Towards efficient constructions of hitting sets that derandomize BPP

A subset H f0; 1g n is a Hitting Set for a class R of boolean functions with n inputs if, for any function f 2 R such that Pr (f = 1) (where 2 (0; 1) is some xed value), there exists an element ~ h 2 H such that f (~ h) = 1. The eecient construction of Hitting Sets for non trivial classes of boolean functions is a fundamental problem in the theory of derandomization. Our paper presents a new method to eeciently construct Hitting Sets for the class of systems of boolean linear functions. Systems of boolean linear functions can be also considered as the algebraic generalization of boolean combinatorial rectangular functions studied by Linial et al in 11]. In the restricted case of boolean rectangular functions, our method (even though completely diierent) achieves equivalent results to those obtained in 11]. Our method gives also an interesting upper bound on the circuit complexity of the solutions of any system of linear equations deened over a nite eld. Furthermore, as preliminary result, we show a new upper bound on the circuit complexity of integer monotone functions that generalizes the upper bound previously obtained by Lupanov in 12].