Generalisations of the bent property of a boolean function are presented, by proposing spectral analysis with respect to a well-chosen set of local unitary transforms. Quadratic boolean functions are related to simple graphs and it is shown that the orbit generated by successive Local Complementations on a graph can be found within the transform spectra under investigation. The flat spectra of a quadratic boolean function are related to modified versions of its associated adjacency matrix.
On design-theoretic aspects of Boolean and vectorial bent functions
There are two construction methods of designs from $(n,m)$ -bent functions, known as translation and addition designs. In this paper we analyze, which equivalence relation for Boolean bent functions, i.e. $(n,1)$ -bent functions, and vectorial bent functions, i.e. $(n,m)$ -bent functions with $2\le m\le n/2$ , is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial $(n,m)$ -bent functions and isomorphism of addition designs are the same concepts …
A general framework for secondary constructions of bent and plateaued functions
In this work, we employ the concept of { \em composite representation} of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying new secondary constructions of bent/plateaued functions. This representation gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects. This framework essentially gives rise to an { \em infinite number} of possibilities to specify such secondary construction methods …
Bent Functions and the Completed Maiorana-McFarland Class
University of Primorska, FAMNIT & IAM（波希米亚大学 FAMNIT 与 IAM）；Otto von Guericke University（奥托·冯·格里克大学）；School of Cyber Engineering, Xidian University（西安电子科技大学网络空间安全学院）
Generic constructions of 5-valued spectra Boolean functions
Whereas the design and properties of bent and plateaued functions have been frequently addressed during the past few decades, there are only a few design methods of so-called 5-valued spectra Boolean functions whose Walsh spectra takes the values in $\{0, \pm 2^{λ_1}, \pm 2^{λ_2}\}$ . Moreover, these design methods mainly regards the specification of these functions in their ANF (algebraic normal form ) domain. In this article, we give a precise characterization of this class of functions in their spectral domain using the concept of a dual of plateaued functions. Both necessary and sufficient…
Designing plateaued Boolean functions in spectral domain and their classification
The design of plateaued functions over $GF(2)^n$ , also known as 3-valued Walsh spectra functions (taking the values from the set $\{0, \pm 2^{\lceil \frac{n+s}{2} \rceil}\}$ ), has been commonly approached by specifying a suitable algebraic normal form which then induces this particular Walsh spectral characterization. In this article, we consider the reversed design method which specifies these functions in the spectral domain by specifying a suitable allocation of the nonzero spectral values and their signs. We analyze the properties of trivial and nontrivial plateaued functions (as affine …
Some New Constructions of Generalized Plateaued Functions
Plateaued functions as an extension of bent functions play a significant role in cryptography, coding theory, sequences and combinatorics. In \cite{Mesnager9} , Mesnager \emph{et al.} introduced generalized plateaued functions in order to study plateaued functions in the general context of generalized $p$ -ary functions. In this paper, we focus on the constructions of generalized $p$ -ary $s$ -plateaued functions from $V_{n}$ to $\mathbb{Z}_{p^k}$ , where $V_{n}$ is an $n$ -dimensional vector space over $\mathbb{F}_{p}$ , $p$ is a prime, $k\geq 1$ and $n+s$ is even when $p=2$ . In particular, …
The Maximum Nonlinearity of 8-variable Balanced Boolean Functions is 116
School of Computer and Electronic Information/School of Artificial Intelligence, Nanjing Normal University（南京师范大学计算机与电子信息学院/人工智能学院）；China Telecom Corporation Limited, Huaibei Branch（中国电信股份有限公司淮北分公司）
Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions
Faculty of Engineering, Özyĕğin University（Özyeğin大学工程学院）；Department of Computer Science, Electrical Engineering and Mathematical Sciences, Western Norway University of Applied Sciences（西挪威应用科学大学计算机科学、电气工程和数学科学系）；Applied Mathematics Department, Naval Postgraduate School（海军研究生院应用数学系）
