A Study of Non Commutative Algebraic Structures and Their Applications in Cryptography
DOI:
https://doi.org/10.31305/rrijm.2022.v07.i04.019Keywords:
cryptography, non-commutative, modular arithmetic, finite fields, abelian groupsAbstract
Non commutative algebraic structures have gained increasing attention in modern cryptography due to their inherent structural complexity and resistance to classical algebraic attacks. Unlike commutative systems, non-commutative structures exhibit asymmetric operational behavior, which can be effectively exploited to construct cryptographic schemes with enhanced security properties. This paper presents a comprehensive theoretical study of non-commutative algebraic structures and examines their relevance and applicability in contemporary cryptographic systems. The study focuses on fundamental non commutative structures such as groups, rings, semigroups, and algebras, emphasizing their algebraic properties and operational characteristics that contribute to cryptographic strength. Particular attention is given to the role of non commutativity in key exchange protocols, encryption mechanisms, authentication schemes, and post quantum cryptography. The absence of commutativity introduces computational hardness assumptions that are difficult to reduce to classical number theoretic problems, thereby providing alternative security foundations. This research adopts a theoretical and analytical approach, synthesizing existing algebraic principles with cryptographic concepts to highlight how non commutative structures support secure communication frameworks. The paper also discusses the advantages, limitations, and potential vulnerabilities associated with non commutative cryptographic constructions. By bridging abstract algebra with applied cryptography, the study aims to contribute to the development of robust cryptographic models suitable for emerging security challenges.
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