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Predicting lattice reduction
 In Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology, EUROCRYPT’08
, 2008
"... Abstract. Despite their popularity, lattice reduction algorithms remain mysterious cryptanalytical tools. Though it has been widely reported that they behave better than their proved worstcase theoretical bounds, no precise assessment has ever been given. Such an assessment would be very helpful to ..."
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Abstract. Despite their popularity, lattice reduction algorithms remain mysterious cryptanalytical tools. Though it has been widely reported that they behave better than their proved worstcase theoretical bounds, no precise assessment has ever been given. Such an assessment would be very helpful to predict the behaviour of latticebased attacks, as well as to select keysizes for latticebased cryptosystems. The goal of this paper is to provide such an assessment, based on extensive experiments performed with the NTL library. The experiments suggest several conjectures on the worst case and the actual behaviour of lattice reduction algorithms. We believe the assessment might also help to design new reduction algorithms overcoming the limitations of current algorithms.
Universally optimal distribution of points on spheres
 Journal of the American Mathematical Society
"... Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a ..."
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Cited by 63 (11 self)
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Abstract. We study configurations of points on the unit sphere that minimize potential energy for a broad class of potential functions (viewed as functions of the squared Euclidean distance between points). Call a configuration sharp if there are m distances between distinct points in it and it is a spherical (2m −1)design. We prove that every sharp configuration minimizes potential energy for all completely monotonic potential functions. Examples include the minimal vectors of the E8 and Leech lattices. We also prove the same result for the vertices of the 600cell, which do not form a sharp configuration. For most known cases, we prove that they are the unique global minima for energy, as long as the potential function is strictly completely monotonic. For certain potential functions, some of these configurations were previously analyzed by Yudin, Kolushov, and Andreev; we build on their techniques. We also generalize our results to other compact twopoint homogeneous spaces, and we
Optimality and Uniqueness of the Leech Lattice Among Lattices
 arXiv:math.MG/04 03263v1 16
, 2004
"... Abstract. We prove that the Leech lattice is the unique densest lattice in R 24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R 24 can exceed the Leech lattice’s density by a factor of ..."
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Cited by 53 (5 self)
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Abstract. We prove that the Leech lattice is the unique densest lattice in R 24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R 24 can exceed the Leech lattice’s density by a factor of more than 1 + 1.65 · 10 −30, and we give a new proof that E8 is the unique densest lattice in R 8. 1.
Kissing Numbers, Sphere Packings, and Some Unexpected Proofs
 NOTICES AMER. MATH. SOC
, 2004
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New conjectural lower bounds on the optimal density of sphere packings
 MATH
, 2006
"... Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100yearold lower bound on the maximal packing density due to Minkowski in ddimensional Euclidea ..."
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Cited by 18 (7 self)
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Sphere packings in high dimensions interest mathematicians and physicists and have direct applications in communications theory. Remarkably, no one has been able to provide exponential improvement on a 100yearold lower bound on the maximal packing density due to Minkowski in ddimensional Euclidean space Rd. The asymptotic behavior of this bound is controlled by 2−d in high dimensions. Using an optimization procedure that we introduced earlier [TS02] and a conjecture concerning the existence of disordered sphere packings in Rd, we obtain a conjectural lower bound on the density whose asymptotic behavior is controlled by 2−0.77865...d, thus providing the putative exponential improvement of Minkowski’s bound. The conjecture states that a hardcore nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in Rd for asymptotically large d if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical packing constructions in low dimensions, known necessary conditions that only have relevance in very low dimensions, and the fact that we can recover the forms of known rigorous lower bounds. A byproduct of our approach is an asymptotic conjectural lower bound on the average kissing number whose behavior is controlled by 20.22134...d, which is to be compared to the best known asymptotic lower bound on the individual kissing number of 20.2075...d. Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies [CE03] to obtain upper bounds on the density, and hence has implications for linear programming bounds. This connection proves that our density estimate can never exceed the CohnElkies upper bound, regardless of the validity of our conjecture.
Jammed HardParticle Packings: From Kepler to Bernal and Beyond
, 2010
"... Understanding the characteristics of jammed particle packings provides basic insights into the structure and bulk properties of crystals, glasses, and granular media, and into selected aspects of biological systems. This review describes the diversity of jammed configurations attainable by frictionl ..."
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Cited by 18 (3 self)
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Understanding the characteristics of jammed particle packings provides basic insights into the structure and bulk properties of crystals, glasses, and granular media, and into selected aspects of biological systems. This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for that purpose it stresses individualpacking geometric analysis. A fundamental feature of that diversity is the necessity to classify individual jammed configurations according to whether they are locally, collectively, or strictly jammed. Each of these categories contains a multitude of jammed configurations spanning a wide and (in the large system limit) continuous range of intensive properties, including packing fraction φ, mean contact number Z, and several scalar order metrics. Application of these analytical tools to spheres in three dimensions (an analog to the venerable Ising model) covers a myriad of jammed states, including maximally dense packings (as Kepler conjectured), lowdensity strictlyjammed tunneled crystals, and a substantial family of amorphous packings. With respect to the last of these, the current approach displaces the
Enumeration of totally real number fields of bounded . . .
, 2008
"... We enumerate all totally real number fields F with root ..."
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Cited by 12 (3 self)
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We enumerate all totally real number fields F with root
CONVOLUTION ROOTS OF RADIAL POSITIVE DEFINITE FUNCTIONS WITH COMPACT SUPPORT
"... Abstract. A classical theorem of Boas, Kac, and Krein states that a characteristic function ϕ with ϕ(x) =0forx  ≥τ admits a representation of the form ϕ(x) = u(y)u(y + x)dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complexvalued with u(x) =0for x  ≥τ/2. The result can be expressed equi ..."
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Cited by 8 (1 self)
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Abstract. A classical theorem of Boas, Kac, and Krein states that a characteristic function ϕ with ϕ(x) =0forx  ≥τ admits a representation of the form ϕ(x) = u(y)u(y + x)dy, x ∈ R, where the convolution root u ∈ L 2 (R) is complexvalued with u(x) =0for x  ≥τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the BoasKac representation under additional constraints: If ϕ is realvalued and even, can the convolution root u be chosen as a realvalued and/or even function? A complete answer in terms of the zeros of the Fourier transform of ϕ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on R d is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with halfsupport. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with halfsupport exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán’s problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if f is a probability density on R d whose characteristic function ϕ vanishes outside the unit ball, then ∫ x  2 f(x)dx = −∆ϕ(0) ≥ 4 j 2 (d−2)/2 where jν denotes the first positive zero of the Bessel function Jν, andtheestimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a realvalued halfsupport convolution root of the spherical correlation function in R 2 does not exist. 1.
Low dimensional strongly perfect lattices. I: The 12dimensional case
 MATH
, 2005
"... It is shown that the CoxeterTodd lattice is the unique strongly perfect lattice in dimension 12. ..."
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Cited by 8 (3 self)
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It is shown that the CoxeterTodd lattice is the unique strongly perfect lattice in dimension 12.
THE NEXTORDER TERM FOR OPTIMAL RIESZ AND LOGARITHMIC ENERGY ASYMPTOTICS ON THE SPHERE
, 2012
"... We survey known results and present estimates and conjectures for the nextorder term in the asymptotics of the optimal logarithmic energy and Riesz senergy of N points on the unit sphere in Rd+1, d ≥ 1. The conjectures are based on analytic continuation assumptions (with respect to s) for the coe ..."
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Cited by 8 (2 self)
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We survey known results and present estimates and conjectures for the nextorder term in the asymptotics of the optimal logarithmic energy and Riesz senergy of N points on the unit sphere in Rd+1, d ≥ 1. The conjectures are based on analytic continuation assumptions (with respect to s) for the coefficients in the asymptotic expansion (as N →∞) of the optimal senergy.