Homomorphism Counting, Fuzzy Group Actions and Conjugacy-Based Cryptography in Finite Algebraic Structures

Michael Nsikan John. (2026). Homomorphism Counting, Fuzzy Group Actions and Conjugacy-Based Cryptography in Finite Algebraic Structures. Ktrend - International Journal of Mathematics and Statistics (IJMS), 1(1), 1-13. https://doi.org/10.5281/zenodo.20506460 

The interaction between finite algebraic structures and cryptographic systems motivates the search for unified frameworks that can connect homomorphism enumeration, quotient constructions, fuzzy algebraic actions, and conjugacy-class methods. In this paper, a structural framework is developed for finite groups acting on near-rings and for homomorphic images that induce modular B-algebras. The paper is motivated by previous works of the author on B-algebras generated by modulo integer groups, conjugacy-class key agreement, fuzzy group actions on near-rings, and homomorphism enumeration from the quaternion group. We define homomorphic complexity indices, fuzzy stabilizer indices, conjugacy complexity indices, and
induced modular B-algebra invariants. Several results are proved: kernels of homomorphisms act trivially under induced actions; cyclic homomorphic images give canonical B-algebras; fuzzy stabilizers are subgroups; orbit-stabilizer relations persist in the fuzzy-invariant setting; and conjugacy-based key agreement can be interpreted through commuting subgroups and algebraic invariants. Examples involving cyclic groups, the quaternion group, dihedral groups, and nilpotent groups illustrate the theory. The framework gives a mathematical basis for combining quotient-based algebra, fuzzy symmetry, and conjugacy structures in the analysis of algebraic cryptographic protocols.