On the linear structures of Balanced functions and quadratic APN functions
This work addresses theoretical problems in cryptography and coding theory, focusing on the structural properties of Boolean functions, and is incremental as it builds on known results to provide new constructions and bounds.
The paper tackles the problem of constructing balanced Boolean functions with trivial linear structures and analyzes the properties of quadratic APN functions, showing that any APN function in even dimension must have a component with trivial linear structures and deriving bounds on the number of bent components.
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.