Hardness of Sparse Sets and Minimal Circuit Size Problem

TL;DR

This paper explores the hardness of sparse sets and the minimal circuit size problem, using polynomial methods over finite fields to establish complexity class separations under certain hardness assumptions.

Contribution

It introduces a polynomial method to amplify hardness of sparse sets in nondeterministic time and links the hardness of MCSP to complexity class separations.

Findings

If certain sparse sets lack efficient randomized streaming algorithms, then NEXP is not equal to BPP.

Hardness of MCSP under polynomial-time reductions implies EXP is not equal to ZPP.

Provides new connections between sparse set hardness and fundamental complexity class separations.

Abstract

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 $NEXP\not=BPP$, 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 $EXP\not=ZPP$.