Characterizations of o-polynomials by the Walsh transform

TL;DR

This paper characterizes o-polynomials, which are significant in cryptography and coding theory, using the Walsh transform of their associated vectorial functions, providing new insights into their structure.

Contribution

It introduces a novel characterization of o-polynomials through Walsh transform analysis, linking geometric objects to cryptographic functions.

Findings

O-polynomials can be characterized by their Walsh transform.

The Walsh transform provides a new tool for analyzing o-polynomials.

This characterization aids in designing cryptographic functions and codes.

Abstract

The notion of o-polynomial comes from finite projective geometry. In 2011 and later, it has been shown that those objects play an important role in symmetric cryptography and coding theory to design bent Boolean functions, bent vectorial Boolean functions, semi-bent functions and to construct good linear codes. In this note, we characterize o-polynomials by the Walsh transform of the associated vectorial functions.