A practical method is described for deciding whether or not a finitedimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalisation of the Parker-Norton `Meataxe' algorithm, but it does not depend for its efficiency on the field being small. The principal tools involved are the calculation of the nullspace and the characteristic polynomial of a matrix over a finite field, and the factorisation of the latter. Related algorithms to determine absolute irreducibility and module isomorphism for irreducibles are also described. Details of an implementation in the GAP system, together with some performance analyses are included. 1991 Mathematics subject classification (Amer. Math. Soc.): 20C40, 20-04. 1 Introduction The purpose of this paper is to describe a practical method for deciding whether or not a finite dimensional FG-module M is irreducible, where F = GF (q) is a finite field and G is a fi...

The rapid expansion of the Internet means that users increasingly want to interact with each other. Due to the openness and unsecure nature of the net, users often have to rely on firewalls to protect their connections. Firewalls, however, make real-time interaction and collaboration more difficult. Firewalls are also complicated to configure and expensive to install and maintain, and are inaccessible to small home offices and mobile users. The Enclaves approach is to transform user machines into "enclaves," which are protected from outside interference and attacks. Using Enclaves, a group of collaborators can dynamically form a secure virtual subnet within which to conduct their joint business. This paper describes the design and implementation of the Enclaves toolkit, and some applications we have built using the toolkit. 1 Motivation Most user interaction and collaboration over the Internet have been primarily via electronic mail. More recently, groupware applications including tel...

Neubiiser asked for an efficient algorithm to decide whether the subgroup of the general linear group GL(d, q) generated by a given set X of non-singular d X d matrices over a finite field ¥q contains the special linear group SL(rf, q).

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We present real, complex, and quaternionic versions of a simple randomized polynomial time algorithm to approximate the permanent of a non-negative matrix and, more generally, the mixed discriminant of positive semidefinite matrices. The algorithm provides an unbiased estimator, which, with high probability, approximates the true value within a factor of O(c n ), where n is the size of the matrix (matrices) and where c 0:28 for the real version, c 0:56 for the complex version and c 0:76 for the quaternionic version. We discuss possible extensions of our method as well as applications of mixed discriminants to problems of combinatorial counting.

The recursive method formalized by Nijenhuis and Wilf [15] and systematized by Flajolet, Van Cutsem and Zimmermann [8], is extended here to floating-point arithmetic. The resulting ADZ method enables one to generate decomposable data structures -- both labelled or unlabelled -- uniformly at random, in expected O(n 1+ffl ) time and space, after a preprocessing phase of O(n 2+ffl ) time, which reduces to O(n 1+ffl ) for context-free grammars.

Abstract. In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.

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Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, Rogers-Ramanujan type identities, potential theory, and various measures on partitions.

We study how several collective operations like broadcast, reduction, scan, etc. can be composed efficiently in complex parallel programs. Our specific contributions are: (1) a formal framework for reasoning about collective operations; (2) a set of optimization rules which save communications by fusing several collective operations into one; (3) performance estimates, which guide the application of optimization rules depending on the machine characteristics; (4) a simple case study with the first results of machine experiments.