Quasi-Random Influences of Boolean Functions

TL;DR

This paper establishes an equivalence among various quasi-random properties of Boolean functions, providing a unified framework and constructing examples that demonstrate the hierarchy of these properties.

Contribution

It introduces a hierarchy of quasi-random properties for Boolean functions and proves their equivalence, along with constructing functions that exemplify and distinguish these properties.

Findings

Proved equivalence of multiple quasi-random properties

Constructed families of Boolean functions with specific properties

Separated levels of the quasi-random hierarchy

Abstract

We examine a hierarchy of equivalence classes of quasi-random properties of Boolean Functions. In particular, we prove an equivalence between a number of properties including balanced influences, spectral discrepancy, local strong regularity, homomorphism enumerations of colored or weighted graphs and hypergraphs associated with Boolean functions as well as the $k$th-order strict avalanche criterion amongst others. We further construct families of quasi-random boolean functions which exhibit the properties of our equivalence theorem and separate the levels of our hierarchy.