On the linear structures of Balanced functions and quadratic APN functions

TL;DR

This paper investigates the linear structures of balanced Boolean functions and quadratic APN functions, constructing functions with trivial linear structures and analyzing bent components in quadratic APN functions.

Contribution

It constructs balanced Boolean functions with trivial linear structures and analyzes the bent components of quadratic APN functions in even dimensions.

Findings

Balanced Boolean functions with trivial linear structures are constructed.

Any APN function in even dimension has a component with trivial linear structures.

Bounds on the number of bent components in quadratic APN functions are established.

Abstract

The set of linear structures of most known balanced Boolean functions is nontrivial. In this paper, some balanced Boolean functions whose set of linear structures is trivial are constructed. We show that any APN function in even dimension must have a component whose set of linear structures is trivial. We determine a general form for the number of bent components in quadratic APN functions in even dimension and some bounds on the number are produced. We also count bent components in any quadratic power functions.