COMPLEX NUMBER THEORY AND QUANTUM COMPUTERS: A NEW ERA OF MATHEMATICAL FACTORIZATION

Abstract

The factorization of composite numbers has occupied an important place in mathematics for centuries and forms the foundation of modern cryptographic systems. This paper analyzes the factorization process using traditional and quantum computers, the operating principles of Shor's algorithm, the impact of quantum technologies on cryptography, and future prospects. It has been determined that properties of quantum computers such as superposition, quantum parallelism, and quantum entanglement can significantly accelerate the factorization of composite numbers, which poses a serious threat to the security of cryptographic algorithms like RSA. According to the research results, the current state of quantum computer development, recent achievements in the field of factorization, and quantum-secure cryptography methods have been analyzed, demonstrating their mathematical and cryptographic significance.