Exploring the properties of prime numbers in cryptography
N Mamatha, Sunitha S.S · World Journal of Advanced Research and Reviews · 2022
Prime numbers are fundamental to modern cryptographic systems, serving as the foundation for public-key encryption protocols. This paper explores the mathematical significance of prime numbers, their unique properties, and their indispensable role in cryptographic algorithms such as RSA, Diffie-Hellman key exchange, and Elliptic Curve Cryptography (ECC). A detailed computational analysis is conducted to examine efficient methods for generating large prime numbers, including probabilistic techniques such as the Miller-Rabin and Solovay-Strassen tests, as well as deterministic approaches like the AKS primality test. The study further evaluates the computational complexity of these methods and their impact on encryption performance and security. Beyond prime number generation, this paper delves into the security implications of prime-based cryptosystems, analyzing potential vulnerabilities such as integer factorization attacks on RSA, discrete logarithm-based attacks on Diffie-Hellman, and the impact of side-channel attacks on cryptographic implementations. Special attention is given to the emerging threat of quantum computing, which poses significant risks to conventional cryptographic schemes by enabling efficient factorization through Shor’s algorithm. Strategies for mitigating these threats, including the adoption of post-quantum cryptographic techniques, are also explored. Figures, tables, and bar charts illustrate the effectiveness of different prime number generation methods, the trade-offs between security and computational efficiency, and the comparative resilience of prime-based cryptosystems under various attack scenarios. This study provides valuable insights into the evolving landscape of cryptographic security and the ongoing need for robust, efficient, and quantum-resistant encryption mechanisms.