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36
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 162 (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 ..."
Perfect space–time block codes
 IEEE Trans. Inform. Theory
, 2006
"... Abstract—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 c ..."
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Cited by 50 (12 self)
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Abstract—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. Index Terms—Cubic shaping, cyclic algebras, division algebras, nonvanishing determinant, perfect codes. I.
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 28 (9 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 ...
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 worstcase /av ..."
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Cited by 12 (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 impl ..."
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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. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...
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 9 (4 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.
Generating Elliptic Curves of Prime Order
 in Cryptographic Hardware and Embedded Systems – CHES 2001, LNCS
, 2001
"... Abstract. Avariation of the Complex Multiplication (CM) method for generating elliptic curves of known order over finite fields is proposed. We give heuristics and timing statistics in the mildly restricted setting of prime curve order. These may be seen to corroborate earlier work of Koblitz in the ..."
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Cited by 5 (0 self)
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Abstract. Avariation of the Complex Multiplication (CM) method for generating elliptic curves of known order over finite fields is proposed. We give heuristics and timing statistics in the mildly restricted setting of prime curve order. These may be seen to corroborate earlier work of Koblitz in the class number one setting. Our heuristics are based upon a recent conjecture by R. Gross and J. Smith on numbers of twin primes in algebraic number fields. Our variation precalculates class polynomials as a separate offline process. Unlike the standard approach, which begins with a prime p and searches for an appropriate discriminant D, we choose a discriminant and then search for appropriate primes. Our online process is quick and can be compactly coded. In practice, elliptic curves with near prime order are used. Thus, our timing estimates and data can be regarded as upper estimates for practical purposes. 1
A question of Stark
 Pacific J. of Math
, 1997
"... One of the programs of Stark’s conjectures is to find as many connections as possible between the values that Artin L–functions or their derivatives take (especially at s =0) and arithmetic information associated to algebraic number fields. The most refined of Stark’s conjectures involves the values ..."
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Cited by 3 (0 self)
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One of the programs of Stark’s conjectures is to find as many connections as possible between the values that Artin L–functions or their derivatives take (especially at s =0) and arithmetic information associated to algebraic number fields. The most refined of Stark’s conjectures involves the values of first derivatives of L–functions at s =0.It was recognized early on that the conjecture should be extended to cover cases where the order of vanishing of the L–functions at s =0is greater than one. In 1980, Stark posed a question along these lines that we will consider in detail here. In particular, we will study his question for relative quadratic extensions and prove that an affirmative answer to his question exists for all cases considered. 1. Introduction. Our aim in this section is to state Stark’s question and see how it is related
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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Cited by 3 (2 self)
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.