On the Lattice Distortion Problem

We introduce and study the \emph{Lattice Distortion Problem} (LDP). LDP asks how "similar" two lattices are. I.e., what is the minimal distortion of a linear bijection between the two lattices? LDP generalizes the Lattice Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply asks whether the minimal distortion is one. As our first contribution, we show that the distortion between any two lattices is approximated up to a $n^{O(\log n)}$ factor by a simple function of their successive minima. Our methods are constructive, allowing us to compute low-distortion mappings that are within a $2^{O(n \log \log n/\log n)}$ factor of optimal in polynomial time and within a $n^{O(\log n)}$ factor of optimal in singly exponential time. Our algorithms rely on a notion of basis reduction introduced by Seysen (Combinatorica 1993), which we show is intimately related to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to within any constant factor (under randomized reductions), by a reduction from the Shortest Vector Problem.Comment: This is the full version of a paper that appeared in ESA 201

On the Quantitative Hardness of CVP

$\newcommand{\eps}{\varepsilon} \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\CVP}{\problem{CVP}} \newcommand{\SVP}{\problem{SVP}} \newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1}$For odd integers $p \geq 1$ (and $p = \infty$), we show that the Closest Vector Problem in the $\ell_p$ norm (\CVP_p) over rank $n$ lattices cannot be solved in 2^{(1-\eps) n} time for any constant \eps > 0 unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to "almost all" values of $p \geq 1$, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of \CVP_2 (i.e., \CVP in the Euclidean norm), for which a $2^{n +o(n)}$-time algorithm is known. In particular, our result applies for any $p = p(n) \neq 2$ that approaches $2$ as $n \to \infty$. We also show a similar SETH-hardness result for \SVP_\infty; hardness of approximating \CVP_p to within some constant factor under the so-called Gap-ETH assumption; and other quantitative hardness results for \CVP_p and \CVPP_p for any $1 \leq p < \infty$ under different assumptions

Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One

We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any $\gamma \leq 1 + O(\log n/n)$, we obtain an efficient dimension-preserving reduction from $\gamma^{O(n/\log n)}$-SVP to $\gamma$-GapSVP and an efficient dimension-preserving reduction from $\gamma^{O(n)}$-CVP to $\gamma$-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when $\gamma = 1$. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction---we reduce $\gamma^{O(n/\log n)}$-SVP to $\gamma$-unique SVP, a potentially easier problem than $\gamma$-GapSVP.Comment: Updated to acknowledge additional prior wor

New Shortest Lattice Vector Problems of Polynomial Complexity

The Shortest Lattice Vector (SLV) problem is in general hard to solve, except for special cases (such as root lattices and lattices for which an obtuse superbase is known). In this paper, we present a new class of SLV problems that can be solved efficiently. Specifically, if for an $n$-dimensional lattice, a Gram matrix is known that can be written as the difference of a diagonal matrix and a positive semidefinite matrix of rank $k$ (for some constant $k$), we show that the SLV problem can be reduced to a $k$-dimensional optimization problem with countably many candidate points. Moreover, we show that the number of candidate points is bounded by a polynomial function of the ratio of the smallest diagonal element and the smallest eigenvalue of the Gram matrix. Hence, as long as this ratio is upper bounded by a polynomial function of $n$, the corresponding SLV problem can be solved in polynomial complexity. Our investigations are motivated by the emergence of such lattices in the field of Network Information Theory. Further applications may exist in other areas.Comment: 13 page

Solving the Closest Vector Problem in 2n2^n Time--- The Discrete Gaussian Strikes Again!

We give a $2^{n+o(n)}$-time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on $n$-dimensional Euclidean lattices. This improves on the previous fastest algorithm, the deterministic $\widetilde{O}(4^{n})$-time and $\widetilde{O}(2^{n})$-space algorithm of Micciancio and Voulgaris. We achieve our main result in three steps. First, we show how to modify the sampling algorithm from [ADRS15] to solve the problem of discrete Gaussian sampling over lattice shifts, $L- t$, with very low parameters. While the actual algorithm is a natural generalization of [ADRS15], the analysis uses substantial new ideas. This yields a $2^{n+o(n)}$-time algorithm for approximate CVP for any approximation factor $\gamma = 1+2^{-o(n/\log n)}$. Second, we show that the approximate closest vectors to a target vector $t$ can be grouped into "lower-dimensional clusters," and we use this to obtain a recursive reduction from exact CVP to a variant of approximate CVP that "behaves well with these clusters." Third, we show that our discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP. The analysis depends crucially on some new properties of the discrete Gaussian distribution and approximate closest vectors, which might be of independent interest

Compute-and-Forward: Finding the Best Equation

Compute-and-Forward is an emerging technique to deal with interference. It allows the receiver to decode a suitably chosen integer linear combination of the transmitted messages. The integer coefficients should be adapted to the channel fading state. Optimizing these coefficients is a Shortest Lattice Vector (SLV) problem. In general, the SLV problem is known to be prohibitively complex. In this paper, we show that the particular SLV instance resulting from the Compute-and-Forward problem can be solved in low polynomial complexity and give an explicit deterministic algorithm that is guaranteed to find the optimal solution.Comment: Paper presented at 52nd Allerton Conference, October 201