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Let $n\geqslant3$ be an odd integer and $u$ an element in the finite field $\gf_{3^n}$. The Ness-Helleseth function is the binomial $f_u(x)=ux^{d_1}+x^{d_2}$ over $\gf_{3^n}$, where $d_1=\frac{3^n-1}{2}-1$ and $d_2=3^n-2$. In 2007, Ness and Helleseth showed that $f_u$ is an APN function when $\chi(u+1)=\chi(u-1)=\chi(u)$, is differentially $3$-uniform when $\chi(u+1)=\chi(u-1)\neq\chi(u)$, and has differential uniformity at most 4 if $ \chi(u+1)\neq\chi(u-1)$ and $u\notin\gf_3$. Here $\chi(\cdot)$ denotes the quadratic character on $\gf_{3^n}$. Recently, Xia et al. determined the differential uniformity of $f_u$ for all $u$ and computed the differential spectrum of $f_u$ for $u$ satisfying $\chi(u+1)=\chi(u-1)$ or $u\in\gf_3$. The remaining problem is the differential spectrum of $f_u$ with $\chi(u+1)\neq\chi(u-1)$ and $u\notin\gf_3$. In this paper, we fill in the gap. By studying differential equations arising from the Ness-Helleseth function $f_u$ more carefully, we express the differential spectrum of $f_u$ for such $u$ in terms of two quadratic character sums. This complements the previous work of Xia et al.

The cross-correlation problem is a classic problem in sequence design. In this paper we compute the cross-correlation distribution of the Niho-type decimation $d=3(p^m-1)+1$ over $\mathrm{GF}(p^{2m})$ for any prime $p \ge 5$. Previously this problem was solved by Xia et al. only for $p=2$ and $p=3$ in a series of papers. The main difficulty of this problem for $p \ge 5$, as pointed out by Xia et al., is to count the number of codewords of "pure weight" 5 in $p$-ary Zetterberg codes. It turns out this counting problem can be transformed by the MacWilliams identity into counting codewords of weight at most 5 in $p$-ary Melas codes, the most difficult of which is related to a K3 surface well studied in the literature and can be computed. When $p \ge 7$, the theory of elliptic curves over finite fields also plays an important role in the resolution of this problem.

Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but which are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters $S\left (2, q+1,(q^n-1)/(q-1)\right )$ is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the $m$-sequences with Welch-type decimation $d=2\cdot 3^{(n-1)/2}+1$, and obtain some infinite family of 2-designs and a family of Steiner systems $S\left (2, 4, (3^n-1)/2\right )$ using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space $\mathrm{PG}(n-1,3)$ and thus is a pseudo-geometric design., Comment: 15 pages. arXiv admin note: text overlap with arXiv:2206.15153, arXiv:2110.03881

Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight of this code supporting 2-designs. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined. The parameters of the related $2$-designs are also given. Moreover, we prove that the codewords with minimum Hamming weight supporting 3-designs, which gives an affirmative solution to Heng's conjecture.

Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary subfield codes $C_{f,g}^{(q)}$ of six different families of linear codes $C_{f,g}$ parametrized by two functions $f, g$ over a finite field $F_{q^m}$ are considered and studied, respectively. The parameters and (Hamming) weight distribution of $C_{f,g}^{(q)}$ and their punctured codes $\bar{C}_{f,g}^{(q)}$ are explicitly determined. The parameters of the duals of these codes are also analyzed. Some of the resultant $q$-ary codes $C_{f,g}^{(q)},$ $\bar{C}_{f,g}^{(q)}$ and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes $C_{f,g}$ are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of $[2^{4m-2},2m+1,2^{4m-3}]$ quaternary Hermitian self-dual code are obtained with $m \geq 2$. As an application, we show that three families of the derived linear codes give rise to several infinite families of $t$-designs ($t \in \{2, 3\}$)., Comment: arXiv admin note: text overlap with arXiv:1804.06003, arXiv:2207.07262 by other authors

Recently, the investigation of Partially APN functions has attracted a lot of attention. In this paper, with the help of resultant elimination and MAGMA, we propose several new infinite classes of 0-APN power functions over $\mathbb{F}_{2^{n}}$. By the main result in [4], these $0$-APN power functions are CCZ-inequivalent to the known ones. Moreover, these infinite classes of 0-APN power functions can explain some exponents for $1\leq n\leq11$ which are not yet ``explained" in the tables of Budaghyan et al. [3].

In the paper, we concentrate on the map $x\mapsto x^{\frac{p^n+1}{2}}$ on $\mathbb{F}_{p^n}$ and using combinatorial and number theory techniques, we compute its detailed $c$-differential spectrum for all values of $c\neq 1$ (the spectrum for $c=1$ is known).

Only three classes of Almost Perfect Nonlinear (for short, APN) power functions over odd characteristic finite fields have been investigated in the literature, and their differential spectra were determined. The differential uniformity of the power function $F(x)=x^{\frac{p^{n}-3}{2}}$ over the finite field $F_{p^n}$ of order $p^n$ (where $p$ is an odd prime), was studied by Helleseth and Sandberg in 1997, where $p^n\equiv3\pmod{4}$ is an odd prime power with $p^n>7$. It was shown that $F$ is PN when $p^n=27$, APN when $5$ is a nonsquare in $F_{p^n}$, and differentially $3$-uniform when $5$ is a square in $F_{p^n}$. In this paper, by investigating some equation systems and certain character sums over $F_{p^n}$, the differential spectrum of $F$ is completely determined. We focusing on the power functions $x^d$ with even $d$ over $F_{p^n}$ ($p$ odd), the power functions $F$ we consider are APN which are of the lowest differential uniformity and the nontrivial differential spectrum. Moreover, we examine the extension of the so-called $c$-differential uniformity by investigating the $c$-differential properties of $F$. Specifically, an upper bound of the $c$-differential uniformity of $F$ is given, and its $c$-differential spectrum is considered in the case where $c=-1$. Finally, we emphasize that, throughout our study of the differential spectrum of the considered power functions, we provide methods for evaluating sums of specific characters with connections to elliptic curves and for determining the number of solutions of specific systems of equations over finite fields., Comment: arXiv admin note: text overlap with arXiv:2210.09822

Permutation polynomials over finite fields are fundamental objects as they are used in various theoretical and practical applications in cryptography, coding theory, combinatorial design, and related topics. This family of polynomials constitutes an active research area in which advances are being made constantly. In particular, constructing infinite classes of permutation polynomials over finite fields with good differential properties (namely, low) remains an exciting problem despite much research in this direction for many years. This article exhibits low differentially uniform power permutations over finite fields of odd characteristic. Specifically, its objective is twofold concerning the power functions $F(x)=x^{\frac{p^n+3}{2}}$ defined over the finite field $F_{p^n}$ of order $p^n$, where $p$ is an odd prime, and $n$ is a positive integer. The first is to complement some former results initiated by Helleseth and Sandberg in \cite{HS} by solving the open problem left open for more than twenty years concerning the determination of the differential spectrum of $F$ when $p^n\equiv3\pmod 4$ and $p\neq 3$. The second is to determine the exact value of its differential uniformity. Our achievements are obtained firstly by evaluating some exponential sums over $F_{p^n}$ (which amounts to evaluating the number of $F_{p^n}$-rational points on some related curves and secondly by computing the number of solutions in $(F_{p^n})^4$ of a system of equations presented by Helleseth, Rong, and Sandberg in ["New families of almost perfect nonlinear power mappings," IEEE Trans. Inform. Theory, vol. 45. no. 2, 1999], naturally appears while determining the differential spectrum of $F$. We show that in the considered case ($p^n\equiv3\pmod 4$ and $p\neq 3$), $F$ is an APN power permutation when $p^n=11$, and a differentially $4$-uniform power permutation otherwise.

Let $q$ be an odd prime power. Let $F_1(x)=x^{d_1}$ and $F_2(x)=x^{d_2}$ be power mappings over $\mathrm{GF}(q^2)$, where $d_1=q-1$ and $d_2=d_1+\frac{q^2-1}{2}=\frac{(q-1)(q+3)}{2}$. In this paper, we study the the boomerang uniformity of $F_1$ and $F_2$ via their differential properties. It is shown that, the boomerang uniformity of $F_i$ ($i=1,2$) is 2 with some conditions on $q$., Comment: 11 pages

In EUROCRYPT 2018, Cid $et\;al.$ introduced a new concept on the cryptographic property of S-boxes to evaluate the subtleties of boomerang-style attacks. This concept was named as boomerang connectivity table (BCT for short) . For a power function, the distribution of BCT can be directly determined by its boomerang spectrum. In this paper, we investigate the boomerang spectra of two classes power functions over even characteristic finite fields via their differential spectra. The boomerang spectrum of the power function $ {x^{{2^{m+1}} - 1}} $ over $ {\mathbb{F}_{{2^{2m}}}} $ is determined, where $2^{m+1}-1$ is a kind of Niho exponent. The boomerang spectrum of the Gold function $G(x)=x^{2^t+1}$ over $ {\mathbb{F}_{{2^n}}} $ is also determined. It is shown that the Gold function has two-valued boomerang spectrum., Comment: Part of the results of our article are covered by sihem BFA's article

Let $n$ be a positive integer and $p$ a prime. The power mapping $x^{p^n-3}$ over $\mathbb{F}_{p^n}$ has desirable differential properties, and its differential spectra for $p=2,\,3$ have been determined. In this paper, for any odd prime $p$, by investigating certain quadratic character sums and some equations over $\mathbb{F}_{p^n}$, we determine the differential spectrum of $x^{p^n-3}$ with a unified approach. The obtained result shows that for any given odd prime $p$, the differential spectrum can be expressed explicitly in terms of $n$. Compared with previous results, a special elliptic curve over $\mathbb{F}_{p}$ plays an important role in our computation for the general case $p \ge 5$.

Very recently, a new concept called multiplicative differential and the corresponding $c$-differential uniformity were introduced by Ellingsen et al. A function $F(x)$ over finite field $\mathrm{GF}(p^n)$ to itself is called $c$-differential uniformity $\delta$, or equivalent, $F(x)$ is differentially $(c,\delta)$ uniform, when the maximum number of solutions $x\in\mathrm{GF}(p^n)$ of $F(x+a)-F(cx)=b$, $a,b,c\in\mathrm{GF}(p^n)$, $c\neq1$ if $a=0$, is equal to $\delta$. The objective of this paper is to study the $-1$-differential uniformity of ternary APN power functions $F(x)=x^d$ over $\mathrm{GF}(3^n)$. We obtain ternary power functions with low $-1$-differential uniformity, and some of them are almost perfect $-1$-nonlinear.

In a prior paper \cite{EFRST20}, two of us, along with P. Ellingsen, P. Felke and A. Tkachenko, 1defined 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 showing that their $c$-differential uniformity increases significantly, in some cases., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2003.13019

Very recently, a new concept called multiplicative differential (and the corresponding $c$-differential uniformity) was introduced by Ellingsen \textit{et al} in [C-differentials, multiplicative uniformity and (almost) perfect c-nonlinearity, IEEE Trans. Inform. Theory, 2020] which is motivated from practical differential cryptanalysis. Unlike classical perfect nonlinear functions, there are perfect $c$-nonlinear functions even for characteristic two. The objective of this paper is to study power function $F(x)=x^d$ over finite fields with low $c$-differential uniformity. Some power functions are shown to be perfect $c$-nonlinear or almost perfect $c$-nonlinear. Notably, we completely determine the $c$-differential uniformity of almost perfect nonlinear functions with the well-known Gold exponent. We also give an affirmative solution to a recent conjecture proposed by Bartoli and Timpanella in 2019 related to an exceptional quasi-planar power function.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth. Let $C_{(1,e)}$ be a cyclic code of length $3^m-1$ over GF(3) with two nonzeros $\alpha$ and $\alpha^e$, where $\alpha$ is a generator of $GF(3^m)^*$ and e is a given integer. It is shown that $C_{(1,e)}$ is optimal with parameters $[3^m-1,3^m-1-2m,4]$ if one of the following conditions is met. 1) $m\equiv0(\mathrm{mod}~ 4)$, $m\geq 4$, and $e=3^\frac{m}{2}+5$. 2) $m\equiv2(\mathrm{mod}~ 4)$, $m\geq 6$, and $e=3^\frac{m+2}{2}+5$.

Linearized Reed-Solomon codes are defined. Higher weight distribution of those codes are determined.

A codeword is associated to a linearized polynomial. The weight distribution of the codewords is determined as the linearized polynomial varies in a family of fixed degree. There is a corresponding result on Wenger graphs from linearized polynomials., Comment: 8 pages

In this paper, a family of five-weight reducible cyclic codes is presented. Furthermore, the weight distribution of these cyclic codes is determined, which follows from the determination of value distributions of certain exponential sums., Comment: arXiv admin note: substantial text overlap with arXiv:1311.3391, and text overlap with arXiv:1302.0952 by other authors

In this paper, a family of six-weight cyclic codes over GF(p) whose duals have two zeros is presented, where p is an odd prime. And the weight distribution of these cyclic codes is determined., Comment: arXiv admin note: text overlap with arXiv:1302.0952, arXiv:1302.0569, arXiv:1301.4824 by other authors

In this paper, we first define a non-degenerate symmetric bilinear form on Fq^4. Then we get an incidence matrix G of Fq^4 by the bilinear form. By its corresponding quadratic form Q, the lines of Fq^4 are classified as isotropic and anisotropic lines. Under this classification, we can get two sub-matrices of G and prove their 2-rank. Key words and phrases: finite field, quadratic form, incidence matrix.