A Simple Generalization of the ElGamal Cryptosystem to Non-Abelian Groups

In this article we study the MOR cryptosystem. We use the group of unitriangular matrices over a finite field as the non-abelian group in the MOR cryptosystem. We show that a cryptosystem similar to the ElGamal cryptosystem over finite fields can be built using the proposed groups and a set of automorphisms of these groups. We also show that the security of this proposed MOR cryptosystem is equivalent to the ElGamal cryptosystem over finite fields.     

Cryptanalysis of the Wu–Dawson Public Key Cryptosystem

A new public key cryptosystem was introduced by Wu and Dawson at the Fourth International Conference on Finite Fields (Fq4). This scheme is similar to the McEliece public key cryptosystem, in the sense that it also can be described in terms of linear error-correcting codes over finite fields. However, in contrast to the McEliece scheme, the security of the Wu–Dawson system is not based on a decoding problem which is assumed to be intractable but on the theory of generalized inverses of matrices over finite fields. The authors compare their scheme with the McEliece scheme and claim that the same level of security can be obtained using smaller codes, therefore reducing the key size. In this note it will be shown that the Wu–Dawson scheme is insecure, i.e., a trapdoor can be computed efficiently from the knowledge of the public key.     

Linearly representable codes over chain rings

In this paper, we consider linear codes over finite chain rings. We present a general mapping which produces codes over smaller alphabets. Under special conditions, these codes are linear over a finite field. We introduce the notion of a linearly representable code and prove that certain MacDonald codes are linearly representable. Finally, we give examples for good linear codes over finite fields obtained from special multisets in projective Hjelmslev planes.     

关于多项式函数的计数

Polynomial functions （in particular, permutation polynomials） play an important role in the design of modern cryptosystem. In this note the problem of counting the number of polynomial functions over finite commutative rings is discussed. Let A be a general finite commutative local ring. Under a certain condition, the counting formula of the number of polynomial functions over A is obtained. Before this paper, some results over special finite commutative rings were obtained by many authors.     

典型群的增广商群

设G 为有限域K 上的一般线性群(特殊线性群、酉群、辛群及正交群), 记整群环ZG 的n 次增广理想为△ⁿ(G). 本文着重研究有限域上的典型群的增广商群Q_n(G) = △ⁿ(G)/△ⁿ⁺¹(G), 并刻画了这些连续商群的结构.     

Shunkov groups saturated with general linear groups

We prove that if a periodic Shunkov group is saturated with degree 2 general linear groups over finite fields then it is isomorphic to the degree 2 general linear group over a suitable locally finite field.     

Cardinal rank metric codes over Galois rings

In 1985, Gabidulin introduced the rank metric in coding theory over finite fields, and used this kind of codes in a McEliece cryptosystem, six years later. In this paper, we consider rank metric codes over Galois rings. We propose a suitable metric for codes over such rings, and show its main properties. With this metric, we define Gabidulin codes over Galois rings, propose an efficient decoding algorithm for them, and hint their cryptographic application.     

Embedding theoems for superalgebras

Abstract

Bounds are given for the diameter of commuting involution graphs of special linear groups over fields of characteristic 2. For 2- and 3-dimensional special linear groups over any finite field the disc sizes are determined. Examples are given of commuting involution graphs which have unbounded diameter.     

Infinite dimensional general linear groups have finite width

A group is said to have finite width whenever it has finite width with respect to each inverse-closed generating set. Bergman showed [1] that infinite symmetric groups have finite width and asked whether the automorphism groups of several classical structures have finite width, mentioning in particular infinite dimensional general linear groups over fields. In this article we prove that infinite dimensional general linear groups over arbitrary division rings have finite width. We consider the problem of finite width for other infinite dimensional classical groups.     

Gauss sums over some matrix groups

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involve classical Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups, i.e., the groups of upper triangular matrices. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds of Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [R. Ferguson, C. Hoffman, F. Luca, A. Ostafe, I.E. Shparlinski, Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. 23 (2010) 501–513].     

Distinguished Representations for Quadratic Extensions

Let K be a quadratic extension of a field k which is either local field or a finite field. Let G be an algebraic group over k. The aim of the present paper is to understand when a representation of G(K) has a G(k) invariant linear form. We are able to accomplish this in the case when G is the group of invertible elements of a division algebra over k of odd index if k is a local field, and for general connected groups over finite fields.     

Congruences for rational points on varieties over finite fields

We prove the existence of rational points on singular varieties over finite fields arising as degenerations of smooth proper varieties with trivial Chow group of 0-cycles. We also obtain congruences for the number of rational points of singular varieties appearing as fibres of a proper family with smooth total and base space and such that the Chow group of 0-cycles of the generic fibre is trivial. In particular this leads to a vast generalization of the classical Chevalley-Warning theorem. The above results are obtained as special cases of our main theorem which can be viewed as a relative version of a theorem of H. Esnault on the number of rational points of smooth proper varieties over finite fields with trivial Chow group of 0-cycles.     

有限域上的逻辑函数与其Chrestenson谱的关系

本文首先给出了有限域上逻辑函数的Chrestenson线性谱的新定义(不同于文献[1]所给出的),如同Chrestenson循环谱一样,重新定义的Chrestenson线性谱也是有限域Fq到复数域的映射,且证明了它们之间在实质意义下可以相互线性表出;最后我们还用重新定义的Chrestenson线性谱给出了有限域上逻辑函数的反演公式.     

GLn(k)和SLn(k)的阶

本文利用矩阵行的初等变换 ,采用递推的方法 ,求出了有限域 k上 n次一般线性群 GLn(k)和 n次特殊线性群 SLn(k)的阶 .     

Quasideterminant Characterization of MDS Group Codes over Abelian Groups

A group code defined over a group G is a subset of Gⁿ which forms a group under componentwise group operation. The well known matrix characterization of MDS (Maximum Distance Separable) linear codes over finite fields is generalized to MDS group codes over abelian groups, using the notion of quasideterminants defined for matrices over non-commutative rings.     

从特殊线性群到一般射影线性群的同态的一个性质

设F,K为域,GLn（F）,SLn（F）分别表示F上的n级一般线生群和n级特殊线性群．PGLn（F）,PSLn（F）分别表示F上的n级射影一般线性群和n级射影特殊线性群．φ：SLn（F）→PGLn（K）,n≥3为非平凡同态．本文确定了当K的持征为2时η的—个性质．     

Forcing Linearity on Greedy Codes

Described in this paper are two different methods of forcing greedy codesto be linear over arbitrary finite fields. Both methods are generalizationsof the binary B-greedy codes as well as the triangular greedycodes over arbitrary fields. One method generalizes to arbitrary orderingsover arbitrary finite fields while the other method generalizes theB-greedy codes to linear codes over arbitrary finite fields.Examples of the first method are computed for triangular greedy codes. Theseexamples give codes similar to triangular codes from B-orderings.Both methods are shown to be substantially different.     

Jacobians of reflection groups

Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for groups with a polynomial ring of invariants over arbitrary fields using a ramification formula of Benson and Crawley-Boevey.

     

A non-commutative cryptosystem based on quaternion algebras

We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen’s method, the key generation and encryption process is approximately 16 / 7 times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions.