Article
Computer Science, Theory & Methods
Rohit Gupta, Luciane Quoos, Qiang Wang
Summary: In this paper, by classifying certain sparse PPs, we provide necessary and sufficient conditions for a polynomial f a,b(x) to be a permutation polynomial over F-q^n.
CRYPTOGRAPHY AND COMMUNICATIONS-DISCRETE-STRUCTURES BOOLEAN FUNCTIONS AND SEQUENCES
(2023)
Article
Mathematics, Applied
Rohit Gupta, Pooja Gahlyan, R. K. Sharma
Summary: In this paper, three new classes of permutation trinomials over Fq3 are proposed, and necessary and sufficient conditions for these trinomials are studied. A conjecture proposed by Gong, Gao, and Liu is proven as a special case, and it is shown that these trinomials are different from known permutation trinomials over Fq3.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
Ariane M. Masuda, Ivelisse M. Rubio, Javier Santiago
Summary: This article presents the existence and nonexistence results for permutation binomials of the form x(r)(x(q-1) + a), with specified conditions. As a consequence, a complete characterization of such permutation binomials is obtained for several cases.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics
Xi Xie, Nian Li, Linjie Xu, Xiangyong Zeng, Xiaohu Tang
Summary: This paper proposes two classes of permutation trinomials for a finite field Fq3 with an arbitrary odd characteristic. The trinomials are constructed using the multivariate method and subtle manipulation of solving equations with low degrees over finite fields. The paper also demonstrates that these trinomials are QM inequivalent to all known permutation polynomials over Fq3. This study is the first to explore the construction of nonlinearized permutation trinomials of Fq3 with at least one coefficient lying in Fq3\Fq.
DISCRETE MATHEMATICS
(2023)
Article
Mathematics, Applied
Xiaoer Qin, Li Yan
Summary: Constructing permutation polynomials in finite fields is a popular topic. This paper investigates the construction of permutation polynomials over F-q3, using the AGW criterion and piecewise method. Several classes of permutation polynomials of the form (x(q2) + x(q) + x + delta)(q3-1/d) +1 + L(x), where d = 2, 3, 4, 6 and L(x) is a linearized polynomial over F-q, are constructed.
Article
Mathematics, Applied
Lijing Zheng, Haibin Kan, Jie Peng, Deng Tang
Summary: This paper investigates two classes of permutation trinomials with Niho-type exponents over the finite field F-22m. The problem is transformed into studying quartic equations over the subfield F-2m, and it is shown that these equations have no solutions in F-2m. Sufficient conditions are established to characterize the coefficients in the permutation polynomials, and it is found that these conditions are also necessary in certain cases.
FINITE FIELDS AND THEIR APPLICATIONS
(2021)
Article
Mathematics, Applied
Daniele Bartoli, Marco Timpanella
Summary: This paper investigates permutation polynomials on F-q2 and reveals their connections with algebraic curves over finite fields. The previously known sufficient conditions are also shown to be necessary.
FINITE FIELDS AND THEIR APPLICATIONS
(2021)
Article
Mathematics
Jose Alves Oliveira, F. E. Brochero Martinez
Summary: This paper establishes the exact number of elements in the finite field F-q with a certain property by utilizing the relation between suitable polynomials and the number of rational points on algebraic curves, as well as results on the Cartin-Manin operator and Riemann Hypothesis for elliptic curves.
DISCRETE MATHEMATICS
(2022)
Article
Mathematics, Applied
Christof Beierle
Summary: This article demonstrates that for every integer d satisfying p < d < 2(p - 1), there exists a generalized almost perfect nonlinear (GAPN) binomial or trinomial over Fp2 with algebraic degree d.
FINITE FIELDS AND THEIR APPLICATIONS
(2023)
Article
Mathematics, Applied
Daniela Oliveira, Lucas Reis
Summary: This paper examines positive integers n for which the irreducible factors of the polynomial x(n) - 1 over a finite field F-q are all binomials and trinomials. Particularly, the integers meeting this condition are fully described for q = 2, 4.
FINITE FIELDS AND THEIR APPLICATIONS
(2021)
Article
Mathematics, Applied
Zhilin Zhang, Lang Tang, Ningjing Huang
Summary: This paper provides a detailed description of the binomial theorem and introduces new classes of combinatorial identities. An application of permutation binomials over finite fields F-q is discussed, under certain conditions.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2022)
Article
Mathematics
Ekram E. Ali, Hari M. Srivastava, Abdel Moneim Y. Lashin, Abeer M. Albalahi
Summary: In this article, two new subclasses (aq, q) and (a, q) of meromorphic functions in the open unit disk U are introduced and studied using the q-binomial theorem. These subclasses refer to analytic functions in the punctured unit disk U-* = U \ {0} = {z : z ? C and 0 < |z| < 1}. The inclusion relations are derived and an integral operator that preserves functions in these function classes is investigated. Additionally, a strict inequality involving a newly introduced linear convolution operator is established, and special cases and corollaries of the main results are considered.
Article
Mathematics, Applied
Daniele Bartoli, Marco Timpanella
Summary: The study focuses on the algebraic curves associated with the polynomial F-A, F-B, F-m, F-n(X) and the investigation of their singularities to provide necessary conditions for the polynomial to be permutations under certain conditions.
FINITE FIELDS AND THEIR APPLICATIONS
(2021)
Article
Mathematics, Applied
Qian Liu, Guifeng Chen, Ximeng Liu, Jian Zou
Summary: This paper studies the permutation property of pentanomials with the form xrh(xpm-1) over Fp2m, where p∈{2, 3}. Based on the solutions of some low-degree equations, eight classes of permutation pentanomials over F22m and F32m are presented. Moreover, several classes of permutation pentanomials and heptanomials are derived from known permutation polynomials on μ2m+1 and μ3m+1, respectively.
FINITE FIELDS AND THEIR APPLICATIONS
(2023)
Article
Mathematics, Applied
Anuj Jakhar, Surender Kumar
Summary: In this paper, explicit conditions involving s, q, a, and b are provided to determine whether K is non-monogenic. In the special case of a = 0 and q = 2, it is shown that if s >= 2 and 32 divides b - 1, then K is not monogenic. Examples are provided to illustrate the results.
JOURNAL OF ALGEBRA AND ITS APPLICATIONS
(2023)
