Paper Details

Name: Shashikumar R
Country: India

Year: 2025
Volume: Volume-12, Issue-2 (July-December)
Page Number: 634-648
DOI: https://doi.org/10.5281/zenodo.22953416

ABSTRACT
The search for cryptographic foundations beyond classical number-theoretic assumptions has stimulated sustained interest in non-Abelian groups, where non-commutativity enables computational problems and protocol structures that do not arise naturally in conventional Abelian settings. This paper critically synthesizes the mathematical, cryptographic, computational, and cryptanalytic literature on non-Abelian group-based cryptography and develops an integrative Hardness–Security–Efficiency–Viability (HSEV) framework. Rather than assuming that an algebraically difficult problem automatically constitutes a secure cryptographic primitive, the study distinguishes four progressively stronger requirements: mathematical problem hardness, computational hardness under realistic parameter distributions, cryptographic security under adversarial interaction, and practical viability under implementation constraints. Major problem families—including the conjugacy search, simultaneous conjugacy search, decomposition, subgroup membership, word, root extraction, and semidirect-product-related problems—are examined across braid groups, polycyclic groups, matrix groups, and related noncommutative structures. Historical evidence from braid-group cryptography demonstrates that apparent algebraic complexity can be weakened by length-based, representation-based, structural, and parameter-specific attacks. More recent research on polycyclic groups, semidirect products, and alternative group platforms illustrates both continuing mathematical potential and unresolved security questions. The analysis further situates group-based cryptography within the post-quantum landscape. Although group-based constructions remain an active research direction, current NIST post-quantum standards are founded principally on lattice- and hash-based constructions, indicating that non-Abelian approaches have not yet achieved comparable standardization maturity. The paper argues that future progress requires a shift from identifying nominally hard algebraic problems toward constructing parameterized problem distributions with demonstrable classical and quantum security, efficient representations, formal security reductions, and reproducible benchmarking. The resulting framework provides a structured research agenda for evaluating when non-commutativity constitutes a meaningful cryptographic resource rather than merely additional algebraic complexity.

Keywords: non-Abelian groups; group-based cryptography; conjugacy search problem; braid groups; polycyclic groups; semidirect products; cryptanalysis; computational hardness; post-quantum cryptography; quantum security.