Why we couldn't prove SETH hardness of the Closest Vector Problem for even norms!

TL;DR

This paper establishes barriers to proving SETH hardness for the Closest Vector Problem in even norms, showing that such reductions imply unlikely complexity class containments, thus explaining why previous proofs for odd norms do not extend to even norms.

Contribution

The paper demonstrates that efficient reductions from SAT to CVP in even norms would imply unlikely complexity class containments, revealing fundamental barriers in proving SETH hardness for these problems.

Findings

Shows a complexity-theoretic barrier for SETH hardness proofs in even norms.

Establishes that such reductions imply coNP ⊆ NP/Poly, which is considered unlikely.

Extends the barrier results to the Shortest Vector Problem and Subset-Sum.

Abstract

Recent work [BGS17,ABGS19] has shown SETH hardness of CVP in the $\ell_p$ norm for any $p$ that is not an even integer. This result was shown by giving a Karp reduction from $k$-SAT on $n$ variables to CVP on a lattice of rank $n$. In this work, we show a barrier towards proving a similar result for CVP in the $\ell_p$ norm where $p$ is an even integer. We show that for any $c>0$, if for every $k > 0$, there exists an efficient reduction that maps a $k$-SAT instance on $n$ variables to a CVP instance for a lattice of rank at most $n^{c}$ in the Euclidean norm, then $\mathsf{coNP} \subset \mathsf{NP/Poly}$. We prove a similar result for CVP for all even norms under a mild additional promise that the ratio of the distance of the target from the lattice and the shortest non-zero vector in the lattice is bounded by $exp(n^{O(1)})$. Furthermore, we show that for any $c> 0$, and any even integer $p$, if for every $k > 0$, there exists an efficient reduction that maps a $k$-SAT instance on $n$ variables to a $SVP_p$ instance for a lattice of rank at most $n^{c}$, then $\mathsf{coNP} \subset \mathsf{NP/Poly}$. The result for SVP does not require any additional promise. While prior results have indicated that lattice problems in the $\ell_2$ norm (Euclidean norm) are easier than lattice problems in other norms, this is the first result that shows a separation between these problems. We achieve this by using a result by Dell and van Melkebeek [JACM, 2014] on the impossibility of the existence of a reduction that compresses an arbitrary $k$-SAT instance into a string of length $\mathcal{O}(n^{k-\epsilon})$ for any $\epsilon>0$. In addition to CVP, we also show that the same result holds for the Subset-Sum problem using similar techniques.