Enhancing the Efficiency and Security of The Rabin Public-Key Cryptosystem Using Gaussian Integers

The Rabin public-key cryptosystem is renowned for its provable security, making it one of the most extensively studied cryptographic schemes. Initially defined over the ring of integers, it has been extended to polynomial rings over finite fields. This work introduces a novel generalization of the Rabin cryptosystem to the domain of Gaussian integers Z[i], incorporating novel arithmetic operations and modifications tailored to this framework. We develop comprehensive encryption and decryption algorithms, supplemented by illustrative examples that clarify these adaptations. A comparative analysis between the classical Rabin scheme and its Gaussian integer extension reveals significant differences in security and computational complexity, demonstrating that the extended scheme significantly enhances resilience against attacks, particularly by mitigating decryption ambiguity and brute-force complexity. Empirical results, supported by computational experiments and graphical analysis, show that the time required for potential attacks increases substantially when transitioning from natural integers to Gaussian integers, a consequence of the richer algebraic structure of Z[i], which imposes additional computational challenges for adversaries. By leveraging these properties, the modified Rabin cryptosystem offers a robust defense against common cryptanalytic techniques, positioning it as a promising alternative for secure communication. This work not only lays the foundation for further exploration of public-key cryptography in alternative algebraic domains but also opens new avenues for both theoretical research and practical applications.