Mind the Gap? Not for SVP Hardness under ETH!
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913049284182016 |
|---|---|
| author | Aggarwal, Divesh Gupta, Rishav Morolia, Aditya Zhang, Chuanqi |
| author_facet | Aggarwal, Divesh Gupta, Rishav Morolia, Aditya Zhang, Chuanqi |
| contents | We prove new hardness results for fundamental lattice problems under the Exponential Time Hypothesis (ETH). Building on a recent breakthrough by Bitansky et al.\ \cite{BHIRW24}, who gave a polynomial-time reduction from $\mathsf{3SAT}$ to the (gap) $\mathsf{MAXLIN}$ problem-a class of CSPs with linear equations over finite fields-we derive ETH hardness for several lattice problems.
First, we show that for any $p \in [1, \infty)$, there exists an explicit constant $γ> 1$ such that $\mathsf{CVP}_{p,γ}$ (the $\ell_p$-norm approximate Closest Vector Problem) does not admit a $2^{o(n)}$-time algorithm unless ETH is false. Our reduction is deterministic and proceeds via a direct reduction from (gap) $\mathsf{MAXLIN}$ to $\mathsf{CVP}_{p,γ}$.
Our main contribution is a randomized ETH hardness result for $\mathsf{SVP}_{p,γ}$ (the $\ell_p$-norm approximate Shortest Vector Problem) for all $p \in (2, \infty)$. This result relies on a novel geometric property of the integer lattice $\mathbb{Z}^n$ in the $\ell_p$ norm, which says that for any $p \in (2, \infty)$, the number of lattice vectors close to $\frac{1}{2}\vec{1}_n$ (in the $\ell_p$ norm) is exponentially larger than the number of short vectors (namely those close to the origin). We establish this property via a new inequality for the Theta function, which we use to get a randomized reduction from $\mathsf{CVP}_{p,γ}$ to $\mathsf{SVP}_{p,γ'}$.
Finally, we also use our ideas to give some minor improvements over prior reductions from $\mathsf{3SAT}$ to $\mathsf{BDD}_{p,α}$ (the Bounded Distance Decoding Problem), yielding better ETH hardness results for $\mathsf{BDD}_{p,α}$ for any $p \in [1, \infty)$ and $α> α_p^{\ddagger}$, where $α_p^{\ddagger}$ is an explicit threshold depending on $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02695 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mind the Gap? Not for SVP Hardness under ETH! Aggarwal, Divesh Gupta, Rishav Morolia, Aditya Zhang, Chuanqi Computational Complexity Cryptography and Security Data Structures and Algorithms We prove new hardness results for fundamental lattice problems under the Exponential Time Hypothesis (ETH). Building on a recent breakthrough by Bitansky et al.\ \cite{BHIRW24}, who gave a polynomial-time reduction from $\mathsf{3SAT}$ to the (gap) $\mathsf{MAXLIN}$ problem-a class of CSPs with linear equations over finite fields-we derive ETH hardness for several lattice problems. First, we show that for any $p \in [1, \infty)$, there exists an explicit constant $γ> 1$ such that $\mathsf{CVP}_{p,γ}$ (the $\ell_p$-norm approximate Closest Vector Problem) does not admit a $2^{o(n)}$-time algorithm unless ETH is false. Our reduction is deterministic and proceeds via a direct reduction from (gap) $\mathsf{MAXLIN}$ to $\mathsf{CVP}_{p,γ}$. Our main contribution is a randomized ETH hardness result for $\mathsf{SVP}_{p,γ}$ (the $\ell_p$-norm approximate Shortest Vector Problem) for all $p \in (2, \infty)$. This result relies on a novel geometric property of the integer lattice $\mathbb{Z}^n$ in the $\ell_p$ norm, which says that for any $p \in (2, \infty)$, the number of lattice vectors close to $\frac{1}{2}\vec{1}_n$ (in the $\ell_p$ norm) is exponentially larger than the number of short vectors (namely those close to the origin). We establish this property via a new inequality for the Theta function, which we use to get a randomized reduction from $\mathsf{CVP}_{p,γ}$ to $\mathsf{SVP}_{p,γ'}$. Finally, we also use our ideas to give some minor improvements over prior reductions from $\mathsf{3SAT}$ to $\mathsf{BDD}_{p,α}$ (the Bounded Distance Decoding Problem), yielding better ETH hardness results for $\mathsf{BDD}_{p,α}$ for any $p \in [1, \infty)$ and $α> α_p^{\ddagger}$, where $α_p^{\ddagger}$ is an explicit threshold depending on $p$. |
| title | Mind the Gap? Not for SVP Hardness under ETH! |
| topic | Computational Complexity Cryptography and Security Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.02695 |