On Derandomized Composition of Boolean Functions

Abstract

The (block-)composition of two Boolean functions $$f : \{0, 1\}^{m} \rightarrow \{0, 1\}, g : \{0, 1\}^{n} \rightarrow \{0, 1\}$$ is the function $$f \diamond g$$ that takes as inputs m strings $$x_{1}, \ldots , x_{m} \in \{0, 1\}^{n}$$ and computes $$(f \diamond g)(x_{1}, \ldots , x_{m}) = f (g(x_{1}), \ldots , g(x_{m})).$$ 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 $$f \diamond g$$ has input length $$m \cdot n$$ 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 $$f \diamond g$$ independent inputs $$x_{1}, \ldots , x_{m},$$ we generate $$x_{1}, \ldots , x_{m}$$ 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.