The Infrastructure of a Global Field of Arbitrary Unit Rank

In this paper, we show a general way to interpret the infrastructure of a global field of arbitrary unit rank. This interpretation generalizes the prior concepts of the giant step operation and f-representations, and makes it possible to relate the infrastructure to the (Arakelov) divisor class group of the global field. In the case of global function fields, we present results that establish that effective implementation of the presented methods is indeed possible, and we show how Shanks' baby-step giant-step method can be generalized to this situation.Comment: Revised version. Accepted for publication in Math. Com

Quantum Algorithm for Computing the Period Lattice of an Infrastructure

We present a quantum algorithm for computing the period lattice of infrastructures of fixed dimension. The algorithm applies to infrastructures that satisfy certain conditions. The latter are always fulfilled for infrastructures obtained from global fields, i.e., algebraic number fields and function fields with finite constant fields. The first of our main contributions is an exponentially better method for sampling approximations of vectors of the dual lattice of the period lattice than the methods outlined in the works of Hallgren and Schmidt and Vollmer. This new method improves the success probability by a factor of at least 2^{n^2-1} where n is the dimension. The second main contribution is a rigorous and complete proof that the running time of the algorithm is polynomial in the logarithm of the determinant of the period lattice and exponential in n. The third contribution is the determination of an explicit lower bound on the success probability of our algorithm which greatly improves on the bounds given in the above works. The exponential scaling seems inevitable because the best currently known methods for carrying out fundamental arithmetic operations in infrastructures obtained from algebraic number fields take exponential time. In contrast, the problem of computing the period lattice of infrastructures arising from function fields can be solved without the exponential dependence on the dimension n since this problem reduces efficiently to the abelian hidden subgroup problem. This is also true for other important computational problems in algebraic geometry. The running time of the best classical algorithms for infrastructures arising from global fields increases subexponentially with the determinant of the period lattice.Comment: 52 pages, 4 figure

On the Probability of Generating a Lattice

We study the problem of determining the probability that m vectors selected uniformly at random from the intersection of the full-rank lattice L in R^n and the window [0,B)^n generate $\Lambda$ when B is chosen to be appropriately large. This problem plays an important role in the analysis of the success probability of quantum algorithms for solving the Discrete Logarithm Problem in infrastructures obtained from number fields and also for computing fundamental units of number fields. We provide the first complete and rigorous proof that 2n+1 vectors suffice to generate L with constant probability (provided that B is chosen to be sufficiently large in terms of n and the covering radius of L and the last n+1 vectors are sampled from a slightly larger window). Based on extensive computer simulations, we conjecture that only n+1 vectors sampled from one window suffice to generate L with constant success probability. If this conjecture is true, then a significantly better success probability of the above quantum algorithms can be guaranteed.Comment: 18 page

Robust Near-Field 3D Localization of an Unaligned Single-Coil Agent Using Unobtrusive Anchors

The magnetic near-field provides a suitable means for indoor localization, due to its insensitivity to the environment and strong spatial gradients. We consider indoor localization setups consisting of flat coils, allowing for convenient integration of the agent coil into a mobile device (e.g., a smart phone or wristband) and flush mounting of the anchor coils to walls. In order to study such setups systematically, we first express the Cram\'er-Rao lower bound (CRLB) on the position error for unknown orientation and evaluate its distribution within a square room of variable size, using 15 x 10cm anchor coils and a commercial NFC antenna at the agent. Thereby, we find cm-accuracy being achievable in a room of 10 x 10 x 3 meters with 12 flat wall-mounted anchors and with 10mW used for the generation of magnetic fields. Practically achieving such estimation performance is, however, difficult because of the non-convex 5D likelihood function. To that end, we propose a fast and accurate weighted least squares (WLS) algorithm which is insensitive to initialization. This is enabled by effectively eliminating the orientation nuisance parameter in a rigorous fashion and scaling the individual anchor observations, leading to a smoothed 3D cost function. Using WLS estimates to initialize a maximum-likelihood (ML) solver yields accuracy near the theoretical limit in up to 98% of cases, thus enabling robust indoor localization with unobtrusive infrastructure, with a computational efficiency suitable for real-time processing.Comment: 7 pages, to be presented at IEEE PIMRC 201

Systematics of Aligned Axions

We describe a novel technique that renders theories of $N$ axions tractable, and more generally can be used to efficiently analyze a large class of periodic potentials of arbitrary dimension. Such potentials are complex energy landscapes with a number of local minima that scales as $\sqrt{N!}$, and so for large $N$ appear to be analytically and numerically intractable. Our method is based on uncovering a set of approximate symmetries that exist in addition to the $N$ periods. These approximate symmetries, which are exponentially close to exact, allow us to locate the minima very efficiently and accurately and to analyze other characteristics of the potential. We apply our framework to evaluate the diameters of flat regions suitable for slow-roll inflation, which unifies, corrects and extends several forms of "axion alignment" previously observed in the literature. We find that in a broad class of random theories, the potential is smooth over diameters enhanced by $N^{3/2}$ compared to the typical scale of the potential. A Mathematica implementation of our framework is available online.Comment: 68 pages, 17 figure