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78
Fading Channels: InformationTheoretic And Communications Aspects
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1998
"... In this paper we review the most peculiar and interesting informationtheoretic and communications features of fading channels. We first describe the statistical models of fading channels which are frequently used in the analysis and design of communication systems. Next, we focus on the information ..."
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Cited by 289 (1 self)
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In this paper we review the most peculiar and interesting informationtheoretic and communications features of fading channels. We first describe the statistical models of fading channels which are frequently used in the analysis and design of communication systems. Next, we focus on the information theory of fading channels, by emphasizing capacity as the most important performance measure. Both singleuser and multiuser transmission are examined. Further, we describe how the structure of fading channels impacts code design, and finally overview equalization of fading multipath channels.
A Subexponential Algorithm for the Determination of Class Groups and Regulators of Algebraic Number Fields
, 1990
"... A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( p log D log log D) c+o(1) where D is the absolute discriminant of F , wh ..."
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Cited by 51 (5 self)
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A new probabilistic algorithm for the determination of class groups and regulators of an algebraic number field F is presented. Heuristic evidence is given which shows that the expected running time of the algorithm is exp( p log D log log D) c+o(1) where D is the absolute discriminant of F , where c 2 R?0 is an absolute constant, and where the o(1)function depends on the degree of F . 1 Introduction Computing the class group and the regulator of an algebraic number field F are two major tasks of algorithmic algebraic number theory. In the last decade, several regulator and class group algorithms have been suggested (e.g. [16],[17],[18],[3]). In [2] the problem of the computational complexity of those algorithms was adressed for the first time. This question was then studied in [2] in great detail. The theoretical results and the computational experience show that computing class groups and regulators is a very difficult problem. More precisely, it turns out that even under the a...
Detecting Perfect Powers In Essentially Linear Time
 Math. Comp
, 1998
"... This paper (1) gives complete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n>1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton's theorem on multiple linear forms in logarithms, th ..."
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Cited by 41 (12 self)
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This paper (1) gives complete details of an algorithm to compute approximate kth roots; (2) uses this in an algorithm that, given an integer n>1, either writes n as a perfect power or proves that n is not a perfect power; (3) proves, using Loxton's theorem on multiple linear forms in logarithms, that this perfectpower decomposition algorithm runs in time (log n) . 1.
Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 40 (3 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Existence of primitive divisors of Lucas and Lehmer numbers
 J. Reine Angew. Math
, 2001
"... We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay. ..."
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Cited by 35 (0 self)
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We prove that for n ? 30, every nth Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor. Whether the mathematicians like it or not, the computer is here to stay.
Extended gcd and Hermite normal form algorithms via lattice basis reduction
 Experimental Mathematics
, 1998
"... Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small ..."
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Cited by 32 (6 self)
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Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small integer multipliers x1,..., xm for the equation d = gcd (d1,..., dm) = x1d1 + · · · + xmdm, where d1,..., dm are given integers. The method generalises to produce small unimodular transformation matrices for computing the Hermite normal form of an integer matrix. 1
Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Cited by 29 (2 self)
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
Computing RiemannRoch spaces in algebraic function fields and related topics
, 2001
"... this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any ..."
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Cited by 21 (0 self)
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this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any "computable" constant field k of any characteristic as long as the required integral closures can be computed, and does not involve constant field extensions
A relative van Hoeij algorithm over number fields
 J. Symbolic Computation
, 2004
"... Abstract. Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as BerlekampZassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatl ..."
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Cited by 19 (1 self)
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Abstract. Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as BerlekampZassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall performance. We then generalize these ideas to factor polynomials over
Factoring into Coprimes in Essentially Linear Time
"... . Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduc ..."
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Cited by 16 (2 self)
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. Let S be a nite set of positive integers. A \coprime base for S" means a set P of positive integers such that (1) each element of P is coprime to every other element of P and (2) each element of S is a product of powers of elements of P . There is a natural coprime base for S. This paper introduces an algorithm that computes the natural coprime base for S in essentially linear time. The best previous result was a quadratictime algorithm of Bach, Driscoll, and Shallit. This paper also shows how to factor S into elements of P in essentially linear time. The algorithms apply to any free commutative monoid with fast algorithms for multiplication, division, and greatest common divisors; e.g., monic polynomials over a eld. They can be used as a substitute for prime factorization in many applications. 1.