Application of Cryptography for Controllability Results of Fractional Neutral Volterra-Fredholm Integro-Differential Equations with State-Dependent Delay

This paper utilizes the Caputo fractional derivative and a semigroup of compact and analytic operators to examine the controllability of fractional Volterra-Fredholm integro-differential equations with state-dependent delay. Controllability results are formulated using Schauder’s fixed point theorem, addressing the inherent difficulties brought about by the fractional dynamics together with state-dependent delays. The theoretical findings are validated through a detailed example and numerical simulations, demonstrating the convergence of solutions. Graphical representations are provided to better understand solution dynamics and highlight system complexity. Additionally, the applicability of the proposed system for cryptographic key generation is explored, showing that it can generate secure, unpredictable keys due to its chaotic behavior, sensitivity to initial conditions, and the interplay between key system parameters.