Mathematical Cryptography and Quantum Resistance: Designing Secure Elliptic Curve Systems for the Post-Quantum Era

Abstract

The rapid advancement of quantum computing has posed a significant threat to traditional public-key cryptographic systems, particularly those based on elliptic curve cryptography (ECC). Although ECC has long been valued for its strong classical security and computational efficiency, quantum algorithms capable of solving the elliptic curve discrete logarithm problem would render these systems vulnerable in the near future. This study investigated how secure elliptic-curve-based cryptographic systems could be redesigned for the post-quantum era by integrating post-quantum cryptographic (PQC) primitives into hybrid security architectures. A design-oriented and analytical research approach was employed to compare classical ECC, standalone PQC, and hybrid ECC–PQC models using quantitative measures of security strength, computational cost, and communication overhead. The results demonstrated that while ECC provided excellent performance, it lacked quantum resistance, whereas PQC offered strong quantum-safe security at the cost of higher computational and bandwidth requirements. The hybrid ECC–PQC approach achieved the most balanced outcome, combining high classical efficiency with robust quantum resistance and strong long-term security. Numerical performance evaluation showed that hybrid systems increased execution time and handshake size moderately compared to PQC alone, yet delivered significantly improved security guarantees. These findings confirmed that hybrid cryptographic architectures represent a practical and reliable pathway for migrating existing elliptic-curve infrastructures toward quantum-resilient security. The study contributed a mathematically grounded and deployment-oriented framework that supports secure, gradual transition strategies for protecting digital communications in a quantum-enabled future.