  Parent-identifying set system is a kind of combinatorial structures with
applications to broadcast encryption. In this paper we investigate the maximum
number of blocks $I_2(n,4)$ in a $2$-parent-identifying set system with ground
set size $n$ and block size $4$. The previous best known lower bound states
that $I_2(n,4)=\Omega(n^{4/3+o(1)})$. We improve this lower bound by showing
that $I_2(n,4)= \Omega(n^{3/2-o(1)})$ using techniques in additive number
theory.
