arXiv subjects

Longjiang Qu

Publications and source records attributed to Longjiang Qu.

At least 19 recordsLinked to original sources

arXiv2026

Minimum Distances of Binary Goppa Codes and Constructions with Prescribed Alternating Automorphism Groups

Goppa codes are a well-known class of linear codes with important applications in cryptography. Determining the minimum distance of Goppa codes and constructing Goppa codes with prescribed automorphism groups are both meaningful and challenging problems in coding theory. In this paper, we first study the minimum distance of binary separable Goppa codes. For the two classes $g(X)=f(X^t)$ and $g(X)=A(X)h(\phi(X))$, we give criteria for attaining the designed distance and derive several infinite families whose minimum distances are determined. We then construct binary Goppa codes and their related codes with $A_4$ or $A_5$ automorphism groups. These constructions also naturally yield binary quasi-cyclic Goppa codes and their related codes. Moreover, by applying the minimum-distance criteria developed above, we determine the parameters of one class of the constructed $A_4$-invariant Goppa codes.

cs.IT↗

arXiv2025

Constructions of binary self-orthogonal singly-even minimal linear codes violating the Aschikhmin-Barg condition with few weights

We first establish a simple yet powerful necessary and sufficient condition for a binary linear code to be SO, leading to a complete characterization of singly-even codes in this family. We further derive necessary and sufficient conditions on Boolean and vectorial Boolean functions for generating such codes via a standard construction method. Building on this foundation, we propose three general frameworks for constructing binary SO singly-even minimal non-AB linear codes with few weights. The first two approaches are based on designing Boolean and vectorial Boolean functions that simultaneously satisfy multiple conditions. The third method generates new SO codes from existing ones. As a result, we obtain many infinite classes of binary self-orthogonal singly-even minimal linear codes violating the AB condition with few weights and fully determined weight distributions. Particularly, numerical results show that some duals of our codes are optimal or near-optimal.

cs.IT↗

arXiv2025

Several new classes of self-orthogonal minimal linear codes violating the Ashikhmin-Barg condition

Linear codes have attracted considerable attention in coding theory and cryptography due to their significant applications in secret sharing schemes, secure two-party computation, Galois geometries, among others. As two special subclasses of linear codes, minimal linear codes and self-orthogonal linear codes are of particular interest. Constructing linear codes that possess both minimality and self-orthogonality is very interesting. The main purpose of this paper is to construct self-orthogonal minimal linear codes that violate the Ashikhmin-Barg (AB for short) condition over the finite field $\mathbb{F}_p$. First, we present several classes of self-orthogonal minimal linear codes violating the AB condition over the finite field $\mathbb{F}_2$ and determine their weight distributions. Next, for any odd prime $p$, we construct two classes of self-orthogonal linear codes from $p$-ary functions, which contain some optimal or almost optimal codes. Finally, based on plateaued functions, we construct two classes of self-orthogonal linear codes that violate the AB condition. Their weight distributions are also provided. To the best of our knowledge, this paper is the first to investigate the constructions of linear codes that violate the AB condition and satisfy self-orthogonality.

cs.IT↗

arXiv2025

Balanced Boolean functions with few-valued Walsh spectra parameterized by $P(x^2+x)$

Boolean functions with few-valued spectra have wide applications in cryptography, coding theory, sequence designs, etc. In this paper, we further study the parametric construction approach to obtain balanced Boolean functions using $2$-to-$1$ mappings of the form $P(x^2+x)$, where $P$ denotes carefully selected permutation polynomials. The key contributions of this work are twofold: (1) We establish a new family of four-valued spectrum Boolean functions. This family includes Boolean functions with good cryptographic properties, e.g., the same nonlinearity as semi-bent functions, the maximal algebraic degree, and the optimal algebraic immunity for dimensions $n \leq 14$. (2) We derive seven distinct classes of plateaued functions, including four infinite families of semi-bent functions and a class of near-bent functions.

cs.IT↗

arXiv2025

Choosing Coordinate Forms for Solving ECDLP Using Shor's Algorithm

Shor's algorithm is well-known for its capability to address the elliptic curve discrete logarithm problem (ECDLP) in polynomial time. The enhancement of its quantum resources continues to be a crucial focus of research. Nevertheless, the application of projective coordinates for quantum resource optimization remains an unresolved issue, mainly because the representation of projective coordinates lacks uniqueness without employing modular division operations. Our study reveals that projective coordinates do not provide the same advantages as affine coordinates when utilizing Shor's method to tackle the ECDLP.

cs.CR↗

arXiv2024

Endomorphism Rings of Supersingular Elliptic Curves and Ternary Quadratic Forms

Let $c<3p/16$ be a prime or $c=1$. Let $E$ be a $\mathbb{Z}[\sqrt{-cp}]$-oriented supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. There exists a $c$-isogeny from $E$ to $E^p$ with kernel $G \subset E[c]$. Given an Eichler order corresponding to the endomorphism ring $\text{End}(E,G)=\{ \theta \in \text{End}(E): \theta(G) \subseteq G \}$, we can compute a ternary quadratic form with discriminant $p$ by solving two square roots in $\mathbb{F}_c$, and the ternary quadratic form corresponds to a maximal order $\mathcal{O} \cong \text{End}(E)$ in $B_{p,\infty}$ by Brandt--Sohn correspondence. Let $D$ be a prime with $D<p$ (resp. $4D<p$). If an imaginary quadratic order with discriminant $-D$ (resp. $-4D$) can be embedded into $\text{End}(E)$, then we can compute a maximal order in $B_{p,\infty}$ corresponding to $\text{End}(E)$ by solving one square root in $\mathbb{F}_D$ and two square roots in $\mathbb{F}_c$. As we know, any isogeny between supersingular elliptic curves can be translated into a kernel ideal of the endomorphism ring. We study the action of the kernel ideal and give a basis of its right order. In general, we propose an efficient algorithm for computing a maximal order from an Eichler order in $B_{p,\infty}$.

math.NT↗

arXiv2024

Constructing rotatable permutations of $\mathbb{F}_{2^m}^3$ with $3$-homogeneous functions

In the literature, there are many results about permutation polynomials over finite fields. However, very few permutations of vector spaces are constructed although it has been shown that permutations of vector spaces have many applications in cryptography, especially in constructing permutations with low differential and boomerang uniformities. In this paper, motivated by the butterfly structure \cite{perrin2016cryptanalysis} and the work of Qu and Li \cite{qu2023}, we investigate rotatable permutations from $\gf_{2^m}^3$ to itself with $d$-homogenous functions. Based on the theory of equations of low degree, the resultant of polynomials, and some skills of exponential sums, we construct five infinite classes of $3$-homogeneous rotatable permutations from $\gf_{2^m}^3$ to itself, where $m$ is odd. Moreover, we demonstrate that the corresponding permutation polynomials of $\gf_{2^{3m}}$ of our newly constructed permutations of $\gf_{2^m}^3$ are QM-inequivalent to the known ones.

math.CO↗

arXiv2023

Oriented Supersingular Elliptic Curves and Eichler Orders

Let $p>3$ be a prime and $E$ be a supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. Let $c$ be a prime with $c < 3p/16$ and $G$ be a subgroup of $E[c]$ of order $c$. The pair $(E,G)$ is called a supersingular elliptic curve with level-$c$ structure, and the endomorphism ring $\text{End}(E,G)$ is isomorphic to an Eichler order with level $c$. We construct two kinds of Eichler orders $\mathcal{O}_c(q,r)$ and $\mathcal{O}'_c(q,r')$ with level $c$. Interestingly, we prove that each $\mathcal{O}_c(q,r)$ or $\mathcal{O}'_c(q,r')$ can represent a primitive reduced binary quadratic form with discriminant $-16cp$ or $-cp$ respectively. If a curve $E$ is $\mathbb{Z}[\sqrt{-cp}]$-oriented or $\mathbb{Z}[\frac{1+\sqrt{-cp}}{2}]$-oriented, then we prove that $\text{End}(E,G)$ is isomorphic to $\mathcal{O}_c(q,r)$ or $\mathcal{O}'_c(q,r')$ respectively. Due to the fact that $\mathbb{Z}[\sqrt{-cp}]$-oriented isogenies between $\mathbb{Z}[\sqrt{-cp}]$-oriented elliptic curves could be represented by quadratic forms, we show that these isogenies are reflected in the corresponding Eichler orders via the composition law for their corresponding quadratic forms.

math.NT↗

arXiv2022

More infinite classes of APN-like Power Functions

In the literature, there are many APN-like functions that generalize the APN properties or are similar to APN functions, e.g. locally-APN functions, 0-APN functions or those with boomerang uniformity 2. In this paper, we study the problem of constructing infinite classes of APN-like but not APN power functions. For one thing, we find two infinite classes of locally-APN but not APN power functions over $\gf_{2^{2m}}$ with $m$ even, i.e., $\mathcal{F}_1(x)=x^{j(2^m-1)}$ with $\gcd(j,2^m+1)=1$ and $\mathcal{F}_2(x)=x^{j(2^m-1)+1}$ with $j = \frac{2^m+2}{3}$. As far as the authors know, our infinite classes of locally-APN but not APN functions are the only two discovered in the last eleven years. Moreover, we also prove that this infinite class $\mathcal{F}_1$ is not only with the optimal boomerang uniformity $2$, but also has an interesting property that its differential uniformity is strictly greater than its boomerang uniformity. For another thing, using the multivariate method, including the above infinite class $\mathcal{F}_1$, we construct seven new infinite classes of 0-APN but not APN power functions.

cs.IT↗

arXiv2022

Generalized Singleton Type Upper Bounds

In this paper, we give upper bounds on the sizes of $(d, L)$ list-decodable codes in the Hamming metric space from covering codes with the covering radius smaller than or equal to $d$. When the list size $L$ is $1$, this gives many new Singleton type upper bounds on the sizes of codes with a given minimum Hamming distance. These upper bounds are stronger than the Griesmer bound when the lengths of codes are large. Some upper bounds on the lengths of general small Singleton defect codes or list-decodable codes attaining the generalized Singleton bound are given. As an application of our generalized Singleton type upper bounds on Hamming metric error-correcting codes, the generalized Singleton type upper bounds on insertion-deletion codes are given, which are much stronger than the direct Singleton bound for insertion-deletion codes when the lengths are large. We also give upper bounds on the lengths of small dimension optimal locally recoverable codes and small dimension optimal $(r, \delta)$ locally recoverable codes with any fixed given minimum distance.

cs.IT↗

arXiv2022

More constructions of $n$-cycle permutations

$n$-cycle permutations with small $n$ have the advantage that their compositional inverses are efficient in terms of implementation. They can be also used in constructing Bent functions and designing codes. Since the AGW Criterion was proposed, the permuting property of several forms of polynomials has been studied. In this paper, characterizations of several types of $n$-cycle permutations are investigated. Three criteria for $ n $-cycle permutations of the form $xh(\lambda(x))$, $ h(\psi(x)) \varphi(x)+g(\psi(x)) $ and $g\left( x^{q^i} -x +\delta \right) +bx $ with general $n$ are provided. We demonstrate these criteria by providing explicit constructions. For the form of $x^rh(x^s)$, several new explicit triple-cycle permutations are also provided. Finally, we also consider triple-cycle permutations of the form $x^t + c\rm Tr_{q^m/q}(x^s)$ and provide one explicit construction. Many of our constructions are both new in the $n$-cycle property and the permutation property.

cs.IT↗

arXiv2022

The Endomorphism Rings of Supersingular Elliptic Curves over $\mathbb{F}_p$ and the Binary Quadratic Forms

It is well known that there is a one-to-one correspondence between supersingular $j$-invariants up to the action of $\text{Gal}(\mathbb{F}_{p^2}/\mathbb{F}_p)$ and type classes of maximal orders in $B_{p,\infty}$ by Deuring's theorem. Interestingly, we establish a one-to-one correspondence between $\mathbb{F}_p$-isomorphism classes of supersingular elliptic curves and primitive reduced binary quadratic forms with discriminant $-p$ or $-16p$. Due to this correspondence and the fact that $\mathbb{F}_p$-isogenies between elliptic curves could be represented by quadratic forms, we show that operations of these isogenies on supersingular elliptic curves over $\mathbb{F}_p$ are compatible with the composition of quadratic forms. Based on these results, we could reduce the security of CSIDH cryptosystem to computing this correspondence explicitly.

math.NT↗

arXiv2022

Characterizations and constructions of n-to-1 mappings over finite fields

$n$-to-$1$ mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of $n$-to-$1$ mappings over finite fields are studied. First, we provide a characterization of general $n$-to-$1$ mappings over $\mathbb{F}_{p^m}$ by means of the Walsh transform. Then, we completely determine $3$-to-$1$ polynomials with degree no more than $4$ over $\mathbb{F}_{p^{m}}$. Furthermore, we obtain an AGW-like criterion for characterizing an equivalent relationship between the $n$-to-$1$ property of a mapping over finite set $A$ and that of another mapping over a subset of $A$. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit $n$-to-$1$ mappings. Especially, three explicit constructions of the form $x^rh\left( x^s \right) $ from the cyclotomic perspective, and several classes of $n$-to-$1$ mappings of the form $ g\left( x^{q^k} -x +\delta \right) +cx$ are provided.

cs.IT↗

arXiv2021

Two new infinite classes of APN functions

In this paper, we present two new infinite classes of APN functions over $\gf_{{2^{2m}}}$ and $\gf_{{2^{3m}}}$, respectively. The first one is with bivariate form and obtained by adding special terms, $\sum(a_ix^{2^i}y^{2^i},b_ix^{2^i}y^{2^i})$, to a known class of APN functions by {G{\"{o}}lo{\v{g}}lu} over $\gf_{{2^m}}^2$. The second one is of the form $L(z)^{2^m+1}+vz^{2^m+1}$ over $\gf_{{2^{3m}}}$, which is a generalization of one family of APN functions by Bracken et al. [Cryptogr. Commun. 3 (1): 43-53, 2011]. The calculation of the CCZ-invariants $\Gamma$-ranks of our APN classes over $\gf_{{2^8}}$ or $\gf_{{2^9}}$ indicates that they are CCZ-inequivalent to all known infinite families of APN functions. Moreover, by using the code isomorphism, we see that our first APN family covers an APN function over $\gf_{{2^8}}$ obtained through the switching method by Edel and Pott in [Adv. Math. Commun. 3 (1): 59-81, 2009].

cs.IT↗

arXiv2020

On two conjectures about the intersection distribution

Recently, S. Li and A. Pott\cite{LP} proposed a new concept of intersection distribution concerning the interaction between the graph $\{(x,f(x))~|~x\in\F_{q}\}$ of $f$ and the lines in the classical affine plane $AG(2,q)$. Later, G. Kyureghyan, et al.\cite{KLP} proceeded to consider the next simplest case and derive the intersection distribution for all degree three polynomials over $\F_{q}$ with $q$ both odd and even. They also proposed several conjectures in \cite{KLP}. In this paper, we completely solve two conjectures in \cite{KLP}. Namely, we prove two classes of power functions having intersection distribution: $v_{0}(f)=\frac{q(q-1)}{3},~v_{1}(f)=\frac{q(q+1)}{2},~v_{2}(f)=0,~v_{3}(f)=\frac{q(q-1)}{6}$. We mainly make use of the multivariate method and QM-equivalence on $2$-to-$1$ mappings. The key point of our proof is to consider the number of the solutions of some low-degree equations.

cs.IT↗

arXiv2020

A complete characterization of the APN property of a class of quadrinomials

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients $a_1,a_2,a_3\in\mathbb{F}_{2^n}$ with $n=2m$ such that $f(x) = {x}^{3\cdot2^m} + a_1x^{2^{m+1}+1} + a_2 x^{2^m+2} + a_3x^3$ is an APN function over $\mathbb{F}_{2^n}$. Our result resolves the first half of an open problem by Carlet in International Workshop on the Arithmetic of Finite Fields, 83-107, 2014.

cs.IT↗

arXiv2020

Binary linear codes with few weights from two-to-one functions

In this paper, we apply two-to-one functions over $\mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $\left(x^{2^t}+x\right)^e$ with $\gcd(t, n)=1$ and $\gcd\left(e, 2^n-1\right)=1$. Based on the study of the Walsh transforms of those functions or their related-ones, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. { Moreover, several examples show that some of our codes are optimal and some have the best known parameters.}

cs.IT↗

arXiv2020

Finding compositional inverses of permutations from the AGW criterion

Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of different forms, we propose a general method for finding these inverses of permutation polynomials constructed by the AGW criterion. As a result, we have reduced the problem of finding the compositional inverse of such a permutation polynomial over a finite field to that of finding the inverse of a bijection over a smaller set. We demonstrate our method by interpreting several recent known results, as well as by providing new explicit results on more classes of permutation polynomials in different types. In addition, we give new criteria for these permutation polynomials being involutions. Explicit constructions are also provided for all involutory criteria.

cs.IT↗