Construction Of Bent Functions Research Articles

Bent partitions

Spread and partial spread constructions are the most powerful bent function constructions. A large variety of bent functions from a 2m-dimensional vector space $${\mathbb {V}}_{2m}^{(p)}$$ over $${\mathbb {F}}_p$$ into $${\mathbb {F}}_p$$ can be generated, which are constant on the sets of a partition of $${\mathbb {V}}_{2m}^{(p)}$$ obtained with the subspaces of the (partial) spread. Moreover, from spreads one obtains not only bent functions between elementary abelian groups, but bent functions from $${\mathbb {V}}_{2m}^{(p)}$$ to B, where B can be any abelian group of order $$p^k$$ , $$k\le m$$ . As recently shown (Meidl, Pirsic 2021), partitions from spreads are not the only partitions of $${\mathbb {V}}_{2m}^{(2)}$$ , with these remarkable properties. In this article we present first such partitions—other than (partial) spreads—which we call bent partitions, for $${\mathbb {V}}_{2m}^{(p)}$$ , p odd. We investigate general properties of bent partitions, like number and cardinality of the subsets of the partition. We show that with bent partitions we can construct bent functions from $${\mathbb {V}}_{2m}^{(p)}$$ into a cyclic group $${\mathbb {Z}}_{p^k}$$ . With these results, we obtain the first constructions of bent functions from $${\mathbb {V}}_{2m}^{(p)}$$ into $${\mathbb {Z}}_{p^k}$$ , p odd, which provably do not come from (partial) spreads.

Two secondary constructions of bent functions without initial conditions

Two secondary constructions of bent functions without initial conditions

A New Family of Boolean Functions with Good Cryptographic Properties

In 2005, Philippe Guillot presented a new construction of Boolean functions using linear codes as an extension of the Maiorana–McFarland’s (MM) construction of bent functions. In this paper, we study a new family of Boolean functions with cryptographically strong properties, such as non-linearity, propagation criterion, resiliency, and balance. The construction of cryptographically strong Boolean functions is a daunting task, and there is currently a wide range of algebraic techniques and heuristics for constructing such functions; however, these methods can be complex, computationally difficult to implement, and not always produce a sufficient variety of functions. We present in this paper a construction of Boolean functions using algebraic codes following Guillot’s work.

Bent and $${{\mathbb {Z}}}_{2^k}$$-Bent functions from spread-like partitions

Bent functions from a vector space $${{\mathbb {V}}}_n$$ over $${{\mathbb {F}}}_2$$ of even dimension $$n=2m$$ into the cyclic group $${{\mathbb {Z}}}_{2^k}$$ , or equivalently, relative difference sets in $${{\mathbb {V}}}_n\times {{\mathbb {Z}}}_{2^k}$$ with forbidden subgroup $${{\mathbb {Z}}}_{2^k}$$ , can be obtained from spreads of $${{\mathbb {V}}}_n$$ for any $$k\le n/2$$ . In this article, existence and construction of bent functions from $${{\mathbb {V}}}_n$$ to $${{\mathbb {Z}}}_{2^k}$$ , which do not come from the spread construction is investigated. A construction of bent functions from $${{\mathbb {V}}}_n$$ into $${{\mathbb {Z}}}_{2^k}$$ , $$k\le n/6$$ , (and more generally, into any abelian group of order $$2^k$$ ) is obtained from partitions of $${{\mathbb {F}}}_{2^m}\times {{\mathbb {F}}}_{2^m}$$ , which can be seen as a generalization of the Desarguesian spread. As for the spreads, the union of a certain fixed number of sets of these partitions is always the support of a Boolean bent function.

An answer to an open problem of Mesnager on bent functions

In 2014, Mesnager proposed two open problems in [4] on the construction of bent functions. One problem has been settled by Tang et al. in 2017. However, the other is still outstanding, which is solved in this letter by considering a class of PS− vectorial bent functions.

A Construction of Bent Functions on a Finite Group

In this paper, we discuss when a class function on a finite group is a bent function. We have found a necessary condition for a class function on a finite abelian group to be bent. Also, we have found a necessary and sufficient condition for a class function on a finite cyclic group to be bent.

A construction of bent functions with optimal algebraic degree and large symmetric group

As maximal, nonlinear Boolean functions, bent functions have many theoretical and practical applications in combinatorics, coding theory, and cryptography. In this paper, we present a construction of bent function \begin{document}$ f_{a,S} $\end{document} with \begin{document}$ n = 2m $\end{document} variables for any nonzero vector \begin{document}$ a\in \mathbb{F}_{2}^{m} $\end{document} and subset \begin{document}$ S $\end{document} of \begin{document}$ \mathbb{F}_{2}^{m} $\end{document} satisfying \begin{document}$ a+S = S $\end{document} . We give a simple expression of the dual bent function of \begin{document}$ f_{a,S} $\end{document} and prove that \begin{document}$ f_{a,S} $\end{document} has optimal algebraic degree \begin{document}$ m $\end{document} if and only if \begin{document}$ |S|\equiv 2 (\bmod 4) $\end{document} . This construction provides a series of bent functions with optimal algebraic degree and large symmetric group if \begin{document}$ a $\end{document} and \begin{document}$ S $\end{document} are chosen properly. We also give some examples of those bent functions \begin{document}$ f_{a,S} $\end{document} and their dual bent functions.

Several infinite families of <i>p</i>-ary weakly regular bent functions

As an optimal combinatorial object, bent functions have been an interesting research object due to their important applications in cryptography, coding theory, and sequence design. The characterization and construction of bent functions are challenging problems in general. The objective of this paper is to present a construction of p-ary weakly regular bent functions from known weakly regular bent functions. This generalizes some earlier constructions of Boolean bent functions and p-ary bent functions, and produces several infinite families of p-ary weakly regular bent functions from known ones. Some infinite families of p-ary rotation symmetric bent functions are obtained as well.

Fourier transforms and bent functions on finite groups

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis $${\widehat{G}}$$ , consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on $${\widehat{G}}$$ , and obtain characterizations of a bent function by its Fourier transforms (as functions on $${\widehat{G}}$$ ). For a function f from G to another finite group, we define a dual function $${\widetilde{f}}$$ on $${\widehat{G}}$$ , and characterize the nonlinearity of f by its dual function $${\widetilde{f}}$$ . Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.

On communication complexity of bent functions from Maiorana–McFarland class

In this article we study two party Communication Complexity of Boolean bent functions from Maiorana–McFarland class. In particular, we describe connections between Maiorana–McFarland construction of bent functions and operations on matrix form of Boolean functions and show that bent functions of 2n variables from Maiorana–McFarland class have deterministic communication complexity equal n + 1. Finally, we show that not all bent functions have high communication complexity lower bound by giving the example of such function.

On the construction of new bent functions from the max-weight and min-weight functions of old bent functions

Given a bent function $$f(\varvec{x})$$ of n variables, its max-weight and min-weight functions are introduced as the Boolean functions $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ whose supports are the sets $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} | w(f \oplus l_{\varvec{a}}) = 2^{n-1}+2^{\frac{n}{2}-1}\}$$ and $$\{\varvec{a} \in {\mathbb {F}}_{2}^{n} | w(f \oplus l_{\varvec{a}}) = 2^{n-1}-2^{\frac{n}{2}-1}\}$$ respectively, where $$w(f \oplus l_{\varvec{a}})$$ denotes the Hamming weight of the Boolean function $$f(\varvec{x}) \oplus l_{\varvec{a}}(\varvec{x})$$ and $$l_{\varvec{a}}(\varvec{x})$$ is the linear function defined by $$\varvec{a} \in {\mathbb {F}}_{2}^{n}$$ . $${f}^{+}(\varvec{x})$$ and $${f}^{-}(\varvec{x})$$ are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple $$(f_{0}(\varvec{x}), f_{1}(\varvec{x}), f_{2}(\varvec{x}), f_{3}(\varvec{x}))$$ of bent functions of n variables such that $$f_{0}(\varvec{x}) \oplus f_{1}(\varvec{x}) \oplus f_{2}(\varvec{x}) \oplus f_{3}(\varvec{x}) = 1$$ , a bent function of $$n+2$$ variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.

Theory of 3-rotation symmetric cubic Boolean functions

Abstract Rotation symmetric Boolean functions have been extensively studied in the last 15 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about such basic questions as when two such functions are affine equivalent. This question in important in applications, because almost all important properties of Boolean functions (such as Hamming weight, nonlinearity, etc.) are affine invariants, so when searching a set for functions with useful properties, it suffices to consider just one function in each equivalence class. This can greatly reduce computation time. Even for quadratic functions, the analysis of affine equivalence was only completed in 2009. The much more complicated case of cubic functions was completed in the special case of affine equivalence under permutations for monomial rotation symmetric functions in two papers from 2011 and 2014. There has also been recent progress for some special cases for functions of degree > 3 ${> 3}$ . In 2007 it was found that functions satisfying a new notion of k-rotation symmetry for k > 1 (where the case k = 1 is ordinary rotation symmetry) were of substantial interest in cryptography and coding theory. Since then several researchers have used these functions for k = 2 and 3 to study such topics as construction of bent functions, nonlinearity and covering radii of various codes. In this paper we develop a detailed theory for the monomial 3-rotation symmetric cubic functions, extending earlier work for the case k = 2 of these functions.

The essential dependence of Kasami bent functions on the products of variables

Kasami bent functions have the most complicated properties in the class of algebraic constructions of bent functions. We prove that the degree t Kasami functions have nonzero order t − 2 derivatives for 4 ≤ t ≤ (n + 3)/3 and nonzero order t − 3 derivatives for (n + 3)/3 < t ≤ n/2. We establish that the order of essential dependence of Kasami bent functions equals either t − 2 or t − 3.

A construction of bent functions from plateaued functions

In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analyzed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent functions having some additional properties that enable the construction of strongly regular graphs are formed, and explicit expressions for bent functions with maximal degree are presented.

On the construction of bent vectorial functions

This paper is devoted to the constructions of bent vectorial functions, that is, maximally non-linear multi-output Boolean functions. Such functions contribute to an optimal resistance to both linear and differential attacks of those cryptosystems in which they are involved as substitution boxes (S-boxes). We survey, study more in details and generalise the known primary and secondary constructions of bent functions, and we introduce new ones.

On the construction of bent functions of $n+2$ variables from bent functions of $n$ variables

In this paper we present a method to construct iteratively new bent functions of $n + 2$ variables from bent functions of $n$ variables using minterms of $n$ variables and minterms of two variables. Also, we provide the number of bent functions of $n+2$ variables that we can obtain with the method here presented.

Constructions of p-ary Quadratic Bent Functions

Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies the constructions of bent functions having the form $\sum_{i=1}^{(n-1)/2}c_{i}{\mathrm{tr}}_{1}^{n}(x^{p^{i}+1})$ for odd n and $\sum_{i=1}^{n/2-1}c_{i}{\mathrm{tr}}_{1}^{n}(x^{p^{i}+1})+c_{n/2}{\mathrm{tr}}_{1}^{n/2}(x^{p^{n/2}+1})$ for even n, over the finite field $\mathbb{F}_{p^{n}}$ of odd characteristic p, where $c_{i}\in \mathbb{F}_{p}$ . Based on the irreducibility of some polynomials on $\mathbb{F}_{p}$ , we focus on characterizing the bent functions for n=p v q r and n=2p v q r , where $v\geq0,\;r\geq1,\;q$ is an odd prime and p a primitive root modulo q 2. Moreover, the enumerations of those functions are also considered.

Monomial bent functions

In this correspondence, we focus on bent functions of the form F(2/sup n/) /spl rarr/ F(2) where x /spl rarr/ Tr(/spl alpha/x/sup d/). The main contribution of this correspondence is, that we prove that for n=4r, r odd, the exponent d=(2/sup r/+1)/sup 2/ allows the construction of bent functions. This open question has been posed by Canteaut based on computer experiments. As a consequence for each of the well understood families of bent functions, we now know an exponent d that yields to bent functions of the given type.

Construction of bent functions via Niho power functions

A Boolean function with an even number n = 2 k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f ( x ) = tr ( α 1 x d 1 + α 2 x d 2 ) , α 1 , α 2 , x ∈ F 2 n , are considered, where the exponents d i ( i = 1 , 2 ) are of Niho type, i.e. the restriction of x d i on F 2 k is linear. We prove for several pairs of ( d 1 , d 2 ) that f is a bent function, when α 1 and α 2 fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F 2 n are bijective.

Normal Boolean functions

Dobbertin (Construction of bent functions and balanced Boolean functions with high nonlinearity, in: Fast Software Encryption, Lecture Notes in Computer Science, Vol. 1008, Springer, Berlin, 1994, pp. 61–74) introduced the normality of bent functions. His work strengthened the interest for the study of the restrictions of Boolean functions on k-dimensional flats providing the concept of k-normality. Using recent results on the decomposition of any Boolean functions with respect to some subspace, we present several formulations of k-normality. We later focus on some highly linear functions, bent functions and almost optimal functions. We point out that normality is a property for which these two classes are strongly connected. We propose several improvements for checking normality, again based on specific decompositions introduced in Canteaut et al. (IEEE Trans. Inform. Theory, 47(4) (2001) 1494), Canteaut and Charpin (IEEE Trans. Inform. Theory). As an illustration, we show that cubic bent functions of 8 variables are normal.