Abstract:
We give deterministic $2^{O(n)}$-time algorithms to solve all the most important computational problems on point lattices in NP, including the Shortest Vector Problem (SVP), Closest Vector Problem (CVP), and Shortest Independent Vectors Problem (SIVP). This improves the $n^{O(n)}$ running time of the best previously known algorithms for CVP (Kannan, Math. Operation Research 12(3):415-440, 1987) and SIVP (Micciancio, Proc. of SODA, 2008), and gives a deterministic alternative to the $2^{O(n)}$-time (and space) randomized algorithm for SVP of (Ajtai, Kumar and Sivakumar, STOC 2001). The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preprocessing function. In the process, we also give algorithms for several other lattice problems, including computing the kissing number of a lattice, and computing the set of all Voronoi relevant vectors. All our algorithms are deterministic, and have $2^{O(n)}$ time and space complexity.