEXP}\not =\mathrm{BPP}$, where a f(n)-sparse set is a language that has at most f(n) strings of length n. We also show that if $\mathrm{MCSP}$ is $\mathrm{ZPP}$-hard under polynomial time truth-table reductions, then $\mathrm{EXP}\not =\mathrm{ZPP}$.