Approximate $\mathrm{CVP}$ in time $2^{0.802 \, n}$ -- now in any norm!

Abstract

We show that a constant factor approximation of the shortest and closest lattice vector problem in any norm can be computed in time $2^{0.802\, n}$. This contrasts the corresponding $2^n$ time, (gap)-SETH based lower bounds for these problems that even apply for small constant approximation. For both problems, $\mathrm{SVP}$ and $\mathrm{CVP}$, 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 body $K$ with an ellipsoid $\mathcal{E}$ so that $2^{\varepsilon n}$ translates of a constant scaling of $\mathcal{E}$ can cover $K$ and vice versa.