A Hybrid Deterministic Mathematical Model for Secure Blockchain Transaction Dynamics Incorporating Lattice-Based Cryptography and Algebraic Semigroup Reconstruction

Blockchain technology has emerged as a foundational infrastructure for decentralized financial systems and secure digital transactions; however, its resilience against evolving cyber threats remains a significant challenge. This study presents a hybrid deterministic mathematical model that integrates lattice-based cryptography with algebraic semigroup structures to investigate the dynamics of blockchain transaction security. The proposed framework partitions blockchain nodes into four interacting classes: secure, vulnerable, compromised, and recovered. The system dynamics are governed by a coupled system of nonlinear ordinary differential equations that incorporates a lattice cryptographic protection coefficient and a semigroup reconstruction parameter. The blockchain security threshold is derived as $R_s=\dfrac{\beta\gamma}{(\lambda+\mu)(\delta+\sigma+\mu)}$, providing a quantitative criterion for evaluating network resilience. Local stability of the equilibrium points is established using Jacobian matrix analysis, while numerical simulations demonstrate the influence of cryptographic protection and algebraic reconstruction on long-term blockchain security. The results indicate that increasing the lattice protection coefficient $\lambda$ and the semigroup reconstruction efficiency $\sigma$ significantly reduces the proportion of compromised nodes while accelerating system recovery. Sensitivity analysis further identifies the parameters governing cryptographic strength and attack transmission as the primary determinants of blockchain stability. The proposed model extends existing deterministic blockchain security models by unifying mathematical modelling, algebraic semigroup theory, and post-quantum lattice cryptography within a single analytical framework, offering practical insights for the design of secure distributed ledger systems capable of resisting emerging computational and quantum-enabled attacks.