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.

Fault Attacks on Hyperelliptic Curve Discrete Logarithm Problem over Finite Fields

In this paper, we present two explicit invalid-curve attacks on the genus 2 hyperelliptic curve over a finite field. First, we propose two explicit attack models by injecting a one-bit fault in a given divisor. Then, we discuss the construction of an invalid curve based on the faulted divisor. Our attacks are based on the fact that the Hyperelliptic Curve Scalar Multiplication (HECSM) algorithm does not utilize the curve parameters and We consider three hyperelliptic curves as the attack targets. For curve with security level 186 (in bits), our attack method can get the weakest invalid curve with security level 42 (in bits); there are 93 invalid curves with security level less than 50. We also estimate the theoretical probability of getting a weak hyperelliptic curve whose cardinality is a smooth integer. Finally, we show that the complexity of the fault attack is subexponential if the attacker can freely inject a fault in the input divisor. Cryptosystems based on the genus 2 hyperelliptic curves cannot work against our attack algorithm in practice.

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.

Carter subgroups of singular classical groups over finite fields

Let F q be a finite field with qelements where q=p~α. In the present paper, the authors study the existence and structure of Carter subgroups of singular symplectic group Sp (n+t,n)(F q), singular unitary group U (n+t,n)(F (q^2)) and singular orthogonal group O (n+t,n)(F q)(n is even) over finite fields F q.

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.

Some Secret Sharing Schemes Based on the Finite Fields

A （t, n）--secret sharing scheme is a method of distribution of information among n participants such that t 〉 1 can reconstruct the secret but （t - 1） cannot. We explore some （k, n）--secret sharing schemes based on the finite fields.

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.

Approximate Synchronization of Multi-Agent Systems over Finite Fields

In this paper,the approximate synchronization of leader-follower multiagent systems(MASs) over finite fields is studied in regard to local and global synchronization.First,the approximately synchronous state set(ASSS) is obtained.Second,combined with ASSS and transient periods,some criteria for the local and global approximate synchronization of systems are given.Moreover,the algorithms for calculating the maximum approximately synchronous basin(MASB) and the maximum control invariant set(MCIS) are presented.Third,the global approximate synchronization of the system is achieved by designing the state feedback control,and a design algorithm of the controller using the truth matrix method is proposed.Moreover,the results of approximate synchronization are degenerated to complete synchronization.Last,two examples are shown to demonstrate the results of this paper.

Zeros of Complete Symmetric Polynomials over Finite Fields

Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields,and proposed a problem whether their bound can be improved.In this paper,the author improves Wan-Zhang's bound from three aspects.The proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric polynomial.And the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound.

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.

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.

Effect of Types and Orders of Electromagnetic Field Finite Element Meshes on Power Communication Harmonic Parameters Calculation Results of Tubular Hydrogenerators

In generator design field,waveform total harmonic distortion(THD)and telephone harmonic factor(THF)are parameters commonly used to measure the impact of generator no-load voltage harmonics on the power communication quality.Tubular hydrogenerators are considered the optimal generator for exploiting low-head,high-flow hydro resources,and they have seen increasingly widespread application in China's power systems recent years.However,owing to the compact and constrained internal space of such generators,their internal magnetic-field harmonics are pronounced.Therefore,accurate calculation of their THD and THF is crucial during the analysis and design stages to ensure the quality of power communication.Especially in the electromagnetic field finite element modeling analysis of such generators,the type and order of the finite element meshes may have a significant impact on the THD and THF calculation results,which warrants in-depth research.To address this,this study takes a real 34 MW large tubular hydrogenerator as an example,and establishes its electromagnetic field finite element model under no-load conditions.Two types of meshes,five mesh densities,and two mesh orders are analyzed to reveal the effect of electromagnetic field finite element mesh types and orders on the calculation results of THD and THF for such generators.

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

In this paper, one construction of Cartesian authentication codes from the normal form of matrices over finite fields are presented and its size parameters are computed. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the P I and P S , which denote the largest probabilities of a successful impersonation attack and of a successful substitution attack respectively, of these codes are also computed.

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.

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.

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.

On observability of Galois nonlinear feedback shift registers over finite fields

Observability ensures that any two distinct initial states can be uniquely determined by their outputs,so the stream ciphers can avoid unobservable nonlinear feedback shift registers(NFSRs)to prevent the occurrence of equivalent keys.This paper discusses the observability of Galois NFSRs over finite fields.Galois NFSRs are treated as logical networks using the semi-tensor product.The vector form of the state transition matrix is introduced,by which a necessary and sufficient condition is proposed,as well as an algorithm for determining the observability of general Galois NFSRs.Moreover,a new observability matrix is defined,which can derive a matrix method with lower computation complexity.Furthermore,the observability of two special types of Galois NFSRs,a full-length Galois NFSR and a nonsingular Galois NFSR,is investigated.Two methods are proposed to determine the observability of these two special types of NFSRs,and some numerical examples are provided to support these results.