Results 1 
2 of
2
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 ..."
Abstract

Cited by 42 (3 self)
 Add to MetaCart
(Show Context)
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.
Uses of Randomness in Computation
, 1994
"... Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worst ..."
Abstract
 Add to MetaCart
Random number generators are widely used in practical algorithms. Examples include simulation, number theory (primality testing and integer factorization), fault tolerance, routing, cryptography, optimization by simulated annealing, and perfect hashing. Complexity theory usually considers the worstcase behaviour of deterministic algorithms, but it can also consider averagecase behaviour if it is assumed that the input data is drawn randomly from a given distribution. Rabin popularised the idea of &quot;probabilistic &quot; algorithms, where randomness is incorporated into the algorithm instead of being assumed in the input data. Yao showed that there is a close connection between the complexity of probabilistic algorithms and the averagecase complexity of deterministic algorithms. We give examples of the uses of randomness in computation, discuss the contributions of Rabin, Yao and others, and mention some open questions.