Integer lattice

Abstract: Our model assumes a population of agents residing at d-dimensional integer lattice locations with one individual per location. We focus on an expectation zero process Xs indexed on this lattice that is assumed to be mixing as detailed below. For simplicity, we also assume the process is stationary: the joint distribution of Xs for a collection of locations is invariant to translation and so, assuming second moments exist, EfXsXs+hg = C(h): The econometrician’s sample consists of realizations of agents’random variables Xs at a collection of locations fsig inside a sample region and measurements of these locations. We use the notation j j to denote the number of agents in our sample region and, for simplicity, assume that all locations in are sampled. When taking limits, we view as one of a sequence of regions indexed by that grow to include the whole lattice, an increasing domain approach to asymptotic approximations. In what follows, we state the notion of mixing coe¢ cients used throughout this Appendix, and provide proofs of the results in Conley and Molinari (2005).

Abstract: In lattice cryptography, worst-case to average-case reductions rely on two problems: Ajtai's SIS and Regev's LWE, which both refer to a very small class of random lattices related to the group $$G=\mathbb {Z}_q^n$$G=Zqn. We generalize worst-case to average-case reductions to all integer lattices of sufficiently large determinant, by allowing G to be any sufficiently large finite abelian group. Our maini¾?tool is a novel generalization of lattice reduction, which we call structural lattice reduction: given a finite abelian group G and a lattice L, it finds a short basis of some lattice $$\bar{L}$$Li¾? such that $$L \subseteq \bar{L}$$L⊆Li¾? and $$\bar{L}/L \simeq G$$Li¾?/Li¾?G. Our group generalizations of SIS and LWE allow us to abstract lattice cryptography, yet preserve worst-case assumptions: as an illustration, we provide a somewhat conceptually simpler generalization of the Alperin-Sheriff-Peikert variant of the Gentry-Sahai-Waters homomorphic scheme. We introduce homomorphic mux gates, which allows us to homomorphically evaluate any boolean function with a noise overhead proportional to the square root of its number of variables, and bootstrap the full scheme using only a linear noise overhead.

Abstract: Results on the finite nonperiodic Toda lattice are extended to some generalizations of the system: The relativistic Toda lattice, the generalized Toda lattice associated with simple Lie groups and the full Kostant-Toda lattice. The areas investigated, include master symmetries, recursion operators, higher Poisson brackets, invariants, and group symmetries for the systems. A survey of previous work on the classical Toda lattice is also included.