Development of a Class of Non-Associative Algebras: Applications in Cryptography and Coding Theory

Abstract

One of the peculiarity of mathematics is that its thorniest contradictions bloom into beau tiful theories. According to Albert Einstein, "Pure mathematics is, in its way the poetry of logical ideas." Mathematics (in particular pure mathematics) has gone through many revolutionary changes over a period of almost one century and it morphed into new shapes with time. Initially, rings and algebra were considered to be associative and commmutative and in some cases associative only. Since the mid of 19th century, many non-associative struc tures have been introduced so far. For instance, Octonions, cayely numbers, Lie algebras, Jordan algebras, Lie structures, alternative rings, loops and loop rings. Non-associative ring theory has flourished as an independent branch of algebra, having links with other branches of mathematics and other fields for instance, biology, physics and other sciences. In 1972, Kazim and Naseeruddin [77] presented a generalization of a commutative semigroup and called it a left almost semigroup (LA-semigroup). An LA-semigroup is a groupoid satisfying the identity: (ab)c = (cb)a, which is known as the left invertive law. An LA-semigroup is non-associative and non-commutative, nevertheless it holds proper ties that are normally found in associative and commutative algebraic structures. Mushtaq and Kamran [100] in the year 1996, extended the idea of an LA-semigroup to a left almost group (LA-group). Despite being a non-associative algebraic structure, an LA-group in terestingly resembles to an abelian group. LA-semigroups and LA-groups are considered by many authors to establish useful results to explore their properties and structures. In 2010, Shah and Rehman [131] combined the two structures to introduce a Left almost ring (LA-ring). It is an additive LA-group and multiplicative LA-semigroup along with the two distributive laws. They generalized a commutative semigroup ring to present an LA-ring, which consists of finitely non-zero functions with domain a commutative semigroup and co-domain an LA-ring. 1 In the current decade, many researchers have put forward their contribution to the de velopment of this particular non-associative non-commutative structure and its generaliza tions. Major contributors include T. Shah and his co-researchers [113, 125, 128, 131–135], who not only studied the structural properties of LA-rings and its generalizations, but ex plored their applications to intuitionistic fuzzy and soft sets. They introduced the concepts of LA-integral domain, LA-field, LA-modules and a generalization of LA-rings called near LA-ring. Furthermore, they discussed the properties of ideals in LA-rings and M-systems in LA-rings. They proved the existence of a non-associative LA-ring and defined a Special LA-ring. Moreover, Shah and Kousar [127] studied the intuitionistic fuzzy normal sub rings in LA-rings. Shah and Razzaque [129, 130], defined soft LA-rings and discussed soft ideals and M-systems in soft LA-rings. Rehman and Razzaque [109, 112], discussed the notions of projective and injective LA-modules, free LA-modules, split sequences in LA-modules and later they extended the applications of soft set theory to LA-rings and presented soft LA-modules and exact sequences of soft LA-modules. Hussain and his co authors [55–60], focused on congruences and the notions of direct product and direct sum in LA-ring, LA-module and their generalizations. They introduced an LA-semiring as a generalization of LA-ring. A number of researchers studied LA-rings from different aspects, a brief look-over to their work is mentioned in this thesis. The aim of this thesis is to explore those areas of LA ring theory which are still to be uncovered. Some notions have just been introduced but are not further investigated in details by any researcher. For instance LA-domain, LA-fields and special LA-rings. We not only promote these concepts but also provide their different applications. The existing literature lacks examples of LA-rings with order greater than 18, we formulate an algorithm to obtain LA-rings of greater orders using LA-rings with small orders. Moreover, we study some new aspects of the soft LA-rings by investigating soft intersection LA-rings. We fuse generalized rough sets with soft sets to define generalized rough soft sets. We also introduce generalized rough and generalized rough soft LA-rings. Further, we explore the applications of LA-rings to Coding theory by introducing DNA codes over a special LA-field and to cryptography by constructing S-boxes over special LA-rings.