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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 190 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
The Golden Code: A 2 × 2 fullrate spacetime code with nonvanishing determinants
 IEEE Transactions on Information Theory
, 2005
"... Abstract — In this paper we present the Golden ..."
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Perfect space–time block codes
 IEEE TRANS. INFORM. THEORY
, 2006
"... In this paper, we introduce the notion of perfect space–time block codes (STBCs). These codes have fullrate, fulldiversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic construct ..."
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Cited by 80 (15 self)
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In this paper, we introduce the notion of perfect space–time block codes (STBCs). These codes have fullrate, fulldiversity, nonvanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We present algebraic constructions of perfect STBCs for 2, 3, 4, and 6 antennas.
Asymptotics for Minimal Discrete Energy on the Sphere
 Trans. Amer. Math. Soc
"... We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1=r s ; where s ? 0 and r is Euclidean distance. With Ed(s; N) denoting the minimal energy for such Npoint arrangements we obtain bounds (vali ..."
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Cited by 43 (10 self)
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We investigate the energy of arrangements of N points on the surface of the unit sphere S d in R d+1 that interact through a power law potential V = 1=r s ; where s ? 0 and r is Euclidean distance. With Ed(s; N) denoting the minimal energy for such Npoint arrangements we obtain bounds (valid for all N) for E d (s; N) in the cases when 0 ! s ! d and 2 d ! s. For s = d, we determine the precise asymptotic behavior of Ed(d; N) as N !1. As a corollary, lower bounds are given for the separation of any pair of points in an Npoint minimal energy configuration, when s d 2. For the unit sphere in R 3 (d = 2), we present two conjectures concerning the asymptotic expansion of E 2 (s; N) that relate to the zeta function iL(s) for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of iL (s) when 0 ! s ! 2 (the divergent case). 1 Introduction and ...
Chaotic eigenfunctions in phase space
, 2008
"... We study individual eigenstates of quantized areapreserving maps on the 2torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states, as well as their stellar parametrization, which encodes states through a mi ..."
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Cited by 39 (0 self)
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We study individual eigenstates of quantized areapreserving maps on the 2torus which are classically chaotic. In order to analyze their semiclassical behavior, we use the Bargmann–Husimi representations for quantum states, as well as their stellar parametrization, which encodes states through a minimal set of points in phase space (the constellation of zeros of the Husimi density). We rigorously prove that a semiclassical uniform distribution of Husimi densities on the torus entails a similar equidistribution for the corresponding constellations. We deduce from this property a universal behavior for the phase patterns of chaotic Bargmann eigenfunctions, which is reminiscent of the WKB approximation for eigenstates of integrable systems (though in a weaker sense). In order to obtain more precise information on “chaotic eigenconstellations”, we then model their properties by ensembles of random states, generalizing former results on the 2sphere to the torus geometry. This approach yields statistical predictions for the constellations, which fit quite well the chaotic data. We finally observe that specific dynamical information, e.g. the presence of high peaks (like scars) in Husimi densities, can be recovered from the knowledge of a few longwavelength Fourier coefficients, which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions.
Quadratic Rational Rotations of the Torus and Dual Lattice Maps
"... We develop a general formalism for computedassisted proofs concerning the orbit structure of certain non ergodic piecewise ane maps of the torus, whose eigenvalues are roots of unity. For a specific class of maps, we prove that if the trace is a quadratic irrational (the simplest nontrivial case, c ..."
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Cited by 17 (10 self)
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We develop a general formalism for computedassisted proofs concerning the orbit structure of certain non ergodic piecewise ane maps of the torus, whose eigenvalues are roots of unity. For a specific class of maps, we prove that if the trace is a quadratic irrational (the simplest nontrivial case, comprising 8 maps), then the periodic orbits are organized into nitely many renormalizable families, with exponentially increasing period, plus a finite number of exceptional families. The proof is based on exact computations with algebraic numbers, where units play the role of scaling parameters. Exploiting a duality existing between these maps and lattice maps representing roundedoff planar rotations, we establish the global periodicity of the latter systems, for a set of orbits of full density.
Some Recent Progress on the Complexity of Lattice Problems
 In Proc. of FCRC
, 1999
"... We survey some recent developments in the study of the complexity of lattice problems. After a discussion of some problems on lattices which can be algorithmically solved efficiently, our main focus is the recent progress on complexity results of intractability. We will discuss Ajtai's worstcas ..."
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Cited by 13 (1 self)
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We survey some recent developments in the study of the complexity of lattice problems. After a discussion of some problems on lattices which can be algorithmically solved efficiently, our main focus is the recent progress on complexity results of intractability. We will discuss Ajtai's worstcase /averagecase connections, NPhardness and nonNPhardness, transference theorems between primal and dual lattices, and the AjtaiDwork cryptosystem. 1 Introduction There have been some exciting developments recently concerning the complexity of lattice problems. Research in the algorithmic aspects of lattice problems has been active in the past, especially following Lovasz's basis reduction algorithm in 1982. The recent wave of activity and interest can be traced in large part to two seminal papers written by Miklos Ajtai in 1996 and in 1997 respectively. In his 1996 paper [1], Ajtai found a remarkable worstcase to averagecase reduction for some versions of the shortest lattice vector probl...
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 RAPPORT DE RECHERCHE 911, INRIA, OCTOBRE
, 1988
"... We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implem ..."
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Cited by 9 (7 self)
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We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number.
Values of the Dedekind eta function at quadratic irrationalities
 Canad. J. Math
, 1999
"... Abstract. Let d be the discriminant of an imaginary quadratic field. Let a, b, c be integers such that b 2 − 4ac = d, a> 0, gcd(a, b, c) = 1. The value of η ( (b + √ d)/2a)  is determined explicitly, where η(z) is Dedekind’s eta function η(z) = e πiz/12 (1 − e m=1 ..."
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Cited by 6 (1 self)
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Abstract. Let d be the discriminant of an imaginary quadratic field. Let a, b, c be integers such that b 2 − 4ac = d, a> 0, gcd(a, b, c) = 1. The value of η ( (b + √ d)/2a)  is determined explicitly, where η(z) is Dedekind’s eta function η(z) = e πiz/12 (1 − e m=1