Alexandr Polujan, Alexander Pott
In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in $n$ variables for all $n\ge 10$. Moreover, we show that for almost all values of $n$, these functions can simultaneously be homogeneous and have no affine derivatives.
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Bent Functions and the Completed Maiorana-McFarland Class
本文从布尔bent函数与完全Maiorana-McFarland类的关联视角，综述该函数设计与分析的基础成果及最新进展。
In a recent paper \cite{Zhang-Xiao}, Zhang and Xiao describe a technique on constructing almost optimal resilient functions on even number of variables. In this paper, we will present an extensive study of the constructions of almost optimal resilient functions by using the generalized Maiorana-McFarland (GMM) construction technique. It is shown that for any given $m$, it is possible to construct infinitely many $n$-variable ($n$ even), $m$-resilient Boolean functions with nonlinearity equal to $2^{n-1}-2^{n/2-1}-2^{k-1}$ where $k<n/2$. A generalized version of GMM construction is further desc…
Design and analysis of bent functions using $\mathcal{M}$-subspaces
本文研究 Maiorana-McFarland bent 函数的 $\mathcal{M}$-子空间，给出唯一半维子空间的置换条件与构造，并据此构造 $\mathcal{MM}^\\#$ 外的 bent 函数。
On Generalizations of Maiorana-McFarland and $\mathcal{PS}_{ap}$ Functions
本文研究了两种经典布尔弯曲函数构造的推广，提出了一类不在经典MM或$\mathcal{PS}_{ap}$类中的广义Maiorana-McFarland弯曲函数，并探讨了广义$\mathcal{PS}_{ap}$函数的分解性质，证明其无法通过已知的二次构造获得。
When does a bent concatenation not belong to the completed Maiorana-McFarland class?
本文研究弯曲级联何时不属于已完成的Maiorana-McFarland类，通过刻画级联形式的$\mathcal{M}$-子空间结构给出充要条件，并提出显式构造不属于该类弯曲函数的方法。
Bent functions satisfying the dual bent condition and permutations with the $(\mathcal{A}_m)$ property
本文推广了Maiorana-McFarland弯曲函数满足对偶弯曲条件的构造，利用具有$(\mathcal{A}_m)$性质的置换递归构造新弯曲函数，并给出齐次三次弯曲函数的显式构造方法。
Almost Maiorana-McFarland bent functions
本文研究了广义Maiorana-McFarland类中的几乎Maiorana-McFarland bent函数，刻画其bent性质、确定对偶，提出构造算法生成${M}^\n#$外函数，解决开放问题并实现类内外函数的相互生成。
On the Maiorana-McFarland Class Extensions
本文研究Maiorana-McFarland类及其闭包和扩展的基数、度量性质，给出显式公式、上界及渐近紧界，并计算平均仿射子空间数，揭示扩展与闭包间的差异。
Association schemes from vectorial generalized Maiorana-McFarland bent functions with non-weakly regular components
本文研究向量广义Maiorana-McFarland弯曲函数，证明其水平集划分的Fourier-自反性，从而首次从非弱正则分量向量弯曲函数构造出多类平移结合方案，并给出融合构造与反例。
Drawing inspiration from Nyberg's paper~\cite{Nyb91} on perfect nonlinearity and the $c$-differential notion we defined in~\cite{EFRST20}, in this paper we introduce the concept of $c$-differential bent functions in two different ways (thus extending Kumar et al.~\cite{Ku85} classical definition). We further extend the notion of perfect $c$-nonlinear introduced in~\cite{EFRST20}, also in two different ways, and show that, in both cases, the concepts of $c$-differential bent and perfect $c$-nonlinear are equivalent (under some natural restriction of the parameters). Some constructions of functi…
Permutations satisfying $(P_1)$ and $(P_2)$ properties and $\ell$-optimal bent functions
本文深入研究满足$(P_1)$和$(P_2)$性质的置换，构造$\boldsymbol{\rho}$-最优bent函数，生成大量非Maiorana-McFarland类bent函数，解释八维bent函数起源。
The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of $f$, which then allows us to derive sufficient conditions that $f$ is generalized bent for any even $q$. The case when $q$ is not a power of two, which has not been addressed previously, is treated separately and a suitable r…
We study compositional inverses of permutation polynomials, complete mappings, mutually orthogonal Latin squares, and bent vectorial functions. Recently it was obtained in [33] the compositional inverses of linearized permutation binomials over finite fields. It was also noted in [29] that computing inverses of bijections of subspaces have applications in determining the compositional inverses of certain permutation classes related to linearized polynomials. In this paper we obtain compositional inverses of a class of linearized binomials permuting the kernel of the trace map. As an applicatio…
Bent functions of the form $\mathbb{F}_2^n\rightarrow\mathbb{Z}_q$, where $q\geqslant2$ is a positive integer, are known as generalized bent (gbent) functions. Gbent functions for which it is possible to define a dual gbent function are called regular. A regular gbent function is said to be self-dual if it coincides with its dual. In this paper we explore self-dual gbent functions for even $q$. We consider several primary and secondary constructions of such functions. It is proved that the numbers of self-dual and anti-self dual gbent functions coincide. We give necessary and sufficient condit…
An asymptotic lower bound on the number of bent functions
本文通过改进 Maiorana–McFarland bent 函数族并结合 latin squares 与 hypercubes 中 transversal 计数进展，建立了 Boolean bent 函数数量的新渐近下界。
We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials is either $1$ or $2$. We then count the resulting sets …
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