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

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