Deterministic Hardness of Approximation of Unique-SVP and GapSVP in $\ell_p$ norms for $p>2$

• URL: http://arxiv.org/abs/2510.16991v1
• Date: Sun, 19 Oct 2025 20:17:26 GMT
• Title: Deterministic Hardness of Approximation of Unique-SVP and GapSVP in $\ell_p$ norms for $p>2$
• Authors: Yahli Hecht, Muli Safra,
• Abstract summary: We establish deterministic hardness of approximation results for the Shortest Vector Problem in $ell_p$ norm ($mathsfSVP_p$) and for Unique-SVP ($mathsfuSVP_p$)<n>For every $p > 2$, we prove constant-ratio hardness: no-time algorithm approximates $mathsfSVP_p$ or $mathsfuSVP_p$ within a ratio of $sqrt2 - o(1)$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\gamma$-$\mathsf{uSVP}_p$ carries over to ${\frac{1}{\gamma}}$-$\mathsf{BDD}_p$ (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.

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