Haiying Liu, Keqin Feng, Rongquan Feng
Several nonexistence results on generalized bent functions $f:Z_{2}^{n} \rightarrow Z_{m}$ presented by using some knowledge on cyclotomic number fields and their imaginary quadratic subfields.
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We obtain new nonexistence results of generalized bent functions from $\{Z^n}_q$ to $\Z_q$ (called type $[n,q]$) in the case that there exist cyclotomic integers in $ \Z[ζ_{q}]$ with absolute value $q^{\frac{n}{2}}$. This result generalize the previous two scattered nonexistence results $[n,q]=[1,2\times7]$ of Pei \cite{Pei} and $[3,2\times 23^e]$ of Jiang-Deng \cite{J-D} to a generalized class. In the last section, we remark that this method can apply to the GBF from $\Z^n_2$ to $\Z_m$.
In this paper, we present some new nonexistence results on $(m,n)$-generalized bent functions, which improved recent results. More precisely, we derive new nonexistence results for general $n$ and $m$ odd or $m \equiv 2 \pmod{4}$, and further explicitly prove nonexistence of $(m,3)$-generalized bent functions for all integers $m$ odd or $m \equiv 2 \pmod{4}$. The main tools we utilized are certain exponents of minimal vanishing sums from applying characters to group ring equations that characterize $(m,n)$-generalized bent functions.
Nonexistence of generalized bent functions and the quadratic norm form equations
本文研究了广义弯曲函数的不存在性，通过证明二次范数形式方程无整数解，得出在特定条件下广义弯曲函数不存在的结论。
In this paper, we study the dual of generalized bent functions $f: V_{n}\rightarrow \mathbb{Z}_{p^k}$ where $V_{n}$ is an $n$-dimensional vector space over $\mathbb{F}_{p}$ and $p$ is an odd prime, $k$ is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The dual of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of \cite{Cesmelioglu5}, we obtain an exp…
A one to one correspondence between regular generalized bent functions from $\F_2^n$ to $\Z_{2^m},$ and $m-$tuples of Boolean bent functions is established. This correspondence maps self-dual (resp. anti-self-dual) generalized bent functions to $m-$tuples of self-dual (resp. anti self-dual) Boolean bent functions. An application to the classification of regular generalized bent functions under the extended affine group is given.
In this paper we introduce generalized hyperbent functions from $F_{2^n}$ to $Z_{2^k}$, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^k}$ consist of components which are generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^{k^\prime}}$ for some $k^\prime < k$. For odd $n$, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$, where the associated Boolean functions are bent.
In this paper we prove that generalized bent (gbent) functions defined on $\mathbb{Z}_2^n$ with values in $\mathbb{Z}_{2^k}$ are regular, and find connections between the (generalized) Walsh spectrum of these functions and their components. We comprehensively characterize generalized bent and semibent functions with values in $\mathbb{Z}_{16}$, which extends earlier results on gbent functions with values in $\mathbb{Z}_4$ and $\mathbb{Z}_8$. We also show that the Gray images of gbent functions with values in $\mathbb{Z}_{2^k}$ are semibent/plateaued when $k=3,4$.
In difference to many recent articles that deal with generalized bent (gbent) functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$ for certain small valued $q\in \{4,8,16 \}$, we give a complete description of these functions for both $n$ even and odd and for any $q=2^k$ in terms of both the necessary and sufficient conditions their component functions need to satisfy. This enables us to completely characterize gbent functions as algebraic objects, namely as affine spaces of bent or semi-bent functions with interesting additional properties, which we in detail describe. We also specify the du…
Codebooks with small inner-product correlation have application in unitary space-time modulations, multiple description coding over erasure channels, direct spread code division multiple access communications, compressed sensing, and coding theory. It is interesting to construct codebooks (asymptotically) achieving the Welch bound or the Levenshtein bound. This paper presented a class of generalized bent $\mathbb{Z}_4$-valued quadratic forms, which contain functions of Heng and Yue (Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$. Cryptogr.…
Octal Bent Generalized Boolean Functions
本文刻画了定义在$\BBZ_2^n$上的值在$\BBZ_8$中的（八进制）Bent广义布尔函数.此外，我们提出了几种结构的广义bent函数的$n$偶数和$n$奇数。
Generalized bent (gbent) functions is a class of functions $f: \mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is a positive integer, that generalizes a concept of classical bent functions through their co-domain extension. A lot of research has recently been devoted towards derivation of the necessary and sufficient conditions when $f$ is represented as a collection of Boolean functions. Nevertheless, apart from the necessary conditions that these component functions are bent when $n$ is even (respectively semi-bent when $n$ is odd), no general construction method has been proposed…
In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$. Explicitly, we describe gbent functions from $\F_2^n$ to $\mathbb{Z}_{2^t}$, which can be seen as a gbent version of Dillon's $PS_{ap}$ class. For the first time, we also introduce the concept of a vectorial gbent function from $\F_2^n$ to $\Z_q^m$, and determine the maximal value which $m$ can attain for the case $q=2^t$. Finally we point to a relation between vectorial gbent functions and relative difference sets.
In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat Walsh-Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph.
On More Bent Functions From Dillon Exponents
本文得到了一类新的由Kloosterman和决定的$p$ -元二项式Bent函数.另外三类函数的双线性性用指数和来刻画，推广了\cite{Linian2013}中的一些结果。在某些特殊情况下，我们还得到了由Kloosterman和决定的Bent函数.
We obtain new non-existence results of generalized bent functions from \ZZ^n_q to \ZZ_q (called type [n,q]). The first case is a class of types where q=2p_1^{r_1}p_2^{r_2}. The second case contains two types [1 <= n <= 3, 2 * 31^e]$ and [1 <= n <= 7,2 * 151^e].
Generalized $p$-ary $\cPS$ Bent Functions
本文利用有限域部分展形构造两类广义$p$元弯曲函数，推广了Lisoněk和Lu的$\cPS^{-}$和$\cPS^{+}$类。
Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those…
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…
A Further Study of Vectorial Dual-Bent Functions
本文进一步研究向量对偶弯曲函数，利用无定形结合方案、线性码、广义Hadamard矩阵和弯曲划分给出新刻画，并建立一般条件下构造结合方案的充要条件，从而构造出更多结合方案。
A Type IV-II Z4-code is a self-dual code over Z4 with the property that all Euclidean weights are divisible by eight and all codewords have even Hamming weight. In this paper we use generalized bent functions for a construction of self-orthogonal codes over Z4 of length $2^m$, for $m$ odd, $m \geq 3$, and prove that for $m \geq 5$ those codes can be extended to Type IV-II Z4-codes. From that family of Type IV-II Z4-codes, we obtain a family of self-dual Type II binary codes by using Gray map. We also consider the weight distributions of the obtained codes and the structure of the supports of t…
Existence of generalized bent functions in the exceptional $q\equiv2\pmod4$, odd-dimensional case
解决Kumar等人的开放问题，在特殊情况\(q\equiv2\pmod4\)且\(d\)为奇数时构造广义弯曲函数，对于满足条件的奇数\(d\)给出显式函数，还表明其傅里叶系数非单位根，否定相关问题。
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, when $k=…
Secondary Constructions of Bent Functions and Highly Nonlinear Resilient Functions
在本文中，我们首先提出了一个新的二级结构的bent函数（建立新的bent函数从两个已经定义的）。此外，我们还应用了以某些特定的bent函数为初始函数的构造，并给出了bent函数的几种具体构造。本文的第二部分致力于弹性函数的构造。给出了构造高非线性弹性函数的间接和构造的一个推广。此外，我们修改了广义结构，以确保高非线性的构造函数。
A new class of hyper-bent Boolean functions in binomial forms
Bent函数，这是最大的非线性布尔函数，偶数的变量和其汉明距离集的所有仿射函数等于$2^{n-1}\pm 2^{\frac{n}{2}-1}$，介绍了Rothaus在1976年时，他认为问题的组合。Bent函数在密码学中有着广泛的应用，如S盒、分组密码和流密码等。此外，它们已被应用于编码理论，扩频和组合设计。超bent函数是Youssef和Gong于2001年提出的一类特殊的bent函数，它具有更强的性质和更少的元素。许多研究集中在bent和hyper-bent函数的构造上。本文考虑由$f_{a,b}:=\mathrm{Tr}_{1}^{n}(ax^{(2^m-1)})+\mathrm{Tr}_{1}^{4}(bx^{\frac{2^n-1}{5}})$在$\mathbb{F}_{2^n}$上定义的函数，其中$n=2m$，$m\equiv 2\pmod 4$，$a\in \mathbb{F}_{2^m}$和$b\in\mathbb{F}_{16}$ .当$a\in \mathbb{F}_{2^m}$和$(b+1)(b^4+b+1)=0$是超线性时，借助于Kloosterman和和$x^5+x+a^{-1}$的因式分解，我们给出了$f_{a,b}$的超线性的一个刻画.进一步，我们利用广义Ramanujan-Nagell方程刻画了$a\in\mathbb{F}_{2^{\frac{m}…
In \cite{Bapic, Tang, Zheng} a new method for the secondary construction of vectorial/Boolean bent functions via the so-called $(P_U)$ property was introduced. In 2018, Qi et al. generalized the methods in \cite{Tang} for the construction of $p$-ary weakly regular bent functions. The objective of this paper is to further generalize these constructions, following the ideas in \cite{Bapic, Zheng}, for secondary constructions of vectorial $p$-ary weakly regular bent and plateaued functions. We also present some infinite families of such functions via the $p$-ary Maiorana-McFarland class. Addition…
Overconstrained character sums over finite abelian groups and decompositions of generalized bent, plateaued and landscape functions
本文研究了有限阿贝尔群上的过约束字符和，提出了一种新的2^ℓ进制表示方法，并探讨了广义弯曲、平坦和景观函数的分解及其性质。
In this paper we define a class of Boolean and generalized Boolean functions defined on $\mathbb{F}_2^n$ with values in $\mathbb{Z}_q$ (mostly, we consider $q=2^k$), which we call landscape functions (whose class containing generalized bent, semibent, and plateaued) and find their complete characterization in terms of their components. In particular, we show that the previously published characterizations of generalized bent and plateaued Boolean functions are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having…
Vectorial Negabent Concepts: Similarities, Differences, and Generalizations
本文澄清了两种向量负弯曲概念的异同，将其推广至广义布尔函数，并利用置换构造了Z_{2^k}-弯曲函数。
On many-to-one property of generalized cyclotomic mappings
本文研究有限域上广义分圆映射的多对一性质，完整刻画了1≤ℓ≤3时的多对一映射，分类了任意因子ℓ下的2对1映射，并构造了多类多对一二项式和三项式。
Constructions and Characterizations of $s$-Plateaued Partitions
本文引入$s$-平铺分划推广弯曲分划，利用其可构造大量$p$元$s$-平铺函数，并刻画了$s$-平铺分划的性质与结构。
Negabent functions as a class of generalized bent functions have attracted a lot of attention recently due to their applications in cryptography and coding theory. In this paper, we consider the constructions of negabent functions over finite fields. First, by using the compositional inverses of certain binomial and trinomial permutations, we present several classes of negabent functions of the form $f(x)=\Tr_1^n(λx^{2^k+1})+\Tr_1^n(ux)\Tr_1^n(vx)$, where $λ\in \F_{2^n}$, $2\leq k\leq n-1$, $(u,v)\in \F^*_{2^n}\times \F^*_{2^n}$, and $\Tr_1^n(\cdot)$ is the trace function from $\F_{2^n}$ to $\…
9-variable Boolean Functions with Nonlinearity 242 in the Generalized Rotation Class
2006年，Kavut，Maitra和Yucel在旋转对称布尔函数（RSBF）类中发现了非线性度为241的9元布尔函数，其非线性度严格大于弯曲级联界240。为了改善这一非线性结果，我们首先定义了n元布尔函数的一些子集作为“k-RSBF和k-DSBF（k-Dihedral Symmetric Boolean Functions）的广义类"，其中k是除n的正整数，k-RSBF是l-RSBF的子集，如果k <1。其次，利用先前用于识别具有非线性241的9变量RSBF的类最速下降迭代启发式搜索算法，我们在3-RSBF和3-DSBF类内进行了搜索。搜索已经完成，找到9个变量的布尔函数与非线性242在这两个类。应该强调的是，尽管3-RSBF类包含具有非线性242的函数;但作为3-RSBF子集的1-RSBF或简单RSBF不包含任何非线性。这个结果还表明一阶Reed-Muller码R（1，9）的覆盖半径至少等于242。第三，由于RSBF在输入向量的特殊置换下是不变的，我们将所有可能的置换分类为布尔函数的线性等价，这些置换是不变的。具体地，对于9变量布尔函数，9！可能的排列被分类为30类;并且搜索算法将这些类中的一些识别为“富”。丰富类产生具有非线性的新布尔函数242，其具有与在广义3-RSBF和3-DSBF类中发现的函数的自相关谱不同的自相关谱。
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