# Approximate $\mathrm {CVP}_{}$ in Time 20.802 n - Now in Any Norm!

@inproceedings{Rothvoss2022Approximate, title={Approximate \$\mathrm \{CVP\}\_\{\}\$ in Time 20.802 n - Now in Any Norm!}, author={Thomas Rothvoss and Moritz Venzin}, booktitle={IPCO}, year={2022} }

We show that a constant factor approximation of the shortest and closest lattice vector problem in any norm can be computed in time 20.802n . This contrasts the corresponding 2n time, (gap)SETH based lower bounds for these problems that even apply for small constant approximation. For both problems, SVP and CVP, we reduce to the case of the Euclidean norm. A key technical ingredient in that reduction is a twist of Milman’s construction of an M-ellipsoid which approximates any symmetric convex…

## References

SHOWING 1-10 OF 57 REFERENCES

### Discrete Gaussian Sampling Reduces to CVP and SVP

- Computer Science, MathematicsSODA
- 2016

There is a simple reduction from CVP to DGS, so this shows that DGS is equivalent to CVP, and it is shown that the CVP result extends to a much wider class of distributions and even to other norms.

### Limits on the Hardness of Lattice Problems in ℓp Norms

- Computer Science, MathematicsTwenty-Second Annual IEEE Conference on Computational Complexity (CCC'07)
- 2007

The results improve prior approximation factors for ℓp norms by up to up to $$\sqrt{n}$$ factors, and provide some evidence that lattice problems in ™p norms (for p > 2) may not be substantially harder than they are in the ™2 norm.

### Enumerative Lattice Algorithms in any Norm Via M-ellipsoid Coverings

- Mathematics, Computer Science2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
- 2011

A novel algorithm for enumerating lattice points in any convex body known as the M-ellipsoid is given, and an expected O(f*(n))^n-time algorithm for Integer Programming, where f*( n) denotes the optimal bound in the so-calledflatnesstheorem, which is conjectured to be f* (n) = O(n).

### A O(1/ε 2) n -Time Sieving Algorithm for Approximate Integer Programming

- MathematicsLATIN
- 2012

A randomized algorithm is given for an approximate version of the Integer Programming Problem for a polytope P⊆ℝn, which correctly decides whether P contains an integer point or whether a (1+e)-scaling of P about its center of gravity is integer free in O(1/e2)n-time and O( 1/e) n-space with overwhelming probability.

### Solving the Shortest Vector Problem in 2n Time Using Discrete Gaussian Sampling: Extended Abstract

- Computer Science, MathematicsSTOC
- 2015

The SVP result follows from a natural reduction from SVP to DGS, and a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate 2n/2 discrete Gaussian samples in just 1.93-approximate decision SVP.

### Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

- Computer Science, Mathematics2015 IEEE 56th Annual Symposium on Foundations of Computer Science
- 2015

A 2n+o(n)-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on n-dimensional Euclidean lattices and it is shown that the approximate closest vectors to a target vector t can be grouped into “lower-dimensional clusters,” and the discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP.

### The shortest vector in a lattice is hard to approximate to within some constant

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

It is shown the shortest vector problem in the l/sub 2/ norm is NP-hard (for randomized reductions) to approximate within any constant factor less than /spl radic/2 and an alternative construction satisfying Ajtai's probabilistic variant of Sauer's lemma is given.

### Faster provable sieving algorithms for the Shortest Vector Problem and the Closest Vector Problem on lattices in 𝓁p norm

- Computer Science, MathematicsAlgorithms
- 2021

In this paper we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in $\ell_p$ norm for $1\leq p\leq\infty$. The running time we…

### Improved Algorithms for the Shortest Vector Problem and the Closest Vector Problem in the Infinity Norm

- Mathematics, Computer ScienceISAAC
- 2018

A new sieving procedure is given that runs in time linear in $N$, thereby significantly improving the running time of the algorithm for SVP in the $\ell_\infty$ norm and it is shown that the heuristic sieving algorithms of Nguyen and Vidick and Wang et al.[WLTB11] can also be analyzed in the $ell_{\ infty}$ norm.

### Sampling short lattice vectors and the closest lattice vector problem

- Computer ScienceProceedings 17th IEEE Annual Conference on Computational Complexity
- 2002

Using the SVP algorithm from (Ajtai et al., 2001), this work obtains a randomized 2[O(1+/spl epsi//sup -1/)n] algorithm to obtain a (1+/)-approximation for the closest lattice vector problem in n dimensions, which improves the existing time bound of O(n!) for CVP.