Finite Fields and Their Applications最新文献

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Finite fields whose members are the sum of a potent and a 4-potent 有限域，其成员是幂次域和4幂次域的和

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-22 DOI: 10.1016/j.ffa.2025.102739

Stephen D. Cohen , Peter V. Danchev , Tomás Oliveira e Silva

We classify those finite fields

F_{q}

whose members are the sum of an n-potent element with

n > 1

and a 4-potent element. It is shown that there are precisely ten non-trivial pairs

(q, n)

for which this is the case. This continues a recent publication by Abyzov et al. (2024) [1] in which the tripotent version was examined in-depth, inasmuch as it extends recent results in this seam of research established by Abyzov and Tapkin (2024) [4].

我们对有限域Fq进行了分类，这些域的成员是一个n强元素和一个4强元素的和。结果表明，恰好有十个非平凡对（q,n）存在这种情况。这是Abyzov等人（2024）[1]最近发表的一篇文章的延续，其中对三能性版本进行了深入研究，因为它扩展了Abyzov和Tapkin(2024)[1]建立的这一研究领域的最新成果。

引用次数: 0

Mean value theorems for short rational exponential sums 短有理指数和的中值定理

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-09-09 DOI: 10.1016/j.ffa.2025.102719

Doowon Koh , Igor E. Shparlinski

We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other ideas.

我们得到了一系列最近关于Weyl和的各种中值定理的结果的有限域类似物。我们的结果不是基于维诺格拉多夫中值定理，而是基于莫德尔的经典论点，并结合了其他几个观点。

引用次数: 0

Further results on permutation pentanomials over Fq3 in characteristic two 特征二上Fq3上排列五反常的进一步结果

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-20 DOI: 10.1016/j.ffa.2025.102743

Tongliang Zhang , Lijing Zheng , Hengtai Wang , Jie Peng , Yanjun Li

Let

q = 2^{m}

. In a recent paper [34], Zhang and Zheng investigated several classes of permutation pentanomials of the form

ϵ_{0} x^{d_{0}} + L (ϵ_{1} x^{d_{1}} + ϵ_{2} x^{d_{2}})

over

F_{q^{3}} (d_{0} = 1, 2, 4)

with a certain linearized polynomial

L (x)

. They applied the multivariate method and specific techniques to analyze the number of solutions of certain equations, and proposed an open problem: the permutation property of some pentanomials of this form remains unproven. In this paper, inspired by the idea of [12], we further characterize the permutation property of such pentanomials over

F_{q^{3}} (d_{0} = 1, 2, 4)

. The techniques presented in this paper will be useful for investigating more new classes of permutation polynomials.

让q = 2 m。在最近的一篇论文[34]中，Zhang和Zheng研究了几种形式为ϵ0xd0+L（ϵ1xd1+ϵ2xd2） / Fq3（d0=1,2,4）的具有一定线性化多项式L(x)的置换五反常。他们运用多元方法和特定技术分析了某些方程的解的个数，并提出了一个开放性问题：一些这种形式的五反常项的置换性质尚未得到证明。在本文中，受[12]思想的启发，我们进一步刻画了Fq3（d0=1,2,4）上这类五反常的置换性质。本文提出的技术将有助于研究更多新的置换多项式类。

引用次数: 0

Continued fractions and indefinite binary quadratic forms over Fq[t] Fq上的连分式和不定二元二次型[t]

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-31 DOI: 10.1016/j.ffa.2025.102747

Jorge Morales

The relation between cycles of indefinite binary quadratic forms over

Z

and continued fractions is classical and well-known. We describe a similar relation for binary quadratic forms over the polynomial ring

F_{q} [t]

, where q is a power of an odd prime. In this context, the cycles of the classical theory are replaced by orbits of the metacyclic group

F_{q}^{⁎} ⋊ Z

acting on the set of reduced forms of a given discriminant, where each orbit corresponds to a proper equivalence class.

Z上不定二元二次型的循环与连分式之间的关系是经典而众所周知的。我们描述了多项式环Fq[t]上二元二次型的类似关系，其中q是奇素数的幂。在这种情况下，经典理论的循环被作用于给定判别式的约简形式集合上的亚环群Fq Z的轨道所取代，其中每个轨道对应于一个适当的等价类。

引用次数: 0

Cardinality-consistent flag codes with larger cardinality 具有较大基数的基数一致的标志代码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-29 DOI: 10.1016/j.ffa.2025.102750

Junfeng Jia, Yanxun Chang

Flag codes, as a generalization of subspace codes, can transmit more information since the subspace channel is used many times. In this paper, we construct optimum distance flag codes of the (generalized) full admissible type

t = (1, \dots, k, n - k, \dots, n - 1)

F_{q}^{n}

with cardinality

\sum_{i = 1}^{s - 1} q^{i k + h} + 1

, where

n = s k + h

with

s \geq 2

and

0 \leq h < k

. Let

A_{q}^{f} (n, D^{(t, n)}, t)

denote the maximum cardinality of such codes. We provide a lower bound for this quantity. We further present a systematic construction of cardinality-consistent flag codes with larger cardinality for general flag distances. By the composition of subspace polynomials, we construct cardinality-consistent cyclic flag codes on

F_{q^{n}}

with larger cardinality than those presented in the literature.

标志码作为子空间码的泛化，由于子空间信道被多次使用，可以传输更多的信息。本文在基数∑i=1s - 1qik+h+1的Fqn上构造了（广义）完全可容许型t=（1，…，k,n - k，…，n - 1）的最优距离标志码，其中n=sk+h, s≥2,0≤h<k。设Aqf（n，D(t,n),t）表示这些码的最大基数。我们给出了这个量的下界。我们进一步提出了对一般旗距具有较大基数的基数一致旗码的系统构造。通过子空间多项式的组合，我们在Fqn上构造了基数一致的循环标志码，其基数大于已有的循环标志码。

引用次数: 0

An unusual family of supersingular curves of genus five in characteristic two 特征二的五属超奇异曲线的一个不寻常的族

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-10 DOI: 10.1016/j.ffa.2025.102736

Dušan Dragutinović

We construct a family of smooth supersingular curves of genus 5 in characteristic 2 with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus 5, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-2 curve. We also provide an explicit parametrization of this family.

构造了特征2上的5属光滑超奇异曲线族，它的维数与5属超奇异轨迹的任意分量的期望维数相匹配，它的成员是具有非平凡自同构群的非超椭圆曲线，族中的每条曲线都在椭圆曲线和2属曲线上具有双覆盖结构。我们还提供了这个家族的显式参数化。

引用次数: 0

The dual codes of two families of BCH codes 两个BCH码族的双码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-09-17 DOI: 10.1016/j.ffa.2025.102721

Haojie Xu , Xia Wu , Wei Lu , Xiwang Cao

In this paper, we present an infinite family of MDS codes over

F_{2^{s}}

and two infinite families of almost MDS codes over

F_{p^{s}}

for any prime p, by investigating the parameters of the dual codes of two families of BCH codes. Notably, these almost MDS codes include two infinite families of near MDS codes over

F_{3^{s}}

, resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost MDS codes and their dual codes hold infinite families of 3-designs over

F_{p^{s}}

for any prime p. Additionally, we study the subfield subcodes of these families of MDS and near MDS codes, and provide several binary, ternary, and quaternary codes with best known parameters.

本文通过研究两族BCH码的对偶码的参数，给出了任意素数p上F2s上的无限族MDS码和Fps上的两个无限族几乎MDS码。值得注意的是，这些近MDS码包括F3s上的两个无限近MDS码族，解决了耿等人在2022年提出的一个猜想。此外，我们证明了这些几乎MDS码和它们的对偶码在任意素数p的Fps上都具有无限族的3-设计。此外，我们研究了这些MDS和近MDS码族的子域子码，并提供了几种具有最已知参数的二进制、三进制和四进制码。

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引用次数: 0

Cryptanalysis of some algebraic variants of the RSA cryptosystem RSA密码系统的一些代数变体的密码分析

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-11-04 DOI: 10.1016/j.ffa.2025.102757

Mohammed Rahmani , Abderrahmane Nitaj , Mhammed Ziane

Let

N = p q

be an RSA modulus, and

n \geq 1

be an integer. Two recently algebraic variants of the RSA cryptosystem use a public exponent e for encryption, and a private exponent d for decryption with

e d \equiv 1 (\mod φ_{n} (N))

, where

φ_{n} (N) = (p^{n} - 1) (q^{n} - 1)

. In this paper, we propose an attack on the two variants using Coppersmith's method and lattice basis reduction. Our attack breaks the systems when d is less than an explicit bound that depends only on n and N. We analyze the security of the RSA variants characterized by the equation

e d - k φ_{n} (N) = 1

. Specifically, we propose a novel attack utilizing lattice-based methods and Coppersmith's technique, when the prime numbers p and q share an amount of their least significant bits. This enables the efficient recovery of the primes p and q in polynomial time.

设N=pq为RSA模，N≥1为整数。RSA密码系统的两个最近的代数变体使用公共指数e进行加密，使用私有指数d进行解密，其中ed≡1(modφn(N))，其中φn(N)=(pn−1)（qn−1）。在本文中，我们提出了一种利用Coppersmith方法和格基约简的方法来对付这两种变体的方法。当d小于仅依赖于n和n的显式边界时，我们的攻击破坏了系统。我们分析了等式ed−kφn(n)=1表征的RSA变体的安全性。具体来说，我们提出了一种利用基于格的方法和Coppersmith技术的新攻击，当素数p和q共享其最低有效位的数量时。这使得在多项式时间内有效地恢复素数p和q。

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引用次数: 0

On sum-free functions 关于无和函数

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-10-21 DOI: 10.1016/j.ffa.2025.102744

Alyssa Ebeling , Xiang-dong Hou , Ashley Rydell , Shujun Zhao

A function from

F_{2^{n}}

F_{2^{n}}

is said to be kth order sum-free if the sum of its values over each k-dimensional

F_{2}

-affine subspace of

F_{2^{n}}

is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function

f_{inv} (x) = x^{- 1}

(with

0^{- 1}

defined to be 0). It is known that

f_{inv}

is 2nd order (equivalently,

(n - 2)

th order) sum-free if and only if n is odd, and it is conjectured that for

3 \leq k \leq n - 3

f_{inv}

is never kth order sum-free. The conjecture has been confirmed for even n but remains open for odd n. In the present paper, we show that the conjecture holds under each of the following conditions: (1)

n = 13

; (2)

3 | n

; (3)

5 | n

; (4) the smallest prime divisor l of n satisfies

(l - 1) (l + 2) \leq (n + 1) / 2

. We also determine the “right” q-ary generalization of the binary multiplicative inverse function

f_{inv}

in the context of sum-freedom. This q-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case.

如果一个从F2n到F2n的函数在F2n的每一个k维的f2仿射子空间上的值的和是非零的，那么这个函数就是无k阶和的。这个概念是最近由C. Carlet引入的，作为APN函数的推广。这个新主题的中心是一个关于乘法反函数finv(x)=x−1（其中0−1定义为0）的和自由度的猜想。已知finv是二阶（即（n−2）阶）自由和当且仅当n为奇数，并且推测当3≤k≤n−3时，finv绝不是第k阶自由和。对于偶数n，该猜想已被证实，但对于奇数n，该猜想仍然是开放的。在本文中，我们证明了该猜想在下列条件下成立：(1)n=13；(2) 3 | n;(3) 5 | n;(4) n的最小素数因子l满足（l−1）(l+2)≤(n+1)/2。我们还确定了二元乘法反函数finv在自由和情况下的“正确”q-ary泛化。这种q-ary泛化不仅保留了其二进制版本的大多数结果，而且还展示了一些在二进制情况下没有观察到的特殊现象。

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引用次数: 0

Brocard-Ramanujan problem for polynomials over finite fields 有限域上多项式的Brocard-Ramanujan问题

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2026-02-01 Epub Date: 2025-09-30 DOI: 10.1016/j.ffa.2025.102731

Wataru Takeda

The Brocard-Ramanujan problem is an unsolved number theory problem to find integer solutions

(x, n)

x^{2} - 1 = n!

. In this paper, we consider this problem over polynomial rings

F_{q} [T]

, where

F_{q}

is a finite field with q elements. We find all solutions to the equation

X^{2} - 1 = Π_{C} (n)

, where

Π_{C} (n)

denotes the Carlitz factorial. More precisely, we characterize all solutions and prove that there are infinitely many solutions if and only if

F_{q}

is an extension of

F_{4}

. This characterization is achieved without using the Mason-Stothers theorem, analogous to the abc conjecture for integers.

Brocard-Ramanujan问题是求解x2−1=n!的整数解（x,n）的未解数论问题。本文考虑多项式环Fq[T]上的这一问题，其中Fq是一个有q个元素的有限域。我们找到方程X2−1=ΠC(n)的所有解，其中ΠC(n)表示Carlitz阶乘。更准确地说，我们刻画了所有解，并证明了当且仅当Fq是F4的扩展时存在无穷多个解。这个特征是不使用梅森-斯托瑟斯定理，类似于整数的abc猜想。

引用次数: 0

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