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Black box exceptional groups of Lie type
 In preparation
, 2002
"... If a black box group is known to be isomorphic to an exceptional simple group of Lie type of rank> 1, other than any 2 F4(q), over a field of known size, a Las Vegas algorithm is used to produce a constructive isomorphism. This yields an upgrade of all known nearly linear time Monte Carlo permuta ..."
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If a black box group is known to be isomorphic to an exceptional simple group of Lie type of rank> 1, other than any 2 F4(q), over a field of known size, a Las Vegas algorithm is used to produce a constructive isomorphism. This yields an upgrade of all known nearly linear time Monte Carlo permutation group algorithms to Las Vegas algorithms when the input group has no composition factor isomorphic to a rank 1 group or to any 2 F4(q). 1
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.
A polynomialtime theory of matrix groups and black box groups
 in these Proceedings
"... We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic proble ..."
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Cited by 2 (0 self)
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We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles. Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log. One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity). As a byproduct, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N, does the group generated by A have a semisimple quotient of order ≥ N? We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.
c ○ TÜB ˙ ITAK Black Box Groups
"... Weproposea uniform approach for recognizing all black box groups of Lietype which is based on theanalysis of thestructureof thecentralizers of involutions. Our approach can be viewed as a computational version of the classification of thefinitesimplegroups. Wepresent an algorithm which constructs a ..."
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Weproposea uniform approach for recognizing all black box groups of Lietype which is based on theanalysis of thestructureof thecentralizers of involutions. Our approach can be viewed as a computational version of the classification of thefinitesimplegroups. Wepresent an algorithm which constructs a long root SL2(q)subgroup in a finitesimplegroup of Lietypeof odd characteristic, then we use the Aschbacher’s “Classical Involution Theorem ” as a model in the recognition algorithm and weconstruct all root SL2(q)subgroups corresponding to the nodes in the extended Dynkin diagram, that is, we construct the extended Curtis PhanTits system of thefinitesimplegroups of Lietypeof odd characteristic. In particular, we construct all subsystem subgroups which can be read from the extended Dynkin diagram. We also present an algorithm which determines whether the pcore(or “unipotent radical”) Op(G) of a black box group G is trivial or not, where G/Op(G) is a finitesimpleclassical group of odd characteristic p, answering a wellknown question of Babai and Shalev. 1.
Efficient recovering of operation tables of black box groups and rings
"... www.math.unizh.ch/aa Abstract—People have been studying the following problem: Given a finite set S with a hidden (black box) binary operation ∗ : S × S → S which might come from a group law, and suppose you have access to an oracle that you can ask for the operation x ∗ y of single pairs (x, y) ∈ ..."
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www.math.unizh.ch/aa Abstract—People have been studying the following problem: Given a finite set S with a hidden (black box) binary operation ∗ : S × S → S which might come from a group law, and suppose you have access to an oracle that you can ask for the operation x ∗ y of single pairs (x, y) ∈ S2 you choose. What is the minimal number of queries to the oracle until the whole binary operation is recovered, i.e. you know x ∗ y for all x, y ∈ S? This problem can trivially be solved by using S2 queries to the oracle, so the question arises under which circumstances you can succeed with a significantly smaller number of queries. In this presentation we give a lower bound on the number of queries needed for general binary operations. On the other hand, we present algorithms solving this problem by using S  queries, provided that ∗ is an abelian group operation. We also investigate black box rings and give lower und upper bounds for the number of queries needed to solve product recovering in this case. I.