Lattice enumeration algorithms are the most basic algorithms for solving hard lattice problems such as the shortest vector problem and the closest vector problem, and are often used in public-key cryptanalysis either as standalone algorithms, or as subroutines in lattice reduction algorithms. Here we revisit these fundamental algorithms and show that surprising exponential speedups can be achie...

The form factors for the semileptonic decays of heavy-light pseudoscalar mesons of the type D ! Kee are studied in quenched lattice QCD at = 6:0 using Wilson fermions. We explore new numerical techniques for improving the signal and study O(a) corrections using three diierent lattice transcriptions of the vector current. We present a detailed discussion of the relation of these lattice currents...

We demonstrate how additive number theory can be used to produce new classes of inequalities in Ehrhart theory. More specifically, we use a classical result of Kneser to produce new inequalities between the coefficients of the Ehrhart δ-vector of a lattice polytope. The inequalities are indexed by the vertices of rational polyhedra Q(r, s) ⊆ R for 0 ≤ r ≤ s. As an application, we deduce all pos...

In this expression, R is a lattice vector between a pair of unit cells: R = ua + vb + wc ; u, v, and w are integers and the dot product k R = kau + kbv + kcw . (In two dimensions, R = ua + vb and k R = kau + kbv .) At this point, we need to clarify the meaning of the vector k and find a way to define the twoand three-dimensional Brillouin zones. To do this, let’s consider the general, oblique, ...

Asymptotic expressions for the optimal scaling factor and resulting minimum distortion , as a function of codebook size N, are found for xed-rate k-dimensional lattice vector quantization of generalized Gaussian sources with decay parameter 1. These expressions are derived by minimizing upper and lower bounds to distortion. It is shown that the optimal scaling factor a N decreases as (lnN) 1== ...

In the present paper we show how to speed up lattice parameter searches for Monte Carlo and quasi–Monte Carlo node sets. The classical measure for such parameter searches is the spectral test which is based on a calculation of the shortest nonzero vector in a lattice. Instead of the shortest vector we apply an approximation given by the LLL algorithm for lattice basis reduction. We empirically ...

We describe how a shortest vector of a 2-dimensional integral lattice corresponds to a best approximation of a unique rational number defined by the lattice. This rational number and its best approximations can be computed with the euclidean algorithm and its speedup by Schönhage (1971) from any basis of the lattice. The described correspondence allows, on the one hand, to reduce a basis of a 2...

For a left vector space V over a totally ordered division ring F, let Co(V ) denote the lattice of convex subsets of V . We prove that every lattice L can be embedded into Co(V ) for some left F-vector space V . Furthermore, if L is finite lower bounded, then V can be taken finite-dimensional, and L embeds into a finite lower bounded lattice of the form Co(V,Ω) = {X ∩Ω | X ∈ Co(V )}, for some f...

We introduce algorithms for lattice basis reduction that are improvements of the famous L 3-algorithm. If a random L 3 {reduced lattice basis b1; : : : ; bn is given such that the vector of reduced Gram{ Schmidt coeecients (fi;jg 1 j < i n) is uniformly distributed in 0; 1) (n 2) , then the pruned enumeration nds with positive probability a shortest lattice vector. We demonstrate the power of t...

Lattice studies of hadronic matrix elements require matching between operators in full continuum QCD and those in the lattice theory being simulated. In the present article we will focus on an effective lattice theory which combines nonrelativistic (NRQCD) heavy quarks and clover light quarks. We wish to match the theories correct through O( p M , α p M , α ap). For heavy meson (e.g. B and B∗) ...