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Haode Yan

Publications and source records attributed to Haode Yan.

At least 19 recordsLinked to original sources
arXiv2026

The Cross-Correlation Distribution of the Niho-Type Decimation $d=4(2^m-1)+1$

The cross-correlation problem is a classical problem in sequence design. In this paper, we determine the cross-correlation distribution of the Niho-type decimation $d=4(2^m-1)+1$ over $\mathbb F_{2^{2m}}$ for every positive integer $m$. With $q=2^m$, this is equivalent to determining the distribution of the number of roots in $U_{q+1}$ of $f_a(x)=x^7+ax^4+\bar a x^3+1$ for $a\in\mathbb F_{q^2}$, where $U_{q+1}=\{x\in\mathbb F_{q^2}:x^{q+1}=1\}$. The main difficulty is to count the four-element subsets of $U_{q+1}$ that may occur as root sets of $f_a$. We normalize such subsets by their product and study the resulting condition through the associated resolvent. This reduces the required enumeration to several equations over $\mathbb F_q$, which are evaluated by character sums and Kloosterman sums. The remaining mixed Kloosterman sum is related to a Kloosterman sum over $\mathbb F_{2^{2m}}$ and is evaluated using a theorem of Carlitz. Consequently, we obtain explicit formulas for all the frequencies in the cross-correlation distribution.

cs.IT↗
arXiv2026

Counterexamples to Charpin's Conjecture on BCH codes

We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed--Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

cs.IT↗
arXiv2026

A class of locally differentially $4$-uniform power functions with Niho exponents

Niho exponents have found important applications in sequence design, coding theory, and cryptography. Determining the differential spectrum of a power function with Niho exponent is a topic of considerable interest. In this paper, we investigate the power function $F(x) = x^{3q - 2}$ over $\mathbb{F}_{q^2}$, where $q = 2^m$ and $m\geq 4$ is an even integer. Notably, the exponent $3q - 2$ is a Niho exponent. By analyzing the properties of certain polynomials over $\mathbb{F}_{q^2}$, we determine the differential spectrum of $F$. Our results show that $F$ is locally differentially $4$-uniform, which complements existing results on the differential spectra of power functions with Niho exponents.

cs.IT↗
arXiv2025

On the Hamming Weight Functions of Linear Codes

Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight functions. Specifically, we develop a general framework that constructs new linear codes from the set of codewords in a given code having a fixed Hamming weight. We analyze the dimension, number of weights, and weight distribution of the constructed codes, and establish connections with the extendability of the original codes as well as the partial weight distribution of the derived codes. As a new tool, this framework enables us to establish an upper bound on the minimum weight of two-weight codes and to characterize all two-weight codes attaining this bound. Moreover, several divisibility properties concerning the parameters of two-weight codes are derived. The proposed method not only generates new families of linear codes but also provides a powerful approach for exploring the intrinsic combinatorial and geometric structures of existing codes.

cs.IT↗
arXiv2025

On covering radius of generalized Zetterberg codes

We employ analytic number theoretic techniques, specifically character sums and Weil type estimates, to study the covering radius of the generalized Zetterberg codes over all finite fields. Although the even and odd field cases require distinct technical treatment, the proofs follow a unified analytic framework that is substantially simpler and more transparent than previous approaches. We prove that the covering radius is at most 3 in all cases, and determine its exact value for a wide range of parameters. In even characteristic, our results fill the gap left by recent studies focused solely on odd characteristic; for odd characteristic, the range of parameters for which the covering radius is exactly determined is considerably broader than previously known. Combined with the corresponding minimum distance results, we obtain infinitely many quasi-perfect and maximal codes within this family.

math.NT↗
arXiv2025

A note on the differential spectrum of a class of locally APN functions

Let $\gf_{p^n}$ denote the finite field containing $p^n$ elements, where $n$ is a positive integer and $p$ is a prime. The function $f_u(x)=x^{\frac{p^n+3}{2}}+ux^2$ over $\gf_{p^n}[x]$ with $u\in\gf_{p^n}\setminus\{0,\pm1\}$ was recently studied by Budaghyan and Pal in \cite{Budaghyan2024ArithmetizationorientedAP}, whose differential uniformity is at most $5$ when $p^n\equiv3~(mod~4)$. In this paper, we study the differential uniformity and the differential spectrum of $f_u$ for $u=\pm1$. We first give some properties of the differential spectrum of any cryptographic function. Moreover, by solving some systems of equations over finite fields, we express the differential spectrum of $f_{\pm1}$ in terms of the quadratic character sums.

cs.IT↗
arXiv2024

A note on the differential spectrum of the Ness-Helleseth function

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.

cs.CR↗
arXiv2023

On correlation distribution of Niho-type decimation $d=3(p^m-1)+1$

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.

cs.IT↗
arXiv2023

Codes and Pseudo-Geometric Designs from the Ternary $m$-Sequences with Welch-type decimation $d=2\cdot 3^{(n-1)/2}+1$

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.

cs.IT↗
arXiv2023

On the parameters of extended primitive cyclic codes and the related designs

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.

cs.IT↗
arXiv2022

Subfield Codes of Several Few-Weight Linear Codes Parametrized by Functions and Their Consequences

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 ($t \in \{2, 3\}$).

cs.IT↗
arXiv2022

Several classes of 0-APN power functions over $\mathbb{F}_{2^n}$

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].

cs.IT↗
arXiv2022

On the differential spectrum of a class of APN power functions over odd characteristic finite fields and their $c$-differential properties

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.

cs.IT↗
arXiv2022

Two low differentially uniform power permutations over odd characteristic finite fields: APN and differentially $4$-uniform functions

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.

cs.IT↗
arXiv2022

Two Classes of Power Mappings with Boomerang Uniformity 2

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 $q$.

cs.IT↗
arXiv2022

Boomerang Spectra of Two Classes of Power Functions via Their Differential Spectra

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.

cs.CR↗
arXiv2021

The Differential Spectrum of the Power Mapping $x^{p^n-3}$

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 $p \ge 5$.

cs.IT↗