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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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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.
Open Problems in Number Theoretic Complexity, II
"... this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new ..."
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Cited by 28 (0 self)
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this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...
The Complete Analysis of a Polynomial Factorization Algorithm Over Finite Fields
, 2001
"... This paper derives basic probabilistic properties of random polynomials over finite fields that are of interest in the study of polynomial factorization algorithms. We show that the main characteristics of random polynomial can be treated systematically by methods of "analytic combinatorics&quo ..."
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This paper derives basic probabilistic properties of random polynomials over finite fields that are of interest in the study of polynomial factorization algorithms. We show that the main characteristics of random polynomial can be treated systematically by methods of "analytic combinatorics" based on the combined use of generating functions and of singularity analysis. Our object of study is the classical factorization chain which is described in Fig. 1 and which, despite its simplicity, does not appear to have been totally analysed so far. In this paper, we provide a complete averagecase analysis.
Factoring Polynomials Over Finite Fields: A Survey
, 2001
"... This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem. ..."
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Cited by 4 (1 self)
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This survey reviews several algorithms for the factorization of univariate polynomials over finite fields. We emphasize the main ideas of the methods and provide an uptodate bibliography of the problem.
On Short Representations of Orders and Number Fields
, 1992
"... We discuss the problem of transforming representations of number fields by generating polynomials and by Qbasis into each other. We prove that for a number field or an order of discriminant \Delta one can determine in polynomial time a representation whose size is polynomially bounded in log j\Delt ..."
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We discuss the problem of transforming representations of number fields by generating polynomials and by Qbasis into each other. We prove that for a number field or an order of discriminant \Delta one can determine in polynomial time a representation whose size is polynomially bounded in log j\Deltaj. 1 Introduction In the literature the time bounds for computing invariants of an algebraic number field K such as the class group or the regulator never depend on the specific representation of the field but only on its discriminant \Delta ( see [1] and [5] ). The algorithms for computing those invariants must therefore receive as input a representation of K whose size is polynomially bounded in log j\Deltaj. In this paper we prove that such a representation can be obtained in polynomial time from any representation of K by a generating polynomial or by means of the multiplication table of a Qbasis. We use the notions of algorithms and complexity in a similar way as in [5]. An algorit...
Deciding Properties of Polynomials without Factoring
, 1997
"... The polynomial time algorithm of Lenstra, Lenstra, and Lovasz [17] for ..."