Superpolynomial Lower Bounds for Monotone Span Programs

In this paper we obtain the first superpolynomial lower bounds for {\it monotone span programs} computing explicit functions. The best previous lower bound was $\Omega(n^{5/2})$ by Beimel, G\''al, Paterson \cite{BGP}; our proof exploits a general combinatorial lower bound criterion from that paper. Our lower bounds are based on an analysis of Paley-type bipartite graphs via Weil''s character sum estimates. We prove an $n^{\Omega ( \log n / \log\log n)}$ lower bound for an explicit family of monotone Boolean functions in $n$ variables, which implies the same lower bound for the size of monotone span programs for the clique problem. Our results give the first superpolynomial lower bounds for linear secret sharing schemes. We demonstrate the surprising power of monotone span programs by exhibiting a function computable in this model in linear size while requiring superpolynomial size monotone circuits and exponential size monotone formulae. We also show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields.