Advanced Analysis of Prime Number Generation Methods in Cryptographic Systems

This thesis analyzes prime number generation methods employed in modern cryptographic systems and examines their underlying mathematical principles. The study explores the generation of large prime numbers, random number generation mechanisms, and the operational characteristics of primality testing algorithms. The performance, efficiency, and practical benefits of the Fermat and Miller–Rabin tests in the prime generation process are evaluated. Furthermore, the concept of safe primes and their significance in public-key cryptographic frameworks, including RSA and Diffie–Hellman systems, are investigated. The findings demonstrate that prime number generation algorithms play a vital role in maintaining the security, reliability, and robustness of cryptographic systems.