A mathematical approach to post-quantum cryptography

Modern daily used cryptosystems like RSA are still safe by the fact that there is no known classical efficient algorithm to factor large integers into primes. However, the soon expected availability of quantum computers threatens the security of these cryptosystems. More concretely, Shor’s quantum algorithm for example solves the integer factorisation problem within polynomial time.

In my PhD thesis, I investigated the group-theoretical generalisation of integer factorisation, called the Hidden Subgroup Problem, and studied the state-of-the-art of the current methods and algorithms which have been found to solve this problem in different cases. This problem has been, for example, already efficiently solved for the case where the corresponding group G is abelian, or Hamiltonian. The semidirect product case has been solved for certain specific instances, but not in general.

In my research I analyze the Hidden Subgroup Problem by reducing more general cases to one of these previous cases, step by step and with the purpose of analyzing how vulnerable a HSP-based cryptosystem is. I also investigate to what extent Shor’s algorithm can be used to attack other existing cryptosystems, for instance those based on group rings.