Under the auspices of the Computational Complexity Foundation (CCF)

REPORTS > DETAIL:

### Paper:

TR18-193 | 14th November 2018 23:17

#### The Hidden Subgroup Problem and MKTP

TR18-193
Authors: Nicollas Sdroievski, Murilo Silva, André Vignatti
Publication: 18th November 2018 07:11
We show that the Hidden Subgroup Problem for black-box groups is in $\mathrm{BPP}^\mathrm{MKTP}$ (where $\mathrm{MKTP}$ is the Minimum $\mathrm{KT}$ Problem) using the techniques of Allender et al (2018). We also show that the problem is in $\mathrm{ZPP}^\mathrm{MKTP}$ provided that there is a \emph{pac overestimator} computable in $\mathrm{ZPP}^\mathrm{MKTP}$ for the logarithm of the order of the given black-box group. This last result implies that for permutation groups, the dihedral group and many types of matrix groups the problem is in $\mathrm{ZPP}^\mathrm{MKTP}$. 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.