A Note on Cyclic Codes from APN Functions

Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in \cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{\infty}$ defined by $s_t=Tr((1+\alpha^t)^e)$, where $\alpha$ is a primitive element in $GF(q)$. These techniques can also be applied into other open problems in \cite{D}

A Generalization of APN Functions for Odd Characteristic

Almost perfect nonlinear (APN) functions on finite fields of characteristic two have been studied by many researchers. Such functions have useful properties and applications in cryptography, finite geometries and so on. However APN functions on finite fields of odd characteristic do not satisfy desired properties. In this paper, we modify the definition of APN function in the case of odd characteristic, and study its properties

On the weight distributions of several classes of cyclic codes from APN monomials

Let $m\geq 3$ be an odd integer and $p$ be an odd prime. % with $p-1=2^rh$, where $h$ is an odd integer. In this paper, many classes of three-weight cyclic codes over $\mathbb{F}_{p}$ are presented via an examination of the condition for the cyclic codes $\mathcal{C}_{(1,d)}$ and $\mathcal{C}_{(1,e)}$, which have parity-check polynomials $m_1(x)m_d(x)$ and $m_1(x)m_e(x)$ respectively, to have the same weight distribution, where $m_i(x)$ is the minimal polynomial of $\pi^{-i}$ over $\mathbb{F}_{p}$ for a primitive element $\pi$ of $\mathbb{F}_{p^m}$. %For $p=3$, the duals of five classes of the proposed cyclic codes are optimal in the sense that they meet certain bounds on linear codes. Furthermore, for $p\equiv 3 \pmod{4}$ and positive integers $e$ such that there exist integers $k$ with $\gcd(m,k)=1$ and $\tau\in\{0,1,\cdots, m-1\}$ satisfying $(p^k+1)\cdot e\equiv 2 p^{\tau}\pmod{p^m-1}$, the value distributions of the two exponential sums T(a,b)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e)} and S(a,b,c)=\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{\Tr(ax+bx^e+cx^s)}, where $s=(p^m-1)/2$, are settled. As an application, the value distribution of $S(a,b,c)$ is utilized to investigate the weight distribution of the cyclic codes $\mathcal{C}_{(1,e,s)}$ with parity-check polynomial $m_1(x)m_e(x)m_s(x)$. In the case of $p=3$ and even $e$ satisfying the above condition, the duals of the cyclic codes $\mathcal{C}_{(1,e,s)}$ have the optimal minimum distance

On construction and (non)existence of c-(almost) perfect nonlinear functions

Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low c-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of APN and PN functions) have been investigated. Here, we provide a characterization of such functions via quadratic polynomials as well as non-existence results.publishedVersio

On construction and (non)existence of cc-(almost) perfect nonlinear functions

Functions with low differential uniformity have relevant applications in cryptography. Recently, functions with low $c$-differential uniformity attracted lots of attention. In particular, so-called APcN and PcN functions (generalization of APN and PN functions) have been investigated. Here, we provide a characterization of such functions via quadratic polynomials as well as non-existence results

Investigations on c-(almost) perfect nonlinear functions

In a prior paper [14], along with P. Ellingsen, P. Felke and A. Tkachenko, we defined a new (output) multiplicative differential, and the corresponding c-differential uniformity, which has the potential of extending differential cryptanalysis. Here, we continue the work, by looking at some APN functions through the mentioned concept and show that their c-differential uniformity increases significantly, in some cases.Comment: arXiv admin note: text overlap with arXiv:1909.0362