Euclidean SVP is deterministically NP-hard to approximate within any constant factor

Abstract

We prove that, for every constant $\rho>1$, the Euclidean shortest vector problem is NP-hard to approximate within any constant factor $\rho$ under a deterministic polynomial-time many-one reduction. This extends our previous deterministic NP-hardness result from $\rho<\sqrt 2$ to arbitrary constants and gives a deterministic version of Khot's randomized arbitrary-constant theorem. Our proof also gives deterministic counterparts of the two classical dimension-dependent regimes of Haviv and Regev: $2^{(\log n)^{1-\varepsilon}}$ under quasipolynomial-time reductions and $n^{c/\log\log n}$ under subexponential-time reductions.