• Ummul Khayer Fatema
• Kawsik Azad Pronoy
• Sanjoy Kumar Roy
• Arjuman Ara

Abstract algebra, historically cultivated as a realm of pure mathematical abstraction, constitutes the fundamental mathematical architecture underpinning 21st-century technological systems. This comprehensive monograph provides an exhaustive theoretical and practical analysis of abstract algebraic structures—including group theory, ring theory, Galois field extensions, Lie groups, and vector spaces—and their direct operational applications in asymmetric cryptography (RSA, ECC, Post-Quantum Ring-LWE), error-correcting algebraic codes (Reed-Solomon, BCH), quantum computing state space dynamics, robotic kinematics (Lie Groups SO(3) and SE(3)), and machine learning architectures. By establishing formal theorems, explicit numerical proofs, algorithmic pipelines, and structural mappings, this paper demonstrates how abstract algebraic invariants guarantee cybernetic security, signal resilience, and precise physical actuation across complex computational paradigms.

Abstract Algebra, Galois Field GF(pⁿ), Elliptic Curve Cryptography, Reed-Solomon Codes, Special Orthogonal Group SO(3), Quantum Operators, Post-Quantum Lattice Cryptography, Boolean Ring.

"Comprehensive Foundations of Abstract Algebraic Structures in Modern Cryptography, Coding Theory, Quantum Computing, and Engineering Kinematics: A Theoretical and Computational Monograph", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.13, Issue 9, page no.a169-a176, September-2026, Available :http://www.jetir.org/papers/JETIR2609018.pdf

"Comprehensive Foundations of Abstract Algebraic Structures in Modern Cryptography, Coding Theory, Quantum Computing, and Engineering Kinematics: A Theoretical and Computational Monograph", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.13, Issue 9, page no. ppa169-a176, September-2026, Available at : http://www.jetir.org/papers/JETIR2609018.pdf