We give a @math -time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on @math -dimensional Euclidean lattices. This improves on the previous fastest algorithm, the deterministic @math -time and @math -space algorithm of Micciancio and Voulgaris. We achieve our main result in three steps. First, we show how to modify the sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian sampling over lattice shifts, @math , with very low parameters. While the actual algorithm is a natural generalization of [ADRS15], the analysis uses substantial new ideas. This yields a @math -time algorithm for approximate CVP for any approximation factor @math . Second, we show that the approximate closest vectors to a target vector @math can be grouped into "lower-dimensional clusters," and we use this to obtain a recursive reduction from exact CVP to a variant of approximate CVP that "behaves well with these clusters." Third, we show that our discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP. The analysis depends crucially on some new properties of the discrete Gaussian distribution and approximate closest vectors, which might be of independent interest.
Finally, after this work was published, @cite_36 showed a dimension-preserving reduction from DGS to CVP, answering a question posed in an earlier version of this paper. Together with our work, this reduction immediately implies a @math -time algorithm for DGS with @math . (Our algorithm works for any parameter @math , but not arbitrarily small @math .) This also provides some (arguably weak) evidence that our technique of using DGS for solving CVP is correct, in the sense that any faster algorithm for CVP necessarily yields a faster algorithm for DGS.
@cite_36
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The discrete Gaussian @math is the distribution that assigns to each vector @math in a shifted lattice @math probability proportional to @math . It has long been an important tool in the study of lattices. More recently, algorithms for discrete Gaussian sampling (DGS) have found many applications in computer science. In particular, polynomial-time algorithms for DGS with very high parameters @math have found many uses in cryptography and in reductions between lattice problems. And, in the past year, Aggarwal, Dadush, Regev, and Stephens-Davidowitz showed @math -time algorithms for DGS with a much wider range of parameters and used them to obtain the current fastest known algorithms for the two most important lattice problems, the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). Motivated by its increasing importance, we investigate the complexity of DGS itself and its relationship to CVP and SVP. Our first result is a polynomial-time dimension-preserving reduction from DGS to CVP. There is a simple reduction from CVP to DGS, so this shows that DGS is equivalent to CVP. Our second result, which we find to be more surprising, is a polynomial-time dimension-preserving reduction from centered DGS (the important special case when @math ) to SVP. In the other direction, there is a simple reduction from @math -approximate SVP for any @math , and we present some (relatively weak) evidence to suggest that this might be the best achievable approximation factor. We also show that our CVP result extends to a much wider class of distributions and even to other norms.