Motivated by the classification of one-to-one rational functions of low degree in terms of their equivalence classes, we determine all many-to-one (including one-to-one) rational functions of degree two or three on the projective line explicitly in terms of their coefficients. Furthermore, we study the linear-fractional equivalence classes and the value sets of these rational functions. As an application, we characterize two classes of many-to-one quadranomials using their coefficients. These one-to-one quadranomials unify and generalize many results in the literature.

Linear codes with few weights have attracted significant interest due to their wide-ranging applications in secret sharing, authentication codes, association schemes, strongly regular graph, and some other fields. This paper focuses on unifying several existing construction methods for few-weight linear codes, extending the works of Wang et al. (2015) [24], Wu et al. (2019) [25], and Fang et al. (2023) [10]. In our code construction, we introduce a novel index set J, whose cardinality and structural properties are shown to critically influence both the length and weight distribution of the resulting few-weight linear codes. By employing cyclotomic mappings and choosing the more general defining sets, several new classes of binary linear codes with at most three weights are constructed. Our framework subsumes all aforementioned constructions as special cases and enlarges the spectrum of attainable parameters. The weight distributions of the corresponding linear codes are also explicitly determined. We also demonstrate that some of the linear codes constructed in this paper are optimal in the sense that they have the best known parameters in the tables maintained by Markus Grassl and/or optimal in the sense that they meet certain bounds on linear codes. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The Rédei function defined over a field of even characteristic has been introduced by Nöbauer in 1986. In this paper, inspired by the work of Fu et al. in odd characteristic, employing the AGW criterion, we present a recursive construction of permutation polynomials in even characteristic using the Rédei function over a field of characteristic 2.

Using arbitrary bases for the finite field Fqn over Fq, we obtain the generalized M & ouml;bius transformations (GMTs), which are a class of bijections between the projective geometry PG(n-1, q) and the set of roots of unity mu qn-1 q-1 where n >= 2 is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of Fq & lowast;n,mu qn-1 q-1 and PG(n-1, q). Through this correspondence and the GMTs, we construct permutation polynomials of index qn-1 q-1 over Fqn . (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/). subset of Fqn ,

We introduce the definition of m-to-1 mappings between two finite sets, which unifies and generalizes the definitions of 2-to-1 and n-to-1 mappings in recent literature. We also characterize these m-to-1 mappings in terms of the generalized local criterion and thus provide three generic constructions of m-to-1 mappings, which unify and generalize the previous known constructions. Using these constructions, the problem whether xrh(xs) is m-to-1 on the multiplicative group F-q(& lowast;) is converted into that whether an associated polynomial x(r1)h(x)(s1q) is m(2)-to-1 on the order I subgroup U-l of F-q(& lowast;), where m(2) = m/(r, s) and I = (q-1)/s. Furthermore, the m(2)-to-1 property of x(r1)h(x)(s1) on U-l is studied in detail in four different cases. In addition, a recursive construction of m-to-1 mappings from m-to-1 mappings is proposed.

We provide a generic construction of permutation polynomials over Fq2 with index q+ 1 from any permutation polynomial of Fq. We also extend our construction using polynomials with coefficients in Fq2 such that they are injective over a subset of Fq & lowast;2, which corresponds to the set mu q +1 of all (q + 1)-th roots of unity. (c) 2026 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

The differential-linear connectivity table (DLCT), introduced by Bar-On et al. at EUROCRYPT'19, is a novel tool that captures the dependency between the two subciphers involved in differential-linear attacks. This paper is devoted to exploring the differential-linear properties of (n, n)-functions. First, by refining specific exponential sums, we propose two classes of power functions over F2nwith low differential-linear uniformity (DLU). Next, we further investigate the differential-linear properties of (n, n)-functions that are polynomials by utilizing power functions with known DLU. Specifically, by combining a cubic function with quadratic functions, and employing generalized cyclotomic mappings, we construct several classes of (n, n)-functions with low DLU, including some that achieve optimal or near-optimal DLU compared to existing results. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we obtain a class of bijections between the projective geometry PG(n-1,q) and the set of roots of unity μ_q^n-1/q-1 in finite field 𝔽_q^n for an arbitrary integer n≥ 2 and any basis of 𝔽_q^n over 𝔽_q. This generalizes the well-studied Möbius transformations for n=2 and a recent result by Qu and Li for n=3 [39]. We also introduce a class of projective polynomials, using the coefficients and properties of which we determine the inverses of these bijections. Moreover, we study the roots of these projective polynomials and explicitly describe the correspondence between a partition of μ_q^n-1/q-1 and a natural partition of PG(n-1,q). As an application, we can generalize many previously known constructions of permutation polynomials over 𝔽_q^2 and thus obtain new classes of permutation polynomials of 𝔽_q^n with index q^n-1/q-1.

Given a finite abelian group G, a finite set D, and a mapping f:D→ G , we find the number of r-subsets S⊆ D where for b∈ G , ∑ _x∈ Sf(x)=b. We count degree n monic polynomials over 𝔽_q with r distinct roots in a set D⊆𝔽_q when the leading terms of degree at least n-ℓ are fixed. We obtain new formulas for ℓ =2 when D is an arbitrary subfield of 𝔽_q with q odd.

Linear codes with few weights have wide applications in secret sharing, authentication codes, strongly regular graphs and association schemes. In this paper, we present linear codes from vectorial dual-bent functions and permutation polynomials, such that their parameters and weight distributions can be explicitly determined. In particular, some of them are three-weight optimal or almost optimal codes. As applications, we extend these codes to construct self-orthogonal codes and show the existence of asymmetric quantum codes. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Experimental results show that, when the order n is odd, there are de Bruijn sequences such that the corresponding complement sequence and the reverse sequence are the same. In this paper, we propose one efficient method to generate such de Bruijn sequences. This solves an open problem asked by Fredricksen forty years ago for showing the existence of such de Bruijn sequences when the odd order n >1 . Moreover, we refine a characterization of de Bruijn sequences with the same complement and reverse sequences and study the number of these de Bruijn sequences, as well as the distribution of de Bruijn sequences of the maximum linear complexity.

We study a new method of constructing Boolean bent functions from cyclotomic mappings. Three generic constructions are obtained by considering different branch functions such as Dillon functions, Niho functions and Kasami functions over multiplicative cosets and additive cosets respectively. As a result, several new explicit infinite families of bent functions and their duals are derived. We demonstrate that some previous constructions are special cases of our simple constructions. In addition, by studying their polynomial forms, we observe that the last construction provides some examples which are EA-inequivalent to five classes of monomials, Dillon type and Niho type polynomials.

We put forward new general criteria to design successor rules that generate binary de Bruijn sequences. Prior fast algorithms based on successor rules in the literature are then shown to be special instances. We implemented the criteria to join the cycles generated by a number of simple feedback shift registers (FSRs) of order $n$. These include the pure cycling register (PCR) and the pure summing register (PSR). For the PCR, we define a transitive relation on its cycles, based on their weights. We also extend the choices of conjugate states by using shift operations. For the PSR, we define three distinct transitive relations on its cycles, namely a run order, a necklace order, and a mixed order. Using the new orders, we propose numerous classes of successor rules. Each class efficiently generates a number, exponential in $n$, of binary de Bruijn sequences. Producing the next bit in each such sequence takes $O(n)$ memory and $O(n)$ time. We implemented computational routines to confirm the claims.

Many-to-one mappings and permutation polynomials over finite fields have important applications in cryptography and coding theory. In this paper, we study the many-to-one property of a large class of polynomials such as f(x) = h(a x^q + b x + c) + u x^q + v x, where h(x) ∈𝔽_q^2[x] and a, b, c, u, v ∈𝔽_q^2. Using a commutative diagram satisfied by f(x) and trace functions over finite fields, we reduce the problem whether f(x) is a many-to-one mapping on 𝔽_q^2 to another problem whether an associated polynomial g(x) is a many-to-one mapping on the subfield 𝔽_q. In particular, when h(x) = x^r and r satisfies certain conditions, we reduce g(x) to polynomials of small degree or linearized polynomials. Then by employing the many-to-one properties of these low degree or linearized polynomials on 𝔽_q, we derive a series of explicit characterization for f(x) to be many-to-one on 𝔽_q^2. On the other hand, for all 1-to-1 mappings obtained in this paper, we determine the inverses of these permutation polynomials. Moreover, we also explicitly construct involutions from 2-to-1 mappings of this form. Our findings generalize and unify many results in the literature.

In this paper, by using the Weil sums, we derive that the [Formula: see text]-differential uniformity of permutation polynomial of the form [Formula: see text] is equal to [Formula: see text] for [Formula: see text] with [Formula: see text] and [Formula: see text], or [Formula: see text] with [Formula: see text], where [Formula: see text]. In particular, several explicit classes of APcN permutations over [Formula: see text] are presented using different choices of the pair [Formula: see text] for the monomial [Formula: see text].

The functional graph of a function $g:X\rightarrow X$ is the directed graph with vertex set $X$ the edges of which are of the form $x\rightarrow g(x)$ for $x\in X$. Functional graphs are heavily studied because they allow one to understand the behavior of $g$ under iteration (i.e., to understand the discrete dynamical system $(X,g)$), which has various applications, especially when $X$ is a finite field $\mathbb{F}_q$. This paper is an extensive study of the functional graphs of so-called index $d$ generalized cyclotomic mappings of $\mathbb{F}_q$, which are a natural and manageable generalization of monomial functions. We provide both theoretical results on the structure of their functional graphs and Las Vegas algorithms for solving fundamental problems, such as parametrizing the connected components of the functional graph by representative vertices, or describing the structure of a connected component given by a representative vertex. The complexity of these algorithms is analyzed in detail, and we make the point that for fixed index $d$ and most prime powers $q$ (in the sense of asymptotic density), suitable implementations of these algorithms have an expected runtime that is polynomial in $\log{q}$ on quantum computers, whereas their expected runtime is subexponential in $\log{q}$ on a classical computer. We also discuss four special cases in which one can devise Las Vegas algorithms with this kind of complexity behavior over most finite fields that solve the graph isomorphism problem for functional graphs of generalized cyclotomic mappings.

We study the complexity (that is, the weight of the multiplication table) of the elliptic normal bases introduced by Couveignes and Lercier. We give an upper bound on the complexity of these elliptic normal bases, and we analyze the weight of some special vectors related to the multiplication table of those bases. This analysis leads us to some perspectives on the search for low complexity normal bases from elliptic periods.

A universal cycle for k-permutations is a cyclic arrangement in which each k-permutation appears exactly once as k consecutive elements. In this paper, we study the enumeration problem of universal cycles for k-permutations (Problem 477 Jackson et al. Discrete Mathematics, 309, 5341–5348, 2009) and obtain exact formulae for k=3, 4 .

The generalized cyclotomic mappings over finite fields $\mathbb{F}_{q}$ are those mappings which induce monomial functions on all cosets of an index $\ell$ subgroup $C_0$ of the multiplicative group $\mathbb{F}_{q}^{*}$. Previous research has focused on the one-to-one property, the functional graphs, and their applications in constructing linear codes and bent functions. In this paper, we devote to study the many-to-one property of these mappings. We completely characterize many-to-one generalized cyclotomic mappings for $1 \le \ell \le 3$. Moreover, we completely classify $2$-to-$1$ generalized cyclotomic mappings for any divisor $\ell$ of $q-1$. In addition, we construct several classes of many-to-one binomials and trinomials of the form $x^r h(x^{q-1})$ on $\mathbb{F}_{q^2}$, where $h(x)^{q-1}$ induces monomial functions on the cosets of a subgroup of $U_{q+1}$.

Additive codes have a wide range of applications. A classical nice and generic way to construct linear codes is via trace functions. In this paper, first, we generalize this method to construct additive codes. Then, we use this method to get some explicit additive codes. Computing Weil-like sums, we obtain parameters of these codes such as the length and weight distribution. We show that our codes have few weights.