偶特征有限域上一类正规基及其对偶基(英文)

设正整数n(≥2),N={α_i|i=0,1,…,n-1)是有限域F_(2n)在F_2的正规基,且t_i=Tr(αα_i)(i=0,1,…,n-1),其中Tr(α)是α∈F_(2n)在F_2上的迹映射.本文讨论了F_(2n)在F_2上的满足如下条件的高斯正规基的存在性:t_0=t_1=t_(n-1),t_i=0(i≠0,1,n-1).给出了这种正规基的对偶基,并由此确定了F_(2n)在F_2上满足上述条件的全部最优正规基.     

有限域上最优正规基的乘法表

本文给出了有限域上最优正规基乘法表的一个计算方法,改进了孙琦的相应结果.在有限域上椭圆曲线密码体制的应用中,本文给出的算法是非常有效的.     

有限域上存在弱自对偶正规基的一个充要条件

对于将有限域上的自对偶基概念推广到了更一般的弱自对偶的情形,给出了有限域上存在这类正规基的一个充要条件设q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αq2| i=0,1,…,n-1}为E在F上的一组正规基.则存在c∈F*及r,0≤r≤n-1,使得β=cαr生成N的对偶基的充要条件是以下三者之一成立(1)q为偶数且n≠0(mod 4);(2)n与q均为奇数;(3)q为奇数,n为偶数,(-1)为F中的非平方元且r为奇数.     

有限域上存在弱自对偶正规基的一个充要条件

对于将有限域上的自对偶基概念推广到了更一般的弱自对偶的情形,给出了有限域上存在这类正规基的一个充妥条件:设q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αqi|i=0,1,…,n-1}为E在F上的一组正规基．则存在c∈F*及r,0≤r≤n-1,使得β=cαr生成N的对偶基的充要条件是以下三者之一成立: (1)q为偶数且n≠0(mod 4);(2) n与q均为奇数;(3)q为奇数,n为偶数,(-1)为F中的非平方元且r为奇数．     

有限域上的弱自对偶正规基

对有限域上的弱自对偶正规基的乘法表的特征进行了刻画,并对其复杂度进行了研究,得到了在几种不同类型的有限域扩张时此类正规基的下界描述.例如,若q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αqi|I=0,1,…,n-1}为E在F上的一组弱自对偶正规基,其对偶基由β=cαr生成,其中c∈F*,0≤r≤n-1,则当r≠0,n/2时,N的复杂度CN为偶数且CN≥4n-2.     

有限域上一类不可约多项式（英文）

设F_q为q元有限域,其中q是素数p的幂,设n是一个正整数.F_q上一个n次首一多项式f(x)的迹定义为x~(n-1)的系数.令N_q(n,t)表示F_q上迹为t∈F_q的n次首一不可约多项式的个数.基于给定多项式的普通分解与其线性化q-相伴式的符号分解之间的关系,本文给出了一种计算N_q(n,t)的新途径.     

有限域上一类高斯正规基复杂度的准确计算公式

通过刻画有限域上分圆数的性质,给出了有限域上一类高斯正规基复杂度的准确计算公式.进而证明了有限域F_(q~n)在F_q上的7型高斯正规基满足所给条件当且仅当n≠4.     

有限域上互反本原正规元的存在性

设q是素数方幂,n是正整数,Fqn是qn个元素的有限域．本文证明了:当正整数n≥32时,对任意的素数方幂q,存在Fqn中的本原元ξ满足ξ和ξ-1都是Fqn 在Fq上的正规元,也即{ξ,ξq,…,ξqn-1}和{ξ-1,ξ-q,…,ξ-qn-1)都构成Fqn在Fq 上的本原正规基．     

有限域上n元n次方程只有零解的充要条件

本文研究了有限域上只有零解的n元n次方程的结构问题.利用对有限域上不可约多元多项式在其扩域中的分解特征的刻画,结合Chevalley定理,得到了有限域上n元n次方程只有零解的一个充要条件,并给出这类方程的一些新的具体构造.     

有限域上幂剩余正规元的存在性

GF(q)是q个元的有限域,q是素数的方幂,n是正整数,GF(qn)为GF(q)的n次扩张.用指数和估计的方法给出了3种情形下幂剩余正规元存在的充分条件,即(1)GF(qn)中存在元ξ为GF(q)上的幂剩余正规元;(2)GF(qn)中存在元ξ与ξ-1同时为GF(q)上幂剩余正规元;(3)对GF(qn)*中任意给定的非零元a和b,GF(qn)中存在元ξ与ξ-1同时为GF(q)上d次幂剩余正规元,且满足Tr(ξ)=a,Tr(ξ-1)=b.     

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.     

Self-Reciprocal Irreducible Pentanomials Over \mathbb{F}_2

Joseph Yucas and Gary Mullen conjectured that there is no self-reciprocal irreducible pentanomial of degree n over if n is divisible by 6. In this note we prove this conjecture for the case n ≡ 0, and disprove the conjecture for the case n ≡ 6 (mod 12) AMS Classifications: 11T55     

On Counting Certain Abelian Varieties Over Finite Fields

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers[Doc. Math., 21, 1607-1643 (2016)],[Taiwanese J. Math., 20(4), 723-741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author[Forum Math., 22(3), 565-582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.     

On the Number of Trace-One Elements in Polynomial Bases for \mathbb{F}_{2^n}

This paper investigates the number of trace-one elements in a polynomial basis for . A polynomial basis with a small number of trace-one elements is desirable because it results in an efficient and low cost implementation of the trace function. We focus on the case where the reduction polynomial is a trinomial or a pentanomial, in which case field multiplication can also be efficiently implemented. Communicated by: P. Wild     

RESEARCH ANNOUNCEMENTS——On Multiplication Tables of Normal Bases and Their Dual-bases Over Finite Fields

Let q be a power of a prime p and n be a positive integer,let K=Fq be the finite fiele with q elements and F=Fqn be the nte extension of K.N=｛αi|i=0,1,…,n-1｝is a normal basis of F over Fq,where αi=α^qi,i=0,1…,n-1.     

Subquadratic-time factoring of polynomials over finite fields

New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree over a finite field of constant cardinality in time . Previous algorithms required time . The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree over the finite field with elements, the algorithms use arithmetic operations in .

The new ``baby step/giant step' techniques used in our algorithms also yield new fast practical algorithms at super-quadratic asymptotic running time, and subquadratic-time methods for manipulating normal bases of finite fields.

     

Irreducibility Criterion Over Finite Fields

The aim of this article is to prove the irreducibility of the polynomial Λ(Y) = Y ^d  + λ _d?1 Y ^d?1 + … + λ₀ over 𝔽 _q [X] where λ _i ∈ 𝔽 _q [X] and deg λ _d?1 > deg λ _i for each i ≠ d ? 1. We discuss in particular connections between the irreducible polynomials Λ and the number of Pisot elements in the case of formal power series.