Polynomial-Factor Deterministic NP-Hardness for SVP in Every lp Norm with p > 2

Abstract

For every constant $2<p<\infty$ and every constant \[ 0<\varepsilon< \min\left\{\frac{p-2}{4p},\frac18\right\}, \] we show that the ${\ell }_p$-shortest vector problem for lattices of rank $M$ is NP hard to approximate within a factor of $M^{\epsilon }$, via a deterministic reduction. For $p=\infty$, the same holds for every constant $0<\epsilon <1/8$. The reduction builds on the polynomial-gap CVP construction of OpenAI [OpenAI 2026] and the direct reduction to SVP for $p>2$ of Hair and Sahai [STOC'26].