Deterministic Hardness of Approximation of Unique-SVP and GapSVP in $\ell_p$ norms for $p>2$
Yahli Hecht, Muli Safra
TL;DR
The paper derives deterministic hardness of approximation results for SVP_p and uSVP_p in ell_p norms for p>2, achieving constant factor hardness within sqrt{2}-o(1) via subexponential deterministic reductions, and, for large p, quasi-polynomial hardness under standard NP and SUBEXP assumptions. It introduces a PCP-inspired, derandomized framework built on lattice encodings, low-degree polynomials, field extensions, and a Hadamard-based rotation to control norms, integrated through Composition-Recursion to scale to quasi-polynomial instance sizes. The results extend to uSVP under Unambiguous-3SAT and yield cryptographic implications: gamma-uSVP hardness transfers to (1/gamma)-BDD hardness, impacting worst-case guarantees for lattice-based cryptosystems. The approach differentiates from prior randomized reductions by delivering deterministic reductions across all p>2, and it lays groundwork for potential extensions to the Euclidean norm and broader cryptographic hardness landscapes. Open questions include extending the deterministic framework to ell2 and sharpening parameters while preserving soundness across all recursion levels.
Abstract
We establish deterministic hardness of approximation results for the Shortest Vector Problem in $\ell_p$ norm ($\mathsf{SVP}_p$) and for Unique-SVP ($\mathsf{uSVP}_p$) for all $p > 2$. Previously, no deterministic hardness results were known, except for $\ell_\infty$. For every $p > 2$, we prove constant-ratio hardness: no polynomial-time algorithm approximates $\mathsf{SVP}_p$ or $\mathsf{uSVP}_p$ within a ratio of $\sqrt{2} - o(1)$, assuming $\textsf{3SAT} \notin \text{DTIME}(2^{O(n^{2/3}\log n)})$, and, $\textsf{Unambiguous-3SAT} \notin \text{DTIME}(2^{O(n^{2/3}\log n)})$. We also show that for any $\varepsilon > 0$ there exists $p_\varepsilon > 2$ such that for every $p \ge p_\varepsilon$: no polynomial-time algorithm approximates $\mathsf{SVP}_p$ within a ratio of $2^{(\log n)^{1- \varepsilon}}$, assuming $\text{NP} \nsubseteq \text{DTIME}(n^{(\log n)^\varepsilon})$; and within a ratio of $n^{1/(\log\log(n))^\varepsilon}$, assuming $\text{NP} \nsubseteq \text{SUBEXP}$. This improves upon [Haviv, Regev, Theory of Computing 2012], which obtained similar inapproximation ratios under randomized reductions. We obtain analogous results for $\mathsf{uSVP}_p$ under the assumptions $\textsf{Unambiguous-3SAT} \not\subseteq \text{DTIME}(n^{(\log n)^\varepsilon})$ and $\textsf{Unambiguous-3SAT} \not\subseteq \text{SUBEXP}$, improving the previously known $1+o(1)$ [Stephens-Davidowitz, Approx 2016]. Strengthening the hardness of $\textsf{uSVP}$ has direct cryptographic impact. By the reduction of Lyubashevsky and Micciancio [Lyubashevsky, Micciancio, CRYPTO 2009], hardness for $γ$-$\mathsf{uSVP}_p$ carries over to ${\frac{1}γ}$-$\mathsf{BDD}_p$ (Bounded Distance Decoding). Thus, understanding the hardness of $\textsf{uSVP}$ improves worst-case guarantees for two core problems that underpin security in lattice-based cryptography.