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

This is a study of the MOR cryptosystem using the special linear group over finite fields. The automorphism group of the special linear group is analyzed for this purpose. At our current state of knowledge, I show that this MOR cryptosystem has better security than the ElGamal cryptosystem over finite fields.     

Cryptosystems in automorphism groups of group rings of Abelian groups

We introduce a new class of public-key cryptosystems generalizing ElGamal cryptosystems to automorphism groups of group rings of Abelian groups. A scheme of the basic variant of such a cryptosystem is presented and some types of attacks to it are considered. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 157–164, 2007.     

Synchronization of chaotic systems using feedback controller: An application to Diffie–Hellman key exchange protocol and ElGamal public key cryptosystem

In this paper, designing an appropriate linear and nonlinear feedback control, the two identical integer order chaotic systems are synchronized by analytically and numerically. It has been realizing that, synchronization using linear feedback control method is efficient than nonlinear feedback control method due to the less computational complexity and the synchronization error. ElGamal public key cryptosystem is described through the proposed Diffie–Hellman key exchange protocol based on the synchronized chaotic systems using linear feedback control and their security are analyzed. The numerical simulations are given to validate the correctness of the proposed synchronization of chaotic systems and the ElGamal cryptosystem.     

Discrete Logarithm Based Cryptosystems in Quadratic Function Fields of Characteristic 2

We present a key exchange scheme similar to that of Diffie and Hellman using the infrastructure of quadratic function fields of even characteristic. This is a modification of the results of Scheidler, Stein and Williams who used quadratic function fields of odd characteristic. We also extend these results to give a digital signature scheme similar to that of ElGamal. These schemes are possible in this structure even though it is not a group. Finally we examine the security of such systems, and give a possible attack based on Pohlig and Hellman's attack on discrete logarithms in finite groups.     

Pell hyperbolas in DLP–based cryptosystems

We present a study on the use of Pell hyperbolas in cryptosystems with security based on the discrete logarithm problem. Specifically, after introducing the group structure over generalized Pell hyperbolas (and also giving the explicit isomorphisms with the classical Pell hyperbolas), we provide a parameterization with both an algebraic and a geometrical approach. The particular parameterization that we propose appears to be useful from a cryptographic point of view because the product that arises over the set of parameters is connected to the Rédei rational functions, which can be evaluated in a fast way. Thus, we exploit these constructions for defining three different public key cryptosystems based on the ElGamal scheme. We show that the use of our parameterization allows to obtain schemes more efficient than the classical ones based on finite fields.     

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.     

Isomorphisms of Kac-Moody groups which preserve bounded subgroups

A subgroup of a Kac-Moody group is called bounded if it is contained in the intersection of two finite type parabolic subgroups of opposite signs. In this paper, we study the isomorphisms between Kac-Moody groups over arbitrary fields of cardinality at least 4, which preserve the set of bounded subgroups. We show that such an isomorphism between two such Kac-Moody groups induces an isomorphism between the respective twin root data of these groups. As a consequence, we obtain the solution of the isomorphism problem for Kac-Moody groups over finite fields of cardinality at least 4.     

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.     

Embedding Problems of Division Algebras

A finite group G is called admissible over a given field if there exists a central division algebra that contains a G-Galois field extension as a maximal subfield. We give a definition of embedding problems of division algebras that extends both the notion of embedding problems of fields as in classical Galois theory, and the question which finite groups are admissible over a field. In a recent work by Harbater, Hartmann, and Krashen, all admissible groups over function fields of curves over complete discretely valued fields with algebraically closed residue field of characteristic zero have been characterized. We show that also certain embedding problems of division algebras over such a field can be solved for admissible groups.     

On the Uniformity of Distribution of the Elliptic Curve ElGamal Signature

We show that, under some natural conditions, the pairs (ρ, σ) produced by the elliptic curve ElGamal signature scheme are uniformly distributed. In particular, this implies that values of ρ and σ are not correlated. The result is based on some new estimates of exponential sums. For the ElGamal signature over a finite field, a similar result has been obtained by the second author.     

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.     

Characterization of One Radical of Group Rings over Finite Prime Fields

We study complete and reduced associative rings (in the sense of L. M. Martynov). We prove a necessary and sufficient test for completeness of a semigroup ring, calculate the greatest complete subring (which is an ideal) of a group ring over finite prime fields, and characterize the reduced group rings of finite groups over finite prime fields.     

Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms

We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of affine spaces over finite fields. In particular, we prove that when such a map is dominant, the set of its fixed closed scheme points is Zariski dense in the affine space.     

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.     

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.     

Ring Isomorphism of Cyclic Cyclotomic Algebras

It is shown that ring isomorphisms between cyclic cyclotomic algebras over cyclotomic number fields are essentially determined by the list of local Schur indices at all rational primes. As a consequence, ring isomorphisms between simple components of the rational group algebras of finite metacyclic groups are determined by the center, the dimension over ?, and the list of local Schur indices at rational primes. An example is given to show that this does not hold for finite groups in general.     

Positively finitely related profinite groups

We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.     

Improvement of cryptosystem based on iterating chaotic map

In this letter we demonstrate that the improvement of cryptosystem based on iterating chaotic map proposed by Yong in 2007 are weak and this cryptosystem can be easily broken using chosen plaintext attack. Then, we give novel improvements to the proposed chaotic cryptosystem. We choose image as plaintext, some experimental tests like sensitivity on initial condition and correlation between two adjacent pixels are presented to show the performances of the new cryptosystem.     

Regular Sets and Geometric Groups

If G is a permutation group acting on a set Ω, a subset Λ of Ω is called a regular set for G if the set-stabilizer of Λ in G is the identity subgroup. We show here that the projective and affine semi-linear groups acting in the natural way as permutation groups on their respective finite geometries, have, in general, for all finite dimensions and all finite fields, regular sets of points. The exceptions to this are found, and an extension of the results to infinite fields is discussed.