Using Normal Form of Nilpotent Matrices over Finite Fields to Construct Cartesian Authentication Codes

Let Fq be a finite field. In this paper, a construction of Cartesian au-thentication codes from the normal form of a class of nilpotent matrices over the field Fq is presented. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the probabilities PI and PS, of a successful im-personation attack and of a successful substitution attack respectively, of these codes are also computed.

AN ANZAHL THEOREM IN SINGULAR PSEUDO-SYMPLECTIC GEOMETRY OVER FINITE FIELDS

This paper obtains formulas for the number of subspaces of a given type contained in a given subspace in singular pseudo-symplectic geometry.

SOME PRIMITIVE POLYNOMIALS OVER FINITE FIELDS

This paper proves that if qn is large enough, for each element a and primitive element b of Fq, there etists a primitive polynomial of degree n ≥5 over the finite field Fq having a as the coefficient of xn-1 and b as the constant term. This proves that if qn is large enongh, for each element a ∈Fq, there exists a primitive polynomial of degree n ≥ 5 over Fq having a as the coefficient of x.

On the Number of Solutions of Some Type Equations in Finite Fields

By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.

Construction of PBIB Desings by Using Subspace of Vector Space over Finite Fields

A new transitivity theorem of the general linear group GLn(Fq)is proved. A kind of new PBIB desings is constructed.

Some New Results about Trigonometry in Finite Fields

In this paper, we study about trigonometry in finite field, we know that , the field with p elements, where p is a prime number if and only if p = 8k + 1 or p = 8k -1. Let F and K be two fields, we say that F is an extension of K, if K⊆F or there exists a monomorphism f: K→F. Recall that , F[x] is the ring of polynomial over F. If (means that F is an extension of K), an element is algebraic over K if there exists such that f(u) = 0 (see [1]-[4]). The algebraic closure of K in F is , which is the set of all algebraic elements in F over K.

Dielectric Constant Calculation of Poly(vinylidene fluoride)Based on Finite Field and Density Functional Theory

In this study,we proposed a novel method that integrates density functional theory(DFT)with the finite field method to accurately estimate the polarizability and dielectric constant of polymers.Our approach effectively accounts for the influence of electronic and geometric conformation changed on the dielectric constant.We validated our method using polyethylene(PE)and polytetrafluoroethylene(PTFE)as benchmark materials,and found that it reliably predicted their dielectric constants.Furthermore,we explored the impact of conformation variations in poly(vinylidene fluoride)(PVDF)on its dielectric constant and polarizability.The resulting dielectric constants ofα-andγ-PVDF(3.0)showed excellent agreement with crystalline PVDF in experiments.Our findings illuminate the relationship between PVDF’s structural properties and its electrical behavior,offering valuable insights for material design and applications.

Note on the Number of Solutions of Cubic Diagonal Equations over Finite Fields

Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.

On Deep Holes of Projective Reed-Solomon Codes over Finite Fields with Even Characteristic

Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic.That is,let F_(q) be finite field with even characteristic,k∈{2,q-2},and let u(x)be the Lagrange interpolation polynomial of the first q components of the received vector u∈F_(q)+1 q Suppose that the(q+1)-th component of u is 0,and u(x)=λx^(k)+f_(≤k-2)(x),λx^(q-2)+f_(≤k-2)(x),where λ∈F^(*)_(q) and f_(≤k-2)(x)is a polynomial over F_(q) with degree no more than k-2.Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(F_(q),k).In fact,our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.

A Class of Quadratic Matrix Equations over Finite Fields

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields.We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials.We also find a generating set for the vanishing ideal of these orbits.

Normal Bases and Their Dual-Bases over Finite Fields

In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual if and only if its multiplication table is symmetric and Tr（α^2） = 1, where α generates N; 3） An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis with q = n = 2 or N is a type-Ⅱ optimal normal basis.

Two Constructions of Optimal Cartesian Authentication Codes from Unitary Geometry over Finite Fields

Two constructions of cartesian authentication codes from unitary geometry are given in this paper. Their size parameters and their probabilities of successful impersonation attack and successful substitution attack are computed. They are optimal under some cases.

On the Number of Solutions of Certain Equations over Finite Fields

Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋···＋ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F＊q,b ∈ F＊q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.

A Construction of Authentication Codes with Arbitration from Vector Spaces over Finite Fields

This paper is devoted to constructing an authentication code with arbitration using subspaces of vector spaces over finite fields.Moreover,if we choose the encoding rules of the transmitter and the decoding rules of the receiver according to a uniform probability distribution,then some parameters and the probabilities of successful attacks are computed.

Exponential Sums over Finite Fields

This is an expository paper on algebraic aspects of exponential sums over finite fields.This is a new direction.Various examples,results and open problems are presented along the way,with particular emphasis on Gauss periods,Kloosterman sums and one variable exponential sums.One main tool is the applications of various p-adic methods.For this reason,the author has also included a brief exposition of certain p-adic estimates of exponential sums.The material is based on the lectures given at the 2020 online number theory summer school held at Xiamen University.Notes were taken by Shaoshi Chen and Ruichen Xu.

Two constructions of A^3-codes from projective geometry in finite fields

An A^3-code is extension of A^2-code in which none of the three participants： transmitter, receiver and arbiter, is assumed trusted. In this article, from projective geometry over finite fields, two A^3 -codes were given, the parameters, and probabilities of successful attacks were computed.

On Primitive Optimal Normal Elements of Finite Fields

Let q be a prime or prime power and Fq^n the extension of q elements finite field Fq with degree n （n 〉 1）. Davenport, Lenstra and School proved that there exists a primitive element α ∈ Fq^n such that α generates a normal basis of Fq^n over Fq. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements α and α^-1 such that both of them generate optimal normal bases of Fq^n over Fq. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.

A New Criterion on k-Normal Elements over Finite Fields

The notion of normal elements for finite fields extension was generalized as k-normal elements by Huczynska et al.(2013).Several methods to construct k-normal elements were presented by Alizadah et al.(2016)and Huczynska et al.(2013),and the criteria on k-normal elements were given by Alizadah et al.(2016)and Antonio et al.(2018).In the paper by Huczynska,S.,Mullen,G.,Panario,D.and Thomson,D.(2013),the number of k-normal elements for a fixed finite field extension was calculated and estimated.In this paper the authors present a new criterion on k-normal elements by using idempotents and show some examples.Such criterion was given for usual normal elements before by Zhang et al.(2015).

Construction of authentication codes with distrust arbitration from polynomials over finite fields

The authentication codes with arbitration are able to solve dispute between the sender and the receiver. The authentication codes with trusted arbitration are called AZ-codes, the authentication codes with distrust arbitration are called A3-codes . As an expansion of Az-cOdes , an A3-code is an authentication system which is closer to the reality environment. Therefore, A3-codes have more extensive application value. In this paper, we construct a class of A3-codes based on polynomials over finite fields, give the parameters of the constructed codes, and calculate a variety of cheating attacks the maximum probabilities of success. Especially, in a special case, the constructed A3-codes are perfect. Compared with a known type of codes, they have almost the same security level, however, our codes need less storage requirements. Therefore, our codes have more advantages.

On Existence of Primitive Normal Elements of Cubic Form over Finite Fields

For a prime p and a positive integer k,let q=p^(k) and F_(q)^(n) be the extension field of F_(q).We derive a sufficient condition for the existence of a primitive element α in F_(q)^(n) such that α^(3)-α+1 is also a primitive element of F_(q)^(n) ,a sufficient condition for the existence of a primitive normal element a in F_(q)^(n) over F_(q) such that α(3)-α+1 is a primitive element of F_(q)^(n) ,and a suficient condition for the existence of a primitive normal element a in F_(q)^(n) over F_(q) such that а^(3)-а+1 is also a primitive normal element of F_(q)^(n) over F_(q).