  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 $EXP\not=ZPP$.
