Almost k-Wise independence and hard Boolean functions

Valentine Kabanets


We construct Boolean functions (computable by polynomial-size circuits) with large lower bounds for read-once branching program (1-b.p.'s): a function in P with the lower bound 2n-polylog(n), a function in quasipolynomial time with the lower bound 2n-O(log n), and a function in LINSPACE with the lower bound 2n-log n-O(1). Our constructions are simpler than those of Andreev et al., as we apply the idea of almost k-wise independence more directly, without the use of discrepancy set generators for large affine subspaces. The simplicity of our constructions also allows us to observe that there exists a Boolean function in AC0[2] (computable by a depth 3, polynomial-size circuit over the basis {∧,⊕,1}) with the optimal lower bound 2n-log n-O(1) for 1-b.p.'s.