In *STOC*, 2015.

Paper Simons talk abstract bibtex 49 downloads

Paper Simons talk abstract bibtex 49 downloads

We give a randomized $2^{n+o(n)}$-time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic $\widetilde{O}(4^n)$-time and $\widetilde{O}(2^n)$-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample $2^{n/2}$ vectors from the discrete Gaussian distribution at any parameter in $2^{n+o(n)}$ time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. We also show that our DGS algorithm implies a $2^{n + o(n)}$-time algorithm that approximates the Closest Vector Problem to within a factor of $1.97$. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate $2^{n/2}$ discrete Gaussian samples in just $2^{n/2+o(n)}$ time and space. Among other things, this implies a $2^{n/2+o(n)}$-time and space algorithm for $1.93$-approximate decision SVP.

@inproceedings{ADRSSolvingShortest15, title = {Solving the {Shortest Vector Problem} in $2^n$ time via discrete {Gaussian} sampling}, abstract = {We give a randomized $2^{n+o(n)}$-time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic $\widetilde{O}(4^n)$-time and $\widetilde{O}(2^n)$-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample $2^{n/2}$ vectors from the discrete Gaussian distribution at any parameter in $2^{n+o(n)}$ time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. We also show that our DGS algorithm implies a $2^{n + o(n)}$-time algorithm that approximates the Closest Vector Problem to within a factor of $1.97$. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate $2^{n/2}$ discrete Gaussian samples in just $2^{n/2+o(n)}$ time and space. Among other things, this implies a $2^{n/2+o(n)}$-time and space algorithm for $1.93$-approximate decision SVP.}, url = {http://arxiv.org/abs/1412.7994}, booktitle = {STOC}, author = {Aggarwal, Divesh and Dadush, Daniel and Regev, Oded and {Stephens-Davidowitz}, Noah}, year = {2015}, url_Simons_talk = {http://www.youtube.com/watch?v=PWy0ZBRAUxA} }

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