arXiv:2608.14529 · v4

Polynomial-Factor Deterministic NP-Hardness for SVP in Every $\ell_p$ Norm with $p>2$

首次提交 2026-08-15 01:47

版本更新 2026-11-07 11:16

雷达观测 2026-11-10 10:11

摘要

We prove that for every constant $2<p<\infty$, the shortest vector problem in the $\ell_p$ norm is NP-hard to approximate within a polynomial factor under a deterministic reduction. More precisely, for lattices of rank $M$, we obtain a factor of $M^\varepsilon$ for every constant $0<\varepsilon<\min\{(p-2)/(4p),1/8\}$. For $p=\infty$, the same holds for every constant $0<\varepsilon<1/8$. Our reduction adapts OpenAI's encoding of a satisfying assignment for the closest vector problem ("Ten Advances in Mathematics and Theoretical Computer Science"). The encoding has one block of coordinates for each evaluation of a polynomial, with one position for each possible value. We allow the entries in a block to be arbitrary field elements, which can cancel when we take weighted sums of powers of these values. We show that on an unsatisfiable formula, a nonzero codeword with small support must have many nonzero entries in each block it uses. Applying a Hadamard matrix to each block, as in a work by Hair and Sahai (STOC 2026), gives the gap in vector length. The main results of this paper were discovered by Codex / ChatGPT 5.6 Sol Ultra using harness components from the UCLA Moonshot Harness Project, in response to a general strategy suggested by the authors. See the end of the introduction for a discussion.