Results 1  10
of
40
Generating random elements of a finite group
 Comm. Algebra
, 1995
"... We present a “practical ” algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are accep ..."
Abstract

Cited by 70 (12 self)
 Add to MetaCart
We present a “practical ” algorithm to construct random elements of a finite group. We analyse its theoretical behaviour and prove that asymptotically it produces uniformly distributed tuples of elements. We discuss tests to assess its effectiveness and use these to decide when its results are acceptable for some matrix groups. 1 1
A polynomialtime theory of blackbox groups I
, 1998
"... We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic o ..."
Abstract

Cited by 41 (6 self)
 Add to MetaCart
We consider the asymptotic complexity of algorithms to manipulate matrix groups over finite fields. Groups are given by a list of generators. Some of the rudimentary tasks such as membership testing and computing the order are not expected to admit polynomialtime solutions due to number theoretic obstacles such as factoring integers and discrete logarithm. While these and other “abelian obstacles ” persist, we demonstrate that the “nonabelian normal structure ” of matrix groups over finite fields can be mapped out in great detail by polynomialtime randomized (Monte Carlo) algorithms. The methods are based on statistical results on finite simple groups. We indicate the elements of a project under way towards a more complete “recognition” of such groups in polynomial time. In particular, under a now plausible hypothesis, we are able to determine the names of all nonabelian composition factors of a matrix group over a finite field. Our context is actually far more general than matrix groups: most of the algorithms work for “blackbox groups ” under minimal assumptions. In a blackbox group, the group elements are encoded by strings of uniform length, and the group operations are performed by a “black box.”
What Do We Know About The Product Replacement Algorithm?
 in: Groups ann Computation III
, 2000
"... . The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an exten ..."
Abstract

Cited by 33 (8 self)
 Add to MetaCart
. The product replacement algorithm is a commonly used heuristic to generate random group elements in a finite group G, by running a random walk on generating ktuples of G. While experiments showed outstanding performance, until recently there was little theoretical explanation. We give an extensive review of both positive and negative theoretical results in the analysis of the algorithm. Introduction In the past few decades the study of groups by means of computations has become a wonderful success story. The whole new field, Computational Group Theory, was developed out of needs to discover and prove new results on finite groups. More recently, the probabilistic method became an important tool for creating faster and better algorithms. A number of applications were developed which assume a fast access to (nearly) uniform group elements. This led to a development of the so called "product replacement algorithm", which is a commonly used heuristic to generate random group elemen...
A Recognition Algorithm for Classical Groups over Finite Fields
 Proc. London Math. Soc
, 1998
"... 2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126 ..."
Abstract

Cited by 31 (0 self)
 Add to MetaCart
2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126
Walks on Generating Sets of Groups
, 1996
"... We study a Markov chain on generating ntuples of a fixed group which arises in algorithms for manipulating finite groups. The main tools are comparison of two Markov chains on different but related state spaces and combinatorics of random paths. The results involve group theoretical parameters such ..."
Abstract

Cited by 22 (0 self)
 Add to MetaCart
We study a Markov chain on generating ntuples of a fixed group which arises in algorithms for manipulating finite groups. The main tools are comparison of two Markov chains on different but related state spaces and combinatorics of random paths. The results involve group theoretical parameters such as the size of minimal generating sets, the number of distinct generating ktuples for different k's and the maximal diameter of the group.
The Product Replacement Algorithm is Polynomial
 In Proc. 41 st IEEE Symposium on Foundations of Computer Science (FOCS
, 2000
"... The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating ktuples of the group, and then output a random component. The algorithm was designed by LeedhamGreen and Soicher ([31]), and further investigated in [12]. It ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
The product replacement algorithm is a heuristic designed to generate random group elements. The idea is to run a random walk on generating ktuples of the group, and then output a random component. The algorithm was designed by LeedhamGreen and Soicher ([31]), and further investigated in [12]. It was found to have an outstanding performance, much better than the the previously known algorithms (see [12, 22, 26]). The algorithm is now included in two major group algebra packages GAP [42] and MAGMA [10]. In spite of the many serious attempts and partial results, (see [6, 14, 15, 21, 22, 32, 39, 40]), the analysis of the algorithm remains difficult at best. For small values of k even graph connectivity becomes a serious obstacle (see [19, 37, 39, 40]). The most general results are due to Diaconis and SaloffCoste [22], who used a state of the art analytic technique to obtain polynomial bounds in special cases, and (sub)exponential bounds in general case. The main result of this pape...
Random matrix theory over finite fields
 Bull. Amer. Math. Soc. (N.S
"... 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 sym ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
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, RogersRamanujan type identities, potential theory, and various measures on partitions.