The Hidden Subgroup Problem and MKTP

Nicollas M. Sdroievski, Murilo V. G. da Silva, André Luís Vignatti

Abstract: We show that the Hidden Subgroup Problem for group families where products and inverses can be computed efficiently is in BPPMKTP (where MKTP is the Minimum KT Problem) using the techniques of Allender et al. (2018) [1]. We also show that the problem is in ZPPMKTP provided that there is a pac overestimator computable in ZPPMKTP for the logarithm of the order of the input group. This last result implies that for permutation groups, the dihedral group and many types of matrix groups the problem is in ZPPMKTP. Lastly, we also show that two decision versions of the problem admit statistical zero knowledge proofs. These results help classify the relative difficulty of the Hidden Subgroup Problem.