Almost k-Wise Independence and Hard Boolean Functions

Valentine Kabanets

Abstract

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. [ABCR97], 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 {\wedge,\oplus,1}) with the optimal lower bound 2n-log n -O(1) for 1-b.p.'s.

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