Conic curve encryption and digital signature based on complex number theory for cybersecurity applications

Secure image transmission requires robust algorithms to ensure authentication, integrity, non-repudiation, and confidentiality. Addressing emerging security challenges necessitates continuous advancements in cryptographic design. This paper presents an authenticated and encrypted image scheme that achieves all essential security services. While Elliptic curve cryptography (ECC) remains a fundamental component of recent encryption schemes, it is vulnerable to side-channel and inherent ECC-specific attacks. To overcome these vulnerabilities, the proposed scheme replaces ECC with Conic curve cryptography (CCC), offering enhanced security and performance. The integration of complex number theory with CCC enables a secure complex key exchange and incorporates a robust Iterative conic curve pseudorandom number generator (ICC-PRNG) to thwart all known attack types. The system is a public key cryptosystem based on multi-hard problems, including the Gaussian conic curve integer factorization problem (GCC-IFP), Conic curve discrete logarithm problem (CC-DLP), and Conic curve integer factorization problem (CC-IFP), combined with XOR operations for image encryption. Additionally, the scheme introduces a novel complex digital signature for encrypted images, leveraging complex arithmetic to enhance security. Experimental results demonstrate high entropy 7.999, correlation near 0.0001, key space [Formula: see text], and average PSNR of 8.51 dB, ensuring resilience against brute-force and statistical attacks. Additionally, the scheme achieves encryption times of 25 ms, making it suitable for real-time applications. Security analysis validates robustness against various attacks, with NIST statistical tests confirming ICC-PRNG effectiveness. By leveraging complex numbers over conic curves, the proposed method improves security and computational efficiency, establishing it as a promising solution for advanced image encryption.