Mind the Gap? Not for SVP Hardness Under ETH!

Abstract

We prove new hardness results for fundamental lattice problems under the Exponential Time Hypothesis (ETH). Building on a recent breakthrough by Bitansky et al. [BHIRW24], who gave a polynomial-time reduction from 3SAT to the (gap) 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 ∈ [1, ∞), there exists an explicit constant γ > 1 such that CVP _{p,γ} (the 𝓁_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) MAXLIN to CVP _{p,γ}.
Our main contribution is a randomized ETH hardness result for SVP _{p,γ} (the 𝓁_p-norm approximate Shortest Vector Problem) for all p ∈ (2, ∞). This result relies on a novel geometric property of the integer lattice ℤⁿ in the 𝓁_p norm, which says that for any p ∈ (2, ∞), the number of lattice vectors close to 1/2 1_n (in the 𝓁_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 CVP _{p,γ} to SVP _{p,γ'}.
Finally, we also use our ideas to give some minor improvements over prior reductions from 3SAT to BDD _{p, α} (the Bounded Distance Decoding Problem), yielding better ETH hardness results for BDD _{p, α} for any p ∈ [1, ∞) and α > α_p^{‡}, where α_p^{‡} is an explicit threshold depending on p.