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15
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 ..."
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Cited by 40 (6 self)
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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. ..."
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Cited by 10 (6 self)
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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.
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
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 ( ..."
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Cited by 8 (8 self)
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) 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...
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 ..."
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Cited by 5 (3 self)
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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.
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 ..."
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Cited by 5 (2 self)
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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 ..."
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Cited by 4 (4 self)
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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 ..."
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Cited by 3 (1 self)
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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
Generalised sifting in blackbox groups
 London Math. Soc. J. Comput. Math
"... Abstract. We present a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. Our procedure is a Monte Carlo algorithm, and i ..."
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Cited by 1 (0 self)
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Abstract. We present a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. Our procedure is a Monte Carlo algorithm, and is presented and analysed in the context of blackbox groups. It is based on a chain of subsets instead of a subgroup chain. Two general versions of the procedure are worked out in detail, and applications are given for membership tests for several of the sporadic simple groups. Our major objective was that the procedures could be proved to be Monte Carlo algorithms, and their costs computed. In addition we explicitly determined suitable subset chains for six of the sporadic groups, and we implemented the algorithms involving these chains in the GAP computational algebra system. It turns out that sample implementations perform well in practice. The implementations will be made available publicly in the form of a GAP package. 1.
An Atlas of characteristic zero representations
, 2005
"... Motivated by the World Wide Web Atlas of Finite Group Representations and the recent classification of lowdimensional representations of quasisimple groups in crosscharacteristic fields by Hiss and Malle, we construct with a computer over 650 representations of finite simple groups. Explicit mat ..."
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Motivated by the World Wide Web Atlas of Finite Group Representations and the recent classification of lowdimensional representations of quasisimple groups in crosscharacteristic fields by Hiss and Malle, we construct with a computer over 650 representations of finite simple groups. Explicit matrices for these representations are available on the Internet and are included on an attached CDROM. Our main tool is a GAP program for decomposing permutation modules. It uses reduction modulo various primes and rational reconstruction to give an acceptable performance. In addition, we define standard generators for the groups under consideration, and exhibit black box algorithms for finding standard generators and checking whether given elements of the group are standard generators. To my parents Acknowledgements I have benefited greatly from the guidance and support of my supervisor, Professor Robert Wilson. I wish to thank him for his encouragement and enthusiasm in this project. I feel privileged to have been one of his students. I am indebted to my examiners Professor Derek Holt and Dr Paul Flavell for their detailed reading of the text and for pointing out several improvements. I thank Dr John Bray who has very helpfully shared his knowledge of computational group theory with me. He has also provided two interesting representations for inclusion here. I also thank Richard Barraclough for helping me with the Monster group programs and Dr Frank Lübeck for helping me with my questions about CHEVIE. My work has been made greatly easier by the GAP computer algebra system. I thank all the developers for their hard work in producing such a marvellous tool and making it freely available. I am very grateful to Sophie Whyte, Elizabeth Wharton and Marijke van Gans, with whom I have shared an office for the past three years. I wish to thank them for many useful conversations, for helping with the crossword, and for patiently putting up with my annoying habits. I also thank the School of Mathematics at the University of Birmingham for providing a stimulating environment for research.