On Derandomized Composition of Boolean Functions

The (block-)composition of two Boolean functions f : {0, 1}^m→{0, 1}, g : {0, 1}^n→{0, 1} is the function f ♢ g that takes as inputs m strings x_1, … , x_m∈{0, 1}^n and computes (f ♢ g)(x_1, … , x_m) = f (g(x_1), … , g(x_m)). This operation has been used several times in the past for amplifying different hardness measures of f and g. This comes at a cost: the function f ♢ g has input length m · n rather than m or n , which is a bottleneck for some applications. In this paper, we propose to decrease this cost by “derandomizing” the composition: instead of feeding into f ♢ g independent inputs x_1, … , x_m, we generate x_1, … , x_m using a shorter seed. We show that this idea can be realized in the particular setting of the composition of functions and universal relations (Gavinsky et al. in SIAM J Comput 46(1):114–131, 2017 ; Karchmer et al. in Computat Complex 5(3/4):191–204, 1995b ). To this end, we provide two different techniques for achieving such a derandomization: a technique based on averaging samplers and a technique based on Reed–Solomon codes.