A survey about Hidden Subgroup Problem from a mathematical and cryptographic perspective

We provide a survey on the Hidden Subgroup Problem (HSP), which plays an important role in studying the security of public-key cryptosystems. We first review the abelian case, where Kitaev's algorithm yields an efficient quantum solution to the HSP, recalling how classical problems (such as order finding, integer factorization, and discrete logarithm) can be formulated as abelian HSP instances. We then examine the current state of the art for non-abelian HSP, where no general efficient quantum solution is known, focusing on some relevant groups including dihedral group (connected to the shortest vector problem), symmetric groups (connected to the graph isomorphism problem), and semidirect product constructions (connected, in a special case, to the code equivalence problem). We also describe the main techniques for addressing the HSP in non-abelian cases, namely Fourier sampling and the black-box approach. Throughout the paper, we highlight the mathematical notions required and exploited in this context, providing a cryptography-oriented perspective.