Internship Report – NEC 2008

This report summarizes the work done at NEC Labs America, Princeton from 28th May 2008 to 29th August 2008. I worked under the supervision of Martin Roetteler and collaborated with Hari Krovi. We worked on the hidden subgroup problem(HSP) for dihedral group. Dihedral HSP will solve f(n)-shortest vector lattice problem for f(n) = n 1 2 ([1]), which imply connection with some cryptographic protocols ([2]). We looked at various approaches for solving dihedral HSP. The first one was a statistical approach by Ettinger-Hoyer ([3]), where they used exponential amount of classical post processing to distinguish between probability distributions of different slopes. In 2003, Kuperberg ([4]) came up with the sieving idea ([5]) to generate state with the irrep label (1, 0, · · · , 0). This gives us the first bit of slope. And repeating it logN (for dihedral group DN ) times (in smaller groups) gives us the complete slope. This approach takes subexponential time (2 √ logN ). Some years back Bacon et. al. ([6]) came up with highly entangled quantum measurements (PGM) to solve DHSP (dihedral HSP). Here the measurement was based on subset sum. So any algorithm which quantum samples from subset sum solution will be able to solve DHSP. We studied all of these approaches and tried to combine them to get a better algorithm. In the process we also encountered groups similar to dihedral group and tried to solve HSP for them. In this article I will summarize different approaches we took for solving these HSP problems. This is just a collection of all the approaches we have worked on and have no intended flow.