Faster tuple lattice sieving using spherical locality-sensitive filters

To overcome the large memory requirement of classical lattice sieving algorithms for solving hard lattice problems, Bai-Laarhoven-Stehl\'{e} [ANTS 2016] studied tuple lattice sieving, where tuples instead of pairs of lattice vectors are combined to form shorter vectors. Herold-Kirshanova [PKC 2017] recently improved upon their results for arbitrary tuple sizes, for example showing that a triple sieve can solve the shortest vector problem (SVP) in dimension $d$ in time $2^{0.3717d + o(d)}$, using a technique similar to locality-sensitive hashing for finding nearest neighbors. In this work, we generalize the spherical locality-sensitive filters of Becker-Ducas-Gama-Laarhoven [SODA 2016] to obtain space-time tradeoffs for near neighbor searching on dense data sets, and we apply these techniques to tuple lattice sieving to obtain even better time complexities. For instance, our triple sieve heuristically solves SVP in time $2^{0.3588d + o(d)}$. For practical sieves based on Micciancio-Voulgaris' GaussSieve [SODA 2010], this shows that a triple sieve uses less space and less time than the current best near-linear space double sieve.Comment: 12 pages + references, 2 figures. Subsumed/merged into Cryptology ePrint Archive 2017/228, available at https://ia.cr/2017/122

Parameter-free Locality Sensitive Hashing for Spherical Range Reporting

We present a data structure for *spherical range reporting* on a point set $S$, i.e., reporting all points in $S$ that lie within radius $r$ of a given query point $q$. Our solution builds upon the Locality-Sensitive Hashing (LSH) framework of Indyk and Motwani, which represents the asymptotically best solutions to near neighbor problems in high dimensions. While traditional LSH data structures have several parameters whose optimal values depend on the distance distribution from $q$ to the points of $S$, our data structure is parameter-free, except for the space usage, which is configurable by the user. Nevertheless, its expected query time basically matches that of an LSH data structure whose parameters have been *optimally chosen for the data and query* in question under the given space constraints. In particular, our data structure provides a smooth trade-off between hard queries (typically addressed by standard LSH) and easy queries such as those where the number of points to report is a constant fraction of $S$, or where almost all points in $S$ are far away from the query point. In contrast, known data structures fix LSH parameters based on certain parameters of the input alone. The algorithm has expected query time bounded by $O(t (n/t)^\rho)$, where $t$ is the number of points to report and $\rho\in (0,1)$ depends on the data distribution and the strength of the LSH family used. We further present a parameter-free way of using multi-probing, for LSH families that support it, and show that for many such families this approach allows us to get expected query time close to $O(n^\rho+t)$, which is the best we can hope to achieve using LSH. The previously best running time in high dimensions was $\Omega(t n^\rho)$. For many data distributions where the intrinsic dimensionality of the point set close to $q$ is low, we can give improved upper bounds on the expected query time.Comment: 21 pages, 5 figures, due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract appearing here is slightly shorter than that in the PDF fil

Robust Proximity Search for Balls using Sublinear Space

Given a set of n disjoint balls b1, . . ., bn in IRd, we provide a data structure, of near linear size, that can answer (1 \pm \epsilon)-approximate kth-nearest neighbor queries in O(log n + 1/\epsilon^d) time, where k and \epsilon are provided at query time. If k and \epsilon are provided in advance, we provide a data structure to answer such queries, that requires (roughly) O(n/k) space; that is, the data structure has sublinear space requirement if k is sufficiently large

Streak camera receiver definition study

Detailed streak camera definition studies were made as a first step toward full flight qualification of a dual channel picosecond resolution streak camera receiver for the Geoscience Laser Altimeter and Ranging System (GLRS). The streak camera receiver requirements are discussed as they pertain specifically to the GLRS system, and estimates of the characteristics of the streak camera are given, based upon existing and near-term technological capabilities. Important problem areas are highlighted, and possible corresponding solutions are discussed

Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space

For a set of $n$ points in $\Re^d$, and parameters $k$ and \eps, we present a data structure that answers (1+\eps,k)-\ANN queries in logarithmic time. Surprisingly, the space used by the data-structure is \Otilde (n /k); that is, the space used is sublinear in the input size if $k$ is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this, we provide a sublinear space data-structure that can estimate the density of a point set under various measures, including: \begin{inparaenum}[(i)] \item sum of distances of $k$ closest points to the query point, and \item sum of squared distances of $k$ closest points to the query point. \end{inparaenum} Our approach generalizes to other distance based estimation of densities of similar flavor. We also study the problem of approximating some of these quantities when using sampling. In particular, we show that a sample of size \Otilde (n /k) is sufficient, in some restricted cases, to estimate the above quantities. Remarkably, the sample size has only linear dependency on the dimension

Tradeoffs for number-squeezing in collisions of Bose-Einstein condensates

We investigate the factors that influence the usefulness of supersonic collisions of Bose-Einstein condensates as a potential source of entangled atomic pairs by analyzing the reduction of the number difference fluctuations between regions of opposite momenta. We show that non-monochromaticity of the mother clouds is typically the leading limitation on number squeezing, and that the squeezing becomes less robust to this effect as the density of pairs grows. We develop a simple model that explains the relationship between density correlations and the number squeezing, allows one to estimate the squeezing from properties of the correlation peaks, and shows how the multi-mode nature of the scattering must be taken into account to understand the behavior of the pairing. We analyze the impact of the Bose enhancement on the number squeezing, by introducing a simplified low-gain model. We conclude that as far as squeezing is concerned the preferable configuration occurs when atoms are scattered not uniformly but rather into two well separated regions.Comment: 13 pages, 13 figures, final versio