Crittanalisi di cifrari post quantum attraverso il Problema del Sottogruppo Nascosto

The hidden subgroup problem (HSP) provides a unified context to pose difficult problems on which public-key ciphers rely; thanks to this generalisation, they turn out to be simple sub-instances. Shor algorithm itself, which was already heading in this direction, is surpassed in generality by HSP. Possessing a solving algorithm for HSP is then equivalent to breaking all ciphers at once. This defense offers, in the first chapter, a preliminary introduction to public-key cryptography; in the second chapter, a detailed description of quantum computing is made, trying to underline physical principles on which it is based and analysing its role in cryptanalysis. Afterwards, following a general view of the state-of-the-art of post-quantum cryptography and the situation of the call for proposals held by NIST, the third chapter analyses, in descending order of deepening, the three main classes of post-quantum public-key schemes: code-based, lattice-based and the apparently unsecure isogeny-based cryptography. Finally, the HSP is introduced, in its two versions: abelian, for which we have a quantum-level efficient algorithm, and non-abelian, still resistant to quantum computers. The first goal we set is to go beyond current results on the reduction of the discrete root problem as HSP. We also asked whether it was possible to reformulate code-based cryptography problems as HSP: we found affirmative answer with the instantiation of scrambler-permutation problem to a non-abelian HSP. The same reduction of the maximum likelihood decoding problem seems to remain an open question: lastly, we provide some ideas on its possible reformulation.