Huck Bennett

Huck Bennett, Kyle Fridberg

A recent line of work motivated by cryptographic applications has studied the complexity of the Lattice Isomorphism Problem (LIP). In this work, we study LIP on self-dual lattices $\cal{L} \subset \mathbb{R}^n$, which appear naturally in many applications. Our main results are a $2^{n/2 + o(n)}$-time randomized algorithm for LIP and a $\mathsf{coNP}$ protocol for LIP on a broad class of self-dual lattices. These results extend recent work on ZLIP, the problem of deciding whether a lattice is isomorphic to $\mathbb{Z}^n$. In particular, the former result extends the $2^{n/2 + o(n)}$-time algorithms for ZLIP of Bennett, Ganju, Peetathawachai, and Stephens-Davidowitz (Eurocrypt, 2023) and of Ducas (Des. Codes Cryptogr., 2024). The latter result extends the $\mathrm{ZLIP} \in \mathsf{coNP}$ result of Hunkenschröder (Math. Prog. Series A, 2024). Our results leverage two key structural properties of self-dual lattices $\cal{L} \subset \mathbb{R}^n$: (1) every such lattice $\cal{L}$ is isomorphic to $\cal{L}_0 \oplus \mathbb{Z}^r$ for some self-dual lattice $\cal{L}_0$ with $λ_1(\cal{L}_0)^2 \geq 2$, and (2) every such lattice $\cal{L}$ has \emph{characteristic vectors}, i.e., there exist vectors $\mathbf{w} \in \cal{L}$ such that for every $\mathbf{v} \in \cal{L}$, $\langle\mathbf{v}, \mathbf{w}\rangle \equiv \langle\mathbf{v}, \mathbf{v}\rangle \pmod{2}$. Our results use a line of work by Elkies and Gaulter on lattices with long shortest characteristic vectors, and can be strengthened assuming a positive answer to a related question of Elkies (Math. Res. Lett., 1995). We also study Permutation Code Equivalence (PCE) on self-dual codes, and we observe that similar structural properties imply a polynomial-time algorithm for PCE on certain such codes. This gives a natural class of codes with large hull for which PCE is easy.

Huck Bennett, Mitchell Black, Amir Nayyeri et al.

The goal of graph inference is to design algorithms for learning properties of a hidden graph using queries to an oracle that returns information about the graph. Graph reconstruction, verification, and property testing are all types of graph inference. In this work, we study graph inference using an oracle that returns the effective resistance (ER) between a pair of vertices. Effective resistance is a distance originating from the study of electrical circuits with many applications. However, ER has received little attention from a graph inference perspective. Indeed, although it is known that an $n$-vertex graph can be uniquely reconstructed from all $\binom{n}{2}$ possible ER queries, little else is known. We address this gap with several new results, including: 1. $O(n)$-query algorithms for testing whether a graph is a tree; deciding whether two graphs are equal assuming one is a subgraph of the other; and testing whether a given vertex (or edge) is a cut vertex (or cut edge). 2. Property testing algorithms, including for testing whether a graph is vertex- or edge-biconnected. We also give a reduction to adapt property testing results from the bounded-degree model to our ER query model. This yields ER-query-based algorithms for testing $k$-connectivity, bipartiteness, planarity, and containment of a fixed subgraph. 3. Graph reconstruction algorithms, including an algorithm for reconstructing a graph from a low-width tree decomposition; a $Θ(k^2)$-query, polynomial-time algorithm for recovering the adjacency matrix $A$ of a hidden graph, given $A$ with $k$ of its entries deleted; and a $k$-query, exponential-time algorithm for the same task. We also compare the power of ER queries and shortest path queries, which are closely related but better studied. Interestingly, we show that the two query models are incomparable in power.

Huck Bennett, Chris Peikert, Yi Tang

We show improved fine-grained hardness of two key lattice problems in the $\ell_p$ norm: Bounded Distance Decoding to within an $α$ factor of the minimum distance ($\mathrm{BDD}_{p, α}$) and the (decisional) $γ$-approximate Shortest Vector Problem ($\mathrm{SVP}_{p,γ}$), assuming variants of the Gap (Strong) Exponential Time Hypothesis (Gap-(S)ETH). Specifically, we show: 1. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for $\mathrm{BDD}_{p, α}$ for any constant $α> α_\mathsf{kn}$, where $α_\mathsf{kn} = 2^{-c_\mathsf{kn}}$ and $c_\mathsf{kn}$ is the $\ell_2$ kissing-number constant, assuming $c_\mathsf{kn} > 0$ and that non-uniform Gap-ETH holds. 2. For all $p \in [1, \infty)$, there is no $2^{o(n)}$-time algorithm for $\mathrm{BDD}_{p, α}$ for any constant $α> α^\ddagger_p$, where $α^\ddagger_p$ is explicit and satisfies $α^\ddagger_p = 1$ for $1 \leq p \leq 2$, $α^\ddagger_p < 1$ for all $p > 2$, and $α^\ddagger_p \to 1/2$ as $p \to \infty$, unless randomized Gap-ETH is false. 3. For all $p \in [1, \infty) \setminus 2 \mathbb{Z}$ and all $C > 1$, there is no $2^{n/C}$-time algorithm for $\mathrm{BDD}_{p, α}$ for any constant $α> α^\dagger_{p, C}$, where $α^\dagger_{p, C}$ is explicit and satisfies $α^\dagger_{p, C} \to 1$ as $C \to \infty$ for any fixed $p \in [1, \infty)$, assuming $c_\mathsf{kn} > 0$ and that non-uniform Gap-SETH holds. 4. For all $p > p_0 \approx 2.1397$, $p \notin 2\mathbb{Z}$, and all $C > C_p$, there is no $2^{n/C}$-time algorithm for $\mathrm{SVP}_{p, γ}$ for some constant $γ> 1$, where $C_p > 1$ is explicit and satisfies $C_p \to 1$ as $p \to \infty$, unless randomized Gap-SETH is false.