Results 1  10
of
25
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 47 (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.”
Prime power graphs for groups of Lie type
 JOURNAL OF ALGEBRA
, 2002
"... We associate a weighted graph (G) to each nite simple group G of Lie type. We show that, with an explicit list of exceptions, (G) determines G up to isomorphism, and for these exceptions, (G) nevertheless determines the characteristic of G. This result was motivated by algorithmic considerations. ..."
Abstract

Cited by 15 (6 self)
 Add to MetaCart
We associate a weighted graph (G) to each nite simple group G of Lie type. We show that, with an explicit list of exceptions, (G) determines G up to isomorphism, and for these exceptions, (G) nevertheless determines the characteristic of G. This result was motivated by algorithmic considerations. We prove that for any nite simple group G of Lie type, input as a black box group with an oracle to compute the orders of group elements, (G) and the characteristic of G can be computed by a Monte Carlo algorithm in time polynomial in the input length. The characteristic is needed as part of the input in a previous constructive recognition algorithm for G.
On sampling generating sets of finite groups and product replacement algorithm. (Extended Abstract)
 Proceedings of ISSAC'99, 9196
, 1999
"... ) IGOR PAK , SERGEY BRATUS y 1 Introduction Let G be a finite group. A sequence of k group elements (g1 ; : : : ; gk ) is called a generating ktuple of G if the elements generate G (we write hg1 ; : : : ; gk i = G). Let Nk (G) be the set of all generating ktuples of G, and let Nk (G) = jNk ( ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
) IGOR PAK , SERGEY BRATUS y 1 Introduction Let G be a finite group. A sequence of k group elements (g1 ; : : : ; gk ) is called a generating ktuple of G if the elements generate G (we write hg1 ; : : : ; gk i = G). Let Nk (G) be the set of all generating ktuples of G, and let Nk (G) = jNk (G)j. We consider two related problems on generating ktuples. Given G and k ? 0, 1) Determine Nk (G) 2) Generate random element of Nk (G), each with probability 1=Nk (G) The problem of determining the structure of Nk (G) is of interest in several contexts. The counting problem goes back to Philip Hall, who expressed Nk (G) as a Mobius type summation of Nk (H) over all maximal subgroups H ae G (see [23]). Recently the counting problem has been studied for large simple groups where remarkable progress has been made (see [25, 27]). In this paper we analyze Nk for solvable groups and products of simple groups. The sampling problem, while often used in theory as a tool for approximate counting...
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
Fast constructive recognition of black box unitary groups
 LMS J. Comput. Math
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
A blackbox group algorithm for recognizing finite symmetric and alternating
 Trans. Amer. Math. Soc
, 2003
"... Abstract. We present a Las Vegas algorithm which, for a given blackbox group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We present a Las Vegas algorithm which, for a given blackbox group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree n of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of n is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of Sn: the conditional probability that a random element σ ∈ Sn is an ncycle, given that σ n =1,is at least 1/10. 1.
Algorithms for matrix groups
 London Math. Soc. Lecture Note Ser
, 2011
"... Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G. 1 ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G. 1
Computing the maximal subgroups of a permutation group I
, 2001
"... We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a "hybrid group" approach ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a "hybrid group" approach
Computing with Matrix Groups
 GROUPS, COMBINATORICS AND GEOMETRY
, 2001
"... A group is usually input into a computer by specifying the group either using a presentation or using a generating set of permutations or matrices. Here we will emphasize the latter approach, referring to [Si3, Si4, Ser1] for details of the other situations. Thus, the basic computational setting dis ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
A group is usually input into a computer by specifying the group either using a presentation or using a generating set of permutations or matrices. Here we will emphasize the latter approach, referring to [Si3, Si4, Ser1] for details of the other situations. Thus, the basic computational setting discussed here is as follows: a group is given, speciﬁed as G = X in terms of some generating set X of its elements, where X is an arbitrary subset of either Sn or GL(d, q ) (a familiar example is the group of Rubik’s cube). The goal is then to ﬁnd properties of G eﬃciently, such as G, the derived series, a composition series, Sylow subgroups, and so on.
The MD6 hash function A proposal to NIST for SHA3
, 2008
"... This report describes and analyzes the MD6 hash function and is part of our submission package for MD6 as an entry in the NIST SHA3 hash function competition 1. Significant features of MD6 include: • Accepts input messages of any length up to 2 64 − 1 bits, and produces message digests of any desir ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
This report describes and analyzes the MD6 hash function and is part of our submission package for MD6 as an entry in the NIST SHA3 hash function competition 1. Significant features of MD6 include: • Accepts input messages of any length up to 2 64 − 1 bits, and produces message digests of any desired size from 1 to 512 bits, inclusive, including