Integer Factorization and Related Cryptographical Problems

In a world increasingly reliant on digital communication, the security of our data hinges on mathematical problems that are surprisingly hard to solve. One of the most important is integer factorization—the task of breaking large numbers into their prime divisors. Factorization is the process of breaking down a number into smaller numbers that multiply together to give the original. For small numbers, this is easy. But when the numbers grow—into hundreds or even thousands of digits— the task becomes incredibly difficult. So difficult, in fact, that no one has yet found a fast and reliable way to do it. This difficulty is the foundation of RSA encryption, one of the most widely used systems for securing digital communication. The assumption is simple: if factoring large numbers is hard, then breaking the encryption is hard too. But there is no formal proof for the hardness of this problem. Itʼs just that decades of effort have failed to crack it. The project's main objective is to explore new methods for integer factorization and develop algorithms that solve the problem faster. The research contributes to a better understanding of the implications for digital security. The project also concerns other mathematical puzzles that play a role in cryptography, such as the subset-sum problem and the shortest vector problem. They have applications in areas like resource allocation and post-quantum cryptography, which aims to protect data even in a future where quantum computers could break todayʼs encryption. The project involves collaboration with national and international experts, including Lilya Budaghyan from the University of Bergen, David Harvey from UNSW in Sydney and Pantelimon Stanica from the Naval Postgraduate School in California.