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18
Fast management of permutation groups I
, 1997
"... We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play ..."
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Cited by 21 (3 self)
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We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worstcase analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.
New Approaches to Designing Public Key Cryptosystems Using OneWay Functions and TrapDoors in Finite Groups
 Journal of Cryptology
"... A symmetric key cryptosystem based on logarithmic signatures for nite permutation groups was described by the rst author in [6], and its algebraic properties were studied in [7]. In this paper we describe two possible approaches to the construction of new public key cryptosystems with message spa ..."
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Cited by 18 (1 self)
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A symmetric key cryptosystem based on logarithmic signatures for nite permutation groups was described by the rst author in [6], and its algebraic properties were studied in [7]. In this paper we describe two possible approaches to the construction of new public key cryptosystems with message space a large nite group G, using logarithmic signatures and their generalizations. The rst approach relies on the fact that permutations of the message space G induced by transversal logarithmic signatures almost always generate the full symmetric group SG on the message space. The second approach could potentially lead to new ElGamal  like systems based on trapdoor, oneway functions induced Research supported in part by National Science Foundation grant CCR9610138 y Research supported in part by NSERC grants IRC #21643196 and RGPIN # 20311498. 1 by logarithmic signaturelike objects we call meshes, which are uniform covers for G. Key words. Trapdoor oneway functions...
Computing in quotient groups
 Proceedings of the 22nd ACM Symposium on Theory of Computing
, 1990
"... We present polynomialtime algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomialtime for permutation groups. Since it is not computationally feasible to represent G/K itself as a permu ..."
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Cited by 17 (6 self)
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We present polynomialtime algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomialtime for permutation groups. Since it is not computationally feasible to represent G/K itself as a permutation group, the methodology for the quotientgroup versions of such problems frequently differ markedly from the procedures that have been observed for the K = 1 subcases. Whereas the algorithms for permutation groups may have rested on elementary notions, procedures underlying the extension to quotient groups often utilize deep knowledge of the structure of the group. In some instances, we present algorithms for problems that were not previously known to be in polynomial time, even for permutation groups themselves (K = 1). These problems apparently required access to quotients. 1.
Simple Groups in Computational Group Theory
 INTERNATIONAL CONGRESS OF MATHEMATICANS
, 1998
"... This note describes recent research using structural properties of finite groups to devise efficient algorithms for group computation. ..."
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Cited by 12 (2 self)
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This note describes recent research using structural properties of finite groups to devise efficient algorithms for group computation.
Algorithms for Matrix Groups and the Tits Alternative
 Proc. 36th IEEE FOCS
, 1999
"... l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals an ..."
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Cited by 11 (2 self)
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l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 00220000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences
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...
Finding Sylow normalizers in polynomial time
 JOURNAL OF ALGORITHMS
, 1990
"... Given a set r of permutations of an nset, let G be the group of permutations generated by r. If p is any prime, it is known that a Sylow psubgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow psubgrou ..."
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Cited by 7 (4 self)
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Given a set r of permutations of an nset, let G be the group of permutations generated by r. If p is any prime, it is known that a Sylow psubgroup P of G can be found in polynomial time. We show that the normalizer of P can also be found in polynomial time. In particular, given two Sylow psubgroups of G, all elements conjugating one to the other can be found (as a coset of the normalizer of one of the Sylow psubgroups). Analogous results are obtained in the case of Hall subgroups of solvable groups.
PolynomialTime Versions of Sylow's Theorem
 JOURNAL OF ALGORITHMS
, 1988
"... Let G be a subgroup of S,, given in terms of a generating set of permutations, and let p be a prime divisor of 1 G 1. If G is solvableand, more generally, if the nonabelian composition factors of G are suitably restrictedit is shown that the following can be found in polynomial time: a Sylow psub ..."
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Cited by 7 (3 self)
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Let G be a subgroup of S,, given in terms of a generating set of permutations, and let p be a prime divisor of 1 G 1. If G is solvableand, more generally, if the nonabelian composition factors of G are suitably restrictedit is shown that the following can be found in polynomial time: a Sylow psubgroup of G containing a given psubgroup, and an element of G conjugating a given Sylow psubgroup to another. Similar results are proved for Hall subgroups of solvable groups and a version of the SchurZassenhaus theorem is obtained.
MemoryBased and DiskBased Algorithms for Very High Degree Permutation Groups
 In Proc. of International Symposium on Symbolic and Algebraic Computation (ISSAC ’03
, 2003
"... Group membership is a fundamental algorithm, upon which most other algorithms of computational group theory depend. Until now, group membership for permutation groups has been limited to ten million points or less. We extend the applicability of group membership algorithms to permutation groups acti ..."
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Cited by 5 (4 self)
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Group membership is a fundamental algorithm, upon which most other algorithms of computational group theory depend. Until now, group membership for permutation groups has been limited to ten million points or less. We extend the applicability of group membership algorithms to permutation groups acting on more than 100,000,000 points. As an example, we experimentally construct a group membership data structure for Thompson's group, acting on 143,127,000 points, in 36 minutes. More significantly, we require approximately 10 GB of RAM for the computation  even though a single permutation of Thompson's group already requires half a gigabyte of storage.