# Asymptotic Nonlinearity of Boolean Functions

Authors
• 1 C.N.R.S. Marseille, Institut de Mathématiques de Luminy, France , (France)
Type
Published Article
Journal
Designs, Codes and Cryptography
Publisher
Publication Date
Jul 01, 2006
Volume
40
Issue
1
Pages
59–70
Identifiers
DOI: 10.1007/s10623-005-6363-8
Source
Springer Nature
Keywords
Boolean functions on the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{F}_{2}^{m}$$\end{document} are not only important in the theory of error-correcting codes, but also in cryptography. In these two cases, the nonlinearity of these functions is a main concept. Carlet, Olejár and Stanek gave an asymptotic lower bound for the nonlinearity of most of them, and I gave an asymptotic upper bound which was strictly larger. In this article, I improve the bounds and get an exact limit for the nonlinearity of most of Boolean functions. This article is inspired by a paper of G. Halász about the related problem of real polynomials with random coefficients.