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

2026-08-14

AI Summary

This study addresses the longstanding challenge of establishing deterministic polynomial-factor NP-hardness for the Shortest Vector Problem (SVP) under ℓ_p norms where p > 2. By constructing a deterministic reduction from 3SAT to GapSVP_p utilizing polynomial-gap CVP constructions and direct reduction techniques, this work achieves the first deterministic polynomial-factor NP-hardness proof for SVP across all p > 2 norms. These results fill a critical gap in the complexity theory of high-norm lattice problems by establishing deterministic NP-hardness for SVP under any p > 2 and the infinity norm. Consequently, this research provides essential theoretical foundations for security analysis in lattice-based cryptography, resolving a significant open problem regarding the computational hardness of SVP in higher norms.

Abstract

For every constant $2<p<\infty$ and every constant \[ 0<\varepsilon< \min\left\{\frac{p-2}{4p},\frac18\right\}, \] we give a deterministic polynomial-time reduction from 3SAT to $M^\varepsilon$-GapSVP$_p$, where $M$ is the lattice rank. For $p=\infty$, the same holds for every constant $0<\varepsilon<1/8$. 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].

Problem

• SVP
• NP-hardness
• lp norm
• deterministic reduction
• GapSVP

Innovation

• Deterministic Reduction
• GapSVP
• Polynomial-Factor Hardness
• lp Norm
• CVP to SVP