Small Pseudo-Random Sets Yield Hard Functions: New Tight Explicit Lower Bounds for Branching Programs

Abstract

In several previous works the construction of a computationally hard function with respect to a certain class of algorithms or Boolean circuits has been used to derive small pseudo-random spaces. In this paper, we revert this connection by presenting two new direct relations between the efficient construction of pseudo-random (both two-sided and one-sided) sets for Boolean affine spaces and the explicit construction of Boolean functions having hard branching program complexity

In the case of 1-read branching programs (1-Br.Pr.), we show that the construction of non trivial (i.e. of cardinality 2^o(n)) discrepancy sets (i.e. two-sided pseudo-random sets) for Boolean affine spaces of dimension greater than n/2 yield a set of explicit Boolean functions having very hard 1-Br.Pr. size. By combining the best known construction of ∈-biased sample spaces for linear tests and a simple “Reduction” Lemma, we derive the required discrepancy set and obtain a Boolean function in P having 1-Br.Pr. size not smaller than $2^{n - O\left( {log^2 n} \right)}$ and a Boolean function in DTIME$\left( {2^{O\left( {log^2 n} \right)} } \right)$ having 1-Br.Pr. size not smaller than 2^{n-O(log n)}. The latter bound is optimal and both of them are exponential improvements over the best previously known lower bound that was $2^{n - 3n^{1/2} }$[21].

As for non deterministic syntactic k-read branching programs (k-Br.Pr.), we introduce a new method to derive explicit, exponential lower bounds that involves the construction of hitting sets (one-sided pseudo-random sets) for affine spaces of dimension o(n=2). Using an appropriate “orthogonal” representation of small Boolean affine spaces, we efficiently construct these hitting sets thus obtaining an explicit Boolean function in P that has k-Br.Pr. size not smaller than $2^{n^{1 - o\left( 1 \right)} }$ for any $k = o\left( {\frac{{\log n}}{{\log \log n}}} \right)$. This improves over the previous best known lower bounds given in [8,11,17] for some range of k