Abstract: Algebra is one of the important fields of mathematics. It concerns with the study and manipulation of mathematical symbols. It also concerns with the study of abstractions such as groups, rings, and fields. Due to the development of these abstractions, it is extended to consider other structures, such as vectors, matrices, and polynomials, which are non-numerical objects. Computer algebra is the implementation of algebraic methods as algorithms and computer programs. Recently, many algebraic cryptosystem protocols are based on non-commutative algebraic structures, such as authentication, key exchange, and encryption-decryption processes are adopted. Cryptography is the science that aimed at sending the information through public channels in such a way that only an authorized recipient can read it. Ring theory is the most attractive category of algebra in the area of cryptography. In this paper, we employ the algebraic structure called skew -Armendariz rings to design a neoteric algorithm for zero knowledge proof. The proposed protocol is established and illustrated through numerical example, and its soundness and completeness are proved.
Abstract: Analysis of real life problems often results in linear
systems of equations for which solutions are sought. The method to
employ depends, to some extent, on the properties of the coefficient
matrix. It is not always feasible to solve linear systems of equations
by direct methods, as such the need to use an iterative method
becomes imperative. Before an iterative method can be employed
to solve a linear system of equations there must be a guaranty that
the process of solution will converge. This guaranty, which must
be determined apriori, involve the use of some criterion expressible
in terms of the entries of the coefficient matrix. It is, therefore,
logical that the convergence criterion should depend implicitly on the
algebraic structure of such a method. However, in deference to this
view is the practice of conducting convergence analysis for Gauss-
Seidel iteration on a criterion formulated based on the algebraic
structure of Jacobi iteration. To remedy this anomaly, the Gauss-
Seidel iteration was studied for its algebraic structure and contrary
to the usual assumption, it was discovered that some property of the
iteration matrix of Gauss-Seidel method is only diagonally dominant
in its first row while the other rows do not satisfy diagonal dominance.
With the aid of this structure we herein fashion out an improved
version of Gauss-Seidel iteration with the prospect of enhancing
convergence and robustness of the method. A numerical section is
included to demonstrate the validity of the theoretical results obtained
for the improved Gauss-Seidel method.
Abstract: Quasigroups are algebraic structures closely related to
Latin squares which have many different applications. The
construction of block cipher is based on quasigroup string
transformation. This article describes a block cipher based
Quasigroup of order 256, suitable for fast software encryption of
messages written down in universal ASCII code. The novelty of this
cipher lies on the fact that every time the cipher is invoked a new set
of two randomly generated quasigroups are used which in turn is
used to create a pair of quasigroup of dual operations. The
cryptographic strength of the block cipher is examined by calculation
of the xor-distribution tables. In this approach some algebraic
operations allows quasigroups of huge order to be used without any
requisite to be stored.
Abstract: The purpose of this research is to study the concepts
of multiple Cartesian product, variety of multiple algebras and to
present some examples. In the theory of multiple algebras, like other
theories, deriving new things and concepts from the things and
concepts available in the context is important. For example, the first
were obtained from the quotient of a group modulo the equivalence
relation defined by a subgroup of it. Gratzer showed that every
multiple algebra can be obtained from the quotient of a universal
algebra modulo a given equivalence relation.
The purpose of this study is examination of multiple algebras and
basic relations defined on them as well as introduction to some
algebraic structures derived from multiple algebras. Among the
structures obtained from multiple algebras, this article studies submultiple
algebras, quotients of multiple algebras and the Cartesian
product of multiple algebras.
Abstract: In this study, we examine multiple algebras and
algebraic structures derived from them and by stating a theory on
multiple algebras; we will show that the theory of multiple algebras
is a natural extension of the theory of universal algebras. Also, we
will treat equivalence relations on multiple algebras, for which the
quotient constructed modulo them is a universal algebra and will
study the basic relation and the fundamental algebra in question.
In this study, by stating the characteristic theorem of multiple
algebras, we show that the theory of multiple algebras is a natural
extension of the theory of universal algebras.
Abstract: A systematic and exhaustive method based on the group
structure of a unitary Lie algebra is proposed to generate an enormous
number of quantum codes. With respect to the algebraic structure,
the orthogonality condition, which is the central rule of generating
quantum codes, is proved to be fully equivalent to the distinguishability
of the elements in this structure. In addition, four types of
quantum codes are classified according to the relation of the codeword
operators and some initial quantum state. By linking the unitary Lie
algebra with the additive group, the classical correspondences of some
of these quantum codes can be rendered.