Perfect Nonlinear Functions Research Articles

Some Constructions of Perfect c-Nonlinear and Pseudo-Perfect c-Nonlinear Functions

The perfect nonlinear functions play an important role in block ciphers and have been widely investigated in the literature. Subsequently, the perfect [Formula: see text]-nonlinear functions were generalized by Ellingsen et al. in 2020. In this paper, we study the [Formula: see text]-differential uniformity of some functions. We first construct some perfect [Formula: see text]-nonlinear functions from known ones. Additionally, pseudo-perfect [Formula: see text]-nonlinear functions are also investigated and a necessary and sufficient condition for a function to be pseudo-perfect [Formula: see text]-nonlinear is presented. Meanwhile, some pseudo-perfect [Formula: see text]-nonlinear functions are constructed. Remarkably, we completely give the difference distribution table of a function with respect to the pseudo [Formula: see text]-derivative.

A further study on the Ness-Helleseth function

Let Fpn be a finite field with pn elements. Ness and Helleseth in [29] first studied a class of functions over Fpn with the form f(x)=uxpn−32+xpn−2,u∈Fpn⁎, which is called the Ness-Helleseth function. The f(x) has been proved to be an almost perfect nonlinear (APN) function by Ness and Helleseth for p=3 in [29] and by Zeng et al. for any odd prime p in [43] under the condition pn≡3(mod4) and η(1+u)=η(u−1). In this paper, we continue to study the Ness-Helleseth functions under the condition that pn≡3(mod4) and η(1+u)≠η(u−1). Firstly, we prove that f(x) is a permutation polynomial with differential uniformity not more than 4 if η(1+u)=η(1−u). Moreover, for some more special u, f is an involution with differential uniformity at most 3. Secondly, we show that f(x) is a locally-APN function for u=±1. In addition, the differential spectrum and boomerang spectrum of f(x) are obtained via judging the number of solutions of some special equations. We obtain the first non-PN function that its boomerang uniformity can attain 0 or 1.

Study of cyclic codes from low differentially uniform functions and its consequences

Cyclic codes have many applications in consumer electronics, communication and data storage systems due to their efficient encoding and decoding algorithms. An efficient approach to constructing cyclic codes is the sequence approach. In their articles (Ding and Zhou 2014 [11]) and (Ding, 2013 [7]), Ding and Zhou constructed several classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented some open problems on cyclic codes from highly nonlinear functions. This article focuses on these exciting works by investigating new insights in this research direction. Specifically, its objective is twofold. The first is to provide a complement with some former results and present correct proofs and statements on some known ones on the cyclic codes from the APN functions. The second is studying the cyclic codes from some known functions processing low differential uniformity. Along with this article, we shall answer some open problems presented in the literature. The first concerns Open Problem 1, proposed by Ding and Zhou (2014) [11]. The others are Open Problems 5.16 and 5.25, raised by Ding (2013) [7].

The Weight Distributions of Two Classes of Linear Codes From Perfect Nonlinear Functions

The Weight Distributions of Two Classes of Linear Codes From Perfect Nonlinear Functions

Note on Budaghyan and Carlet's almost perfect nonlinear functions

Almost perfect nonlinear (APN) functions have good properties and are widely applied in sequence design and coding theory. Budaghyan and Carlet (2008) [5] constructed a family of APN hexanomials F3 over F22m with a certain technical condition. In this article, we give the number of APN hexanomials F3 and support a determination theorem for APN hexanomials F3 if i=1. Moreover, we construct a family of APN functions in bivariate form and show it is CCZ-equivalent to F3.

Hardware architecture of Dillon’s APN permutation for different primitive polynomials

Cryptographically strong functions used as S-boxes in block cyphers are fundamental for the cypher’s security. Their representation as lookup tables is possible for functions of small dimension. For larger dimensions, this is infeasible, especially if resources are limited. An alternative is representing the function as a polynomial over a finite field, and constructing a circuit evaluating this polynomial. We study how the choice of primitive polynomial affects the efficiency of hardware implementations. We take Dillon’s permutation on 6 bits (the only known permutation in even dimension from the cryptographically optimal Almost Perfect Nonlinear functions) as a relevant example, and present hardware architectures, polynomial representations and hardware theoretical complexities for all primitive polynomials of degree six. To compare the efficiency, we report on results obtained from FPGA (Field Programmable Gate Array) implementations. To the best of our knowledge, no similar study has been given in the literature. We observe that using the primitive trinomial y6+y+1 reduces the number of 2-input XOR gates up to 11.7% and the number of XOR gates × Delay metrics up to 13.2% with respect to the worst-case implementation. Therefore, the choice of primitive polynomial can profoundly impact the efficiency of such an implementation, and should be carefully considered.

Value Distributions of Perfect Nonlinear Functions

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent.

A new class of generalized almost perfect nonlinear monomial functions

In this short note, we present a new class of GAPN power functions of the type xk2p2i+k1pi+k0 over finite fields Fpn with p odd and gcd(n,i)=1 (up to EA-equivalence).

P℘N functions, complete mappings and quasigroup difference sets

AbstractWe investigate pairs of permutations of such that is a permutation for every . We show that, in that case, necessarily for some complete mapping of , and call the permutation a perfect nonlinear (PN) function. If , then is a PcN function, which have been considered in the literature, lately. With a binary operation on involving , we obtain a quasigroup, and show that the graph of a PN function is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for PN functions, respectively, for the difference sets in the corresponding quasigroup.

Further constructions and characterizations of generalized almost perfect nonlinear functions

Further constructions and characterizations of generalized almost perfect nonlinear functions

Investigating rational perfect nonlinear functions

Perfect nonlinear (PN) functions over a finite field, whose study is also motivated by practical applications to Cryptography, have been the subject of several recent papers where the main problems, such as effective constructions and non-existence results, are considered. So far, all contributions have focused on PN functions represented by polynomials, and their constructions. Unfortunately, for polynomial PN functions, the approach based on Hasse–Weil type bounds applied to function fields can only provide non-existence results in a small degree regime. In this paper, we investigate the non-existence problem of rational perfect nonlinear functions over a finite field. Our approach makes it possible to use deep results about the number of points of algebraic varieties over finite fields. Our main result is that no PN rational function f/g with f,gin mathbb {F}_q[X] exists when certain mild arithmetical conditions involving the degree of f(X) and g(X) are satisfied.

Revisiting some results on APN and algebraic immune functions

<p style='text-indent:20px;'>We push a little further the study of two recent characterizations of almost perfect nonlinear (APN) functions. We state open problems about them, and we revisit in their perspective a well-known result from Dobbertin on APN exponents. This leads us to a new result about APN power functions and more general APN polynomials with coefficients in a subfield <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{2^k} $\end{document}</tex-math></inline-formula>, which eases the research of such functions. It also allows to construct automatically many differentially uniform functions from them (this avoids calculations for proving their differential uniformity as done in a recent paper, which are tedious and specific to each APN function). In a second part, we give simple proofs of two important results on Boolean functions, one of which deserves to be better known but needed clarification, while the other needed correction.</p>

Secondary constructions of bm$p$-ary (non-)weakly regular bent functions

Bent functions can be used in many areas, such as symmetric cryptography, sequence design, and association schemes. In this paper, a generic secondary 构造 of bent functions from indirect sum and semi-direct sum methods is presented. Our method can be used to construct high algebraic degree (non-)weakly regular bent functions by employing suitable initial (vectorial) bent functions or their combinations. More precisely, we present some classes of (weakly) regular bent functions by using vectorial $\mathcal{M}$-$\mathcal{M}$ (Maiorana-McFarland) and $\mathcal{PS}$ (partial spread) bent functions, respectively. In particular, the explicit expressions for the dual of these (weakly) regular bent functions are obtained. Based on the vectorial perfect nonlinear ($\mathrm{PN}$) functions, infinite families of non-weakly regular bent functions can also be produced.

Biprojective Almost Perfect Nonlinear Functions

In this paper, we introduce the concept of biprojectivity in studying the cryptographically important almost perfect nonlinear (APN) functions. Although several known families of biprojective APN functions exist in the literature, the concept itself has not been explicitly observed and studied in detail. We give a survey of known APN families and functions that fall into the biprojective setting, then provide a method for finding such families and functions. We finally give two new infinite families of biprojective APN functions <inline-formula> <tex-math notation="LaTeX">${\mathsf {CCZ}}$ </tex-math></inline-formula>-inequivalent to any known APN function and study their properties.

Subfield Codes of Linear Codes from Perfect Nonlinear Functions and Their Duals

Let $\mathbb{F}_{p^m}$ be a finite field with $p^m$ elements, where $p$ is an odd prime and $m$ is a positive integer. Recently, \cite{Hengar} and \cite{Wang2020} determined the weight distributions of subfield codes with the form $$\mathcal{C}_f=\left\{\left(\left( {\rm Tr}_1^m(a f(x)+bx)+c\right)_{x \in \mathbb{F}_{p^m}}, {\rm Tr}_1^m(a)\right)\, : \, a,b \in \mathbb{F}_{p^m}, c \in \mathbb{F}_p\right\}$$ for $f(x)=x^2$ and $f(x)=x^{p^k+1}$, respectively, where $k$ is a nonnegative integer. In this paper, we further investigate the subfield code $\mathcal{C}_f$ for $f(x)$ being a known perfect nonlinear function over $\mathbb{F}_{p^m}$ and generalize some results in \cite{Hengar,Wang2020}. The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields. In addition, the parameters of the duals of these codes are also determined. Several examples show that some of our codes and their duals have the best known parameters with respect to the code tables in \cite{MGrassl}. The duals of some proposed codes are optimal with respect to the Sphere Packing bound if $p\geq 5$.

Shortened Linear Codes From APN and PN Functions

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes.

A new class of generalized almost perfect nonlinear power function

As a generalization of almost perfect nonlinear functions, generalized almost perfect nonlinear (GAPN) functions defined over finite fields of odd characteristic have also important applications in cryptography and finite geometry. In this paper, we mainly focus on GAPN power functions, and a new class of such function (up to EA equivalence) is obtained. Furthermore, using our method, almost all the known GAPN power functions can be confirmed in a simple way.

On monomial generalized almost perfect nonlinear functions

Generalized almost perfect nonlinear (GAPN) functions are a generalization of APN functions to finite fields of odd characteristic p introduced in 2017 by Kuroda and Tsujie. In this paper we deal with GAPN functions of monomial type. To this aim, we connect the GAPN property for a monomial function over Fpn to the existence of suitable rational points of an algebraic curve defined over Fpn. We give necessary conditions for a monomial function to be GAPN, providing the converse of recent results by Özbudak and Sălăgean and by Zha, Hu and Zhang.

Plateaued functions on finite abelian groups and partial geometric difference sets

AbstractAs a generalization of plateaued functions on finite fields and bent functions (perfect nonlinear functions) on finite abelian groups, plateaued functions on finite abelian groups were introduced in [B. Xu, Plateaued functions, partial geometric difference sets, and partial geometric designs, J. Combin. Des. 27 (2019), 756–783]. In this paper, we continue the research in the paper mentioned above. We will obtain various characterizations of plateaued functions; these characterizations establish close connections between plateaued functions and some combinatorial objects: partial geometric difference sets and related partial geometric difference families. Then we introduce the complementary matrix and Cayley matrix for a subset of a finite group and use them to characterize partial geometric difference sets. As applications, we will show how to construct directed strongly regular graphs from partial geometric difference sets and establish a natural relation between partial geometric difference sets and partial geometric designs. The tensor product of a group algebra and a cyclotomic field is an important tool for our discussions.

The second-order zero differential spectra of almost perfect nonlinear functions and the inverse function in odd characteristic

The second-order zero differential spectra of almost perfect nonlinear functions and the inverse function in odd characteristic