  Separating hash families are useful combinatorial structures which generalize
several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$
denote the maximum size of universe for a $t$-perfect hash family of length $N$
over an alphabet of size $q$. In this paper, we show that $q^{2-o(1)}<p_t(t,
q)=o(q^2)$ for all $t\geq 3$, which answers an open problem about separating
hash families raised by Blackburn et al. in 2008 for certain parameters.
Previously, this result was known only for $t=3, 4$. Our proof is obtained by
establishing the existence of a large set of integers avoiding nontrivial
solutions to a set of correlated linear equations.
