# [Resource Topic] 2009/094: On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions

Welcome to the resource topic for 2009/094

Title:
On the Lower Bounds of the Second Order Nonlinearity of some Boolean Functions

Authors: Sugata Gangopadhyay, Sumanta Sarkar, Ruchi Telang

Abstract:

The r-th order nonlinearity of a Boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers. It is also important in coding theory as it is related to the covering radius of the Reed-Muller code \mathcal{R}(r, n). In this paper we deduce the lower bounds of the second order nonlinearity of the two classes of Boolean functions of the form \begin{enumerate} \item f_{\lambda}(x) = Tr_1^n(\lambda x^{d}) with d=2^{2r}+2^{r}+1 and \lambda \in \mathbb{F}_{2^{n}} where n = 6r. \item f(x,y)=Tr_1^t(xy^{2^{i}+1}) where x,y \in \mathbb{F}_{2^{t}}, n = 2t, n \ge 6 and i is an integer such that 1\le i < t, \gcd(2^t-1, 2^i+1) = 1. \end{enumerate} For some \lambda, the first class gives bent functions whereas Boolean functions of the second class are all bent, i.e., they achieve optimum first order nonlinearity.

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