A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

We give deterministic $\tilde{O}(2^{2n})$-time $\tilde{O}(2^n)$-space 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 [R. Kannan, Math. Oper. Res. , 12 (1987), pp. 415--440] and SIVP [D. Micciancio, Proceedings of the $19$th Annual ACM-SIAM Symposium on Discrete Algorithms , 2008, pp. 84--93] and gives a deterministic and asymptotically faster alternative to the $2^{O(n)}$-time (and space) randomized algorithm for SVP of Ajtai, Kumar, and Sivakumar [ Proceedings of the $33$rd Annual ACM Symposium on Theory of Computing , 2001, pp. 266--275]. 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. A direct consequence of our results is a derandomization of the best current polynomial time approximation algorithms for SVP and CVP, achieving a $2^{O(n \log\log n / \log n)}$ approximation factor.