Galois环上的本原多项式的一个判别准则

本文给出Ｇａｌｏｉｓ环Ｒ上的基本不可约多项式ｆ（ｘ）的根的具体表达式和其阶的联系；由此，对本原多项式和次本原多项式分别推导出代数判别式，其主要部分分别由ｆ（ｘ）ｍｏｄｐ和ｆ（ｘ）ｍｏｄｐ２的系数所确定．     

QF-环上有限生成模嵌入自由模的秩数和最小嵌入的算法

一个模嵌入自由模相当于用坐标来表示元素,这在数学上一直有重要意义．理论上, ＱＦ－环上的模都可以嵌入自由模,但没有好的算法,影响了它的应用．本文给出ＱＦ－环上有限生成模的最小嵌入的一种算法和所嵌入自由模的秩数的估计．     

唯一分解整环R上不可约多项式的一个判别准则

本获得了一个判别唯一分解整环R及其商域Q上的n次（n＞2)多项式不可约的充分条件。     

Rearrangements of access structures and their realizations in secret sharing schemes

Firstly, the definitions of the secret sharing schemes (SSS), i.e. perfect SSS, statistical SSS and computational SSS are given in an uniform way, then some new schemes for several familiar rearrangements of access structures with respect to the above three types of SSS are constructed from the old schemes. It proves that the new schemes and the old schemes are of the same security. A method of constructing the SSS which realizes the general access structure by rearranging some basic access structures is developed. The results of this paper can be used to key managements and access controls.     

New Colored Visual Secret Sharing Schemes

Visual secretsharing (VSS) schemes are used to protect the visual secret bysending n transparencies to different participantsso that k-1 or fewer of them have no informationabout the original image, but the image can be seen by stackingk or more transparencies. However, the revealedsecret image of a conventional VSS scheme is just black and white.The colored k out of n VSS scheme sharinga colored image is first introduced by Verheul and Van Tilborg[1]. In this paper, a new construction for the colored VSS schemeis proposed. This scheme can be easily implemented on basis ofa black & white VSS scheme and get much better block lengththan the Verheul-Van Tilborg scheme.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

A FINITE STRUCTURE THEOREM BETWEEN PRIMITIVE RINGS AND ITS APPLICATION TO GALOIS THEORY

设\[\mathfrak{M} = \sum {F{u_i}} \]是除环F上向量空间，P是F的一个子除环且在F中是Galois，即存 在F的一个自同构群G使\[I(G) = P\].记Ф是F的中心，\[{G_0}\]是属于G的内自同构群， \[{G_0}\]的元素记为\[{I_r},r \in F\]；，记\[{E^'} = \sum\limits_{{I_{{r_j}}} \in {G_0}} {{\Phi _{{r_j}}}} \]是G的代数，\[P' = {C_F}({E^'})\]是\[{E^'}\]在F中的中心化子.记\[\mathfrak{U}(F,\mathfrak{M})\]是\[\mathfrak{M}\]的F-线性变换完全环，\[{T_v}(F,\mathfrak{M})\]是\[\mathfrak{U}(F,\mathfrak{M})\]中所有秩小于\[\mathcal{X}{_v}\]的元素集合，那末我们有如下主要结果： (1)\[{[F:P']_L} = n\]有限当且仅当\[{T_v}(P',\mathfrak{M}) = \sum\limits_{j = 1}^n \oplus {r_{jL}}{T_v}(F,\mathfrak{M})\]，其中\[{r_j} \in {E^'}\]，\[{r_{jL}}\]表示元素\[{r_j}\]的标量左乘. （2）\[{[P':P]_L} = t\]有限当且仅当凡\[{T_v}(P,\mathfrak{M}) = \sum\limits_{j = 1}^t \oplus {S_j}{T_v}(P',\mathfrak{M})\],其中\[{S_j}\]表示\[\mathfrak{M}\]的F-半线变换自同构，它的伴随同构\[{\psi _j} \in G\]. ⑶如有某个序数v使\[{T_v}(P,\mathfrak{M})\]，\[{T_v}(P',\mathfrak{M})\]及\[{T_v}(F,\mathfrak{M})\]满足⑴及(2)中的关系 式，那末对任何\[{T_\mu }(P,\mathfrak{M})\]，\[{T_\mu }(P',\mathfrak{M})\]及\[{T_\mu }(F,\mathfrak{M})\]皆满足(1)及(2)中的关系式.特别 对\[\mathfrak{U}(P,\mathfrak{M})\]，\[\mathfrak{U}(P',\mathfrak{M})\]及\[\mathfrak{U}(F,\mathfrak{M})\]也是如此. ⑷如果\[{[F:P]_L}\]有限，那末必有\[{C_p}({C_F}(E')) = E'\]，\[{[F:P']_L} = \dim E'\]，\[{[P':P']_L} = [G/{G_0}]\]，其中dim E'表示E'在\[\Phi \]上的维数，\[[G/{G_0}]\]表示\[{G_0}\]在G中的指数,特别\[G\]是 Galois 群，则 \[{C_F}(P') = {C_F}(P) = E'\]. (5)若\[{\tilde G}\]是F的另一自同构群且\[I(G) = I(\tilde G)\]，那末必有\[[G/{G_0}] = [\tilde G/{{\tilde G}_0}]\]， \[\dim {\kern 1pt} {\kern 1pt} E' = \dim {\kern 1pt} {\kern 1pt} \tilde E'\]. 其中\[{\kern 1pt} \tilde E'\]表示\[{\tilde G}\]的代数. 如果P取为F的中心时，于是从上述结果(1)就得出熟知的定理：\[[F:\Phi ]\]是有限的当 且仅当\[\mathfrak{U}(\Phi ,\mathfrak{M}) = \mathfrak{U}(F,\mathfrak{M}){ \otimes _\Phi }{F_L}\]. 另方面，运用我们上述的结果，可导出除环F的有限Galois理论.     

Basic Properties of the (t, n)-Threshold Visual Secret Sharing Scheme with Perfect Reconstruction of Black Pixels

In this paper we consider the (t, n)-threshold visual secret sharing scheme (VSSS) in which black pixels in a secret black-white images is reproduced perfectly as black pixels when we stack arbitrary t shares. This paper provides a new characterization of the (t, n)-threshold visual secret sharing scheme with such a property (hereafter, we call such a VSSS the (t, n)-PBVSSS for short). We use an algebraic method to characterize basis matrices of the (t, n)-PBVSSS in a certain class of matrices. We show that the set of all homogeneous polynomials each element of which yields basis matrices of the (t, n)-PBVSSS becomes a set of lattice points in an (n−t+1)-dimensional linear space. In addition, we prove that the optimal basis matrices in the sense of maximizing the relative difference among all the basis matrices in the class coincides with the basis matrices given by Blundo, Bonis and De Santis [3] for all n≥ t ≥ 2.     

基于线性码的新的秘密分享方案

A secret sharing system can be damaged when the dealer cheating occurs. In this paper,two kinds of secret sharing schemes based on linear code are proposed. One is a verifiable scheme which each participant can verify his own share from dealer‘s distribution and ensure each participant to receive valid share. Another does not have a trusted center, here, each participant plays a dual-role as the dealer and shadow(or share) provider in the whole scheme.     

A CHARACTERIZATION OF THE GENERALIZED INVERSES OVER DIVISION RINGS

Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.     

正则环上矩阵分解

利用幂等矩阵和单边可逆阵,给出了正则环上具有群逆的矩阵结构,并证明了具有奇数特征的单边单位正则环上的矩阵都可分解为两个单边可逆矩阵和形式。     

ON SECONDARY MODULES OVER PULLBACK RINGS

Abstract

In this paper we describe the derivations of orthosymplectic Lie superalgebras over a superring. In particular, we derive sufficient conditions under which the derivations can be expressed as a semidirect product of inner and outer derivations. We then present some examples for which these conditions hold.     

ON POLE ASSIGNABILITY AND FEEDEBACK CYCLISATION FOR SYSTEMS OVER RINGS OF FINITE DIMENSION

ABSTRACT

It is shown that the rank of the projective added to the state space of a system in dynamic feedback can be decreased by assigning some poles to the original system first and by using these to obtain a new system with state space of lower rank on which dynamic feedback can be applied.