Generalized Repunit Hybrid Quaternions: Structural and Pre-Cryptographic Insights

In this study, we introduce the generalized Repunit sequence and its hybrid quaternion extension derived from a parametric recurrence relation that preserves the base-10 structure of classical Repunit numbers. Fundamental properties of the proposed sequences, including the characteristic equation, generating function, and Binet-type formula, are systematically investigated. Several algebraic identities, such as bilinear index-reduction formulas, are established to demonstrate the internal structure and consistency of the construction. Numerical experiments and graphical analyses are conducted to examine the structural behavior of the generalized Repunit sequence and its hybrid quaternion counterpart. While the scalar Repunit sequence exhibits regular and predictable growth, the hybrid quaternion extension displays significantly higher structural complexity and variability. Density distributions, contour plots, histogram representations, and discrete variation measures confirm the presence of enhanced diffusion and local irregularity in the quaternion-based structure. These statistical, graphical, and numerical findings indicate that generalized Repunit hybrid quaternion sequences possess properties that are relevant to encoding, masking, and preprocessing mechanisms in applied mathematical and computational frameworks. However, this work does not propose a complete cryptographic algorithm, nor does it claim compliance with established cryptographic security standards such as NIST SP 800-22. The results should therefore be interpreted as pre-cryptographic indicators that motivate further research toward rigorous security evaluation, algorithmic development, and broader applications in areas such as coding theory, signal processing, and nonlinear dynamical systems.