Zero-difference Balanced Functions Research Articles

The preimage distributions of a class of zero-difference balanced functions and their partitioned difference families

By further exploring some ideas in Ding and Yin, the results on preimage distributions of a class of zero-difference balanced functions are extended and the parameters of a class of partitioned difference families are determined by means of self-conjugate Gauss sums over finite fields.

A Note on the Zero-Difference Balanced Functions with New Parameters

A Note on the Zero-Difference Balanced Functions with New Parameters

Partitioned difference families versus zero-difference balanced functions

In a recent paper by the first author in this journal it was pointed out that the literature on zero-difference balanced functions is often repetitive and of little value. Indeed it was shown that some papers published in the last decade on this topic reproduced in a very convoluted way simple results on difference families which were known since the 90s or even earlier. In spite of this fact, unfortunately, a new paper of the same kind has recently appeared in this journal. Its main result was indeed already obtained by Furino in 1991 and here it will be shown that it is only a very special case of a much more general result by the first author. We take this opportunity to make a comparison between the equivalent notions of a partitioned difference family (PDF) and a zero-difference balanced function (ZDBF), explaining the reasons for which we prefer to adopt the terminology and notation of PDFs. Finally, “playing” with some known results on difference families, we produce a plethora of disjoint difference families with new parameters. Each of them can be viewed as a PDF with many blocks of size 1; therefore, even though the ZDBF community do not appear concerned about this, they are not so relevant from the design theory perspective. The main goal of this note is to explain the relationships between ZDBFs and the prior research, giving an example of how seemingly novel ZBDF results can be readily obtained from well known results on difference families.

A new construction of zero-difference balanced functions and two applications

Zero-difference balanced (ZDB) functions are a generalization of perfect nonlinear functions, and have received a lot of attention due to their important applications in coding theory, cryptography, combinatorics and some engineering areas. In this paper, based on cyclotomy and generalized cyclotomy, a construction of a partitioned difference family is presented, and then a class of ZDB functions is obtained. In addition, these ZDB functions are applied to construct optimal constant composition codes and optimal and perfect difference systems of sets.

New classes of generalized zero-difference balanced functions and their applications

ABSTRACTGeneralized zero-difference balanced (G-ZDB) function is a generalization of zero-difference balanced (ZDB) function, which has many important applications in coding and cryptography. In this paper, three new classes of G-ZDB functions are presented based on the generalized cyclotomic classes. Our constructions give new parameters of G-ZDB functions not covered in the literature. As applications, frequency-hopping sequences with optimal Hamming autocorrelation are obtained by using the newly constructed G-ZDB functions.

More Optimal Difference Systems of Sets and Frequency-Hopping Sequences From Zero-Difference Functions

Difference systems of sets (DSS) and frequency-hopping sequences (FHS) are two objects with many applications in wireless communication. Zero-difference balanced function (ZDBF) and near zero-difference balanced functions (NZDBF) are two types of functions which can be used to obtain optimal DSSs and FHSs. In order to obtain more optimal DSSs and FHSs, zero-difference function (ZDF) as a generalization of ZDBF and NZDBF was recently proposed. In this paper, four classes of ZDFs with good applications are given from some known ZDBFs. It is noticed that these ZDFs are neither ZDBFs nor NZDBFs. As a result, more optimal DSSs and FHSs with new flexible parameters are obtained.

A Note on Two Constructions of Zero-Difference Balanced Functions

Notes on two constructions of zero-difference balanced (ZDB) functions are made in this letter. Then ZDB functions over $\mathbb{Z}_{e}\times \prod_{i=0}^{k}{\mathbb{F}_{q_i}}$ are obtained. And it shows that all the known ZDB functions using cyclotomic cosets over $\mathbb{Z}_{n}$ are special cases of a generic construction. Moreover, applications of these ZDB functions are presented.

New Constructions of Zero-Difference Balanced Functions

New Constructions of Zero-Difference Balanced Functions

Optimal FHSs and DSSs via near zero-difference balanced functions

Zero-difference balanced (ZDB) functions were introduced by Ding in connection with constructions of optimal constant composition codes and optimal and perfect difference systems of sets. Based on such functions, people have constructed optimal constant weight codes and optimal frequency-hopping sequences. In order to obtain more optimal cryptographic objects, the zero-difference balanced (ZDB) function is generalized to the near zero-difference balanced (N-ZDB) function in the present paper, whose characterizations are partially given. Furthermore, we prove that near zero-difference balanced (N-ZDB) functions are equivalent to partitioned almost difference families (PADFs) in design theory. As the main contribution of this paper, three classes of the N-ZDB functions are proposed by means of the partition of Zn, where n is an odd positive integer. Employing these N-ZDB functions, we obtain at the same time optimal frequency-hopping sequences and optimal difference systems of sets with flexible parameters.

A new class of zero-difference balanced functions

In this short paper, we generalize the cyclotomic constructions of zero-difference balanced (ZDB) functions proposed by Zha and Hu [7] to generate ZDB functions with higher definition domain.

A generic method to construct zero-difference balanced functions

Zero-difference balanced (ZDB) function plays an important role in communication field. In this paper, we propose a generic method to construct ZDB functions on generic algebraic rings. Using this method, we construct many new ZDB functions and retrieve some existing ZDB functions in a much simpler way. Moreover, new applications of the constructed ZDB functions, such as constructing optimal constant weight codes and optimal frequency-hopping sequences, are presented.

Zero-Difference Balanced Functions With New Parameters and Their Applications

As an optimal combinatorial object, zero-difference balanced (ZDB) functions introduced by Ding in 2008, are a generalization of the well-known perfect nonlinear functions. ZDB functions have received much attention in recent years due to its important applications in coding theory and sequence design. One objective of this paper is to present a construction of ZDB functions based on a kind of generalized cyclotomy. It generates ZDB functions over cyclic group with new parameters which can not be produced by earlier constructions. Another objective of this paper is to employ these ZDB functions to obtain at the same time: 1) optimal constant-composition codes; 2) perfect difference systems of sets; and 3) optimal frequency-hopping sequences, all with new parameters.

Two Classes of Optimal Constant Composition Codes from Zero Difference Balanced Functions

Two Classes of Optimal Constant Composition Codes from Zero Difference Balanced Functions

Two Classes of New Zero Difference Balanced Functions from Difference Balanced Functions and Perfect Ternary Sequences

Two Classes of New Zero Difference Balanced Functions from Difference Balanced Functions and Perfect Ternary Sequences

Some New Constructions for Generalized Zero-Difference Balanced Functions

Zero-difference balanced (ZDB) functions have many applications in coding theory, cryptography and communications engineering and so on. Recently, the authors [10, 11] generalized the definition of ZDB functions to be G-ZDB functions. In this paper, based on p-cyclotomic cosets, two classes of ZDB functions are obtained, and several new classes of G-ZDB functions are constructed. Furthermore, some frequency-hopping sequences are obtained directly from G-ZDB functions.

ON GENERALIZED ZERO-DIFFERENCE BALANCED FUNCTIONS

In the present paper, by generalizing the definition of the zero-difference balanced (ZDB) function to be the G-ZDB function, several classes of G-ZDB functions are constructed based on properties of cyclotomic numbers. Furthermore, some special constant composition codes are obtained directly from G-ZDB functions.

Cyclotomic Constructions of Zero-Difference Balanced Functions With Applications

Based on a recent work of Ding, Wang, and Xiong, we present some cyclotomic constructions of zero-difference balanced (ZDB) functions, which interpret the previous construction proposed by Cai, Zeng, Helleseth, Tang, and Yang. The resulted ZDB functions can be used to generate optimal constant composition codes and perfect difference systems of sets with new parameters.

Zero-Difference Balanced Function Derived from Fermat Quotients and Its Applications

Zero-Difference Balanced Function Derived from Fermat Quotients and Its Applications

Quadratic zero-difference balanced functions, APN functions and strongly regular graphs

Let $F$ be a function from $\mathbb{F}_{p^n}$ to itself and $\delta$ a positive integer. $F$ is called zero-difference $\delta$-balanced if the equation $F(x+a)-F(x)=0$ has exactly $\delta$ solutions for all non-zero $a\in\mathbb{F}_{p^n}$. As a particular case, all known quadratic planar functions are zero-difference 1-balanced; and some quadratic APN functions over $\mathbb{F}_{2^n}$ are zero-difference 2-balanced. In this paper, we study the relationship between this notion and differential uniformity; we show that all quadratic zero-difference $\delta$-balanced functions are differentially $\delta$-uniform and we investigate in particular such functions with the form $F=G(x^d)$, where $\gcd(d,p^n-1)=\delta +1$ and where the restriction of $G$ to the set of all non-zero $(\delta +1)$-th powers in $\mathbb{F}_{p^n}$ is an injection. We introduce new families of zero-difference $p^t$-balanced functions. More interestingly, we show that the image set of such functions is a regular partial difference set, and hence yields strongly regular graphs; this generalizes the constructions of strongly regular graphs using planar functions by Weng et al. Using recently discovered quadratic APN functions on $\mathbb{F}_{2^8}$, we obtain $15$ new $(256, 85, 24, 30)$ negative Latin square type strongly regular graphs.

Three New Families of Zero-Difference Balanced Functions With Applications

Zero-difference balanced (ZDB) functions integrate a number of subjects in combinatorics and algebra, and have many applications in coding theory, cryptography and communications engineering. In this paper, three new families of ZDB functions are presented. The first construction, inspired by the recent work \cite{Cai13}, gives ZDB functions defined on the abelian groups $(\gf(q_1) \times \cdots \times \gf(q_k), +)$ with new and flexible parameters. The other two constructions are based on $2$-cyclotomic cosets and yield ZDB functions on $\Z_n$ with new parameters. The parameters of optimal constant composition codes, optimal and perfect difference systems of sets obtained from these new families of ZDB functions are also summarized.