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14
Genericcase complexity and decision problems in group theory, preprint
, 2003
"... Abstract. We give a precise definition of “genericcase complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory the word, conjugacy and membership problems all have lineartime genericcase complexity. We prove such theorems by ..."
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Cited by 69 (26 self)
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Abstract. We give a precise definition of “genericcase complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory the word, conjugacy and membership problems all have lineartime genericcase complexity. We prove such theorems by using the theory of random walks on regular graphs. Contents 1. Motivation
Divisibility Theory and Complexity of Algorithms for Free Partially Commutative
 Groups, Contemp. Math. 378 (Groups, Languages, Algorithms
, 2005
"... Nota Bene. The original version of the paper was published ..."
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Cited by 19 (12 self)
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Nota Bene. The original version of the paper was published
Random subgroups and analysis of the lengthbased and quotient attacks
 Journal of Mathematical Cryptology
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The nonamenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups
, 2002
"... Abstract. We show that if H is a quasiconvex subgroup of infinite index in a nonelementary hyperbolic group G then the Schreier coset graph for G relative to H is nonamenable. 1. ..."
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Cited by 7 (2 self)
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Abstract. We show that if H is a quasiconvex subgroup of infinite index in a nonelementary hyperbolic group G then the Schreier coset graph for G relative to H is nonamenable. 1.
Quotient tests and random walks in computational group theory
, 2005
"... For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few sim ..."
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Cited by 3 (1 self)
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For many decision problems on a finitely presented group G, we can quickly weed out negative solutions by using much quicker algorithms on an appropriately chosen quotient group G/K of G. However, the behavior of such “quotient tests” can be sometimes paradoxical. In this paper, we analyze a few simple case studies of quotient tests for the classical identity, word, conjugacy problems in groups. We attempt to combine a rigorous analytic study with the assessment of algorithms from the practical point of view. It appears that, in case of finite quotient groups G/K, the efficiency of the quotient test very much depends on the mixing times for random walks on the Cayley graph of G/K.
Rightinvariance: A property for probabilistic analysis of cryptography based on infinite groups
 In ASIACRYPT
, 2004
"... Abstract. Infinite groups have been used for cryptography since about twenty years ago. However, it has not been so fruitful as using finite groups. An important reason seems the lack of research on building a solid mathematical foundation for the use of infinite groups in cryptography. As a first s ..."
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Cited by 2 (0 self)
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Abstract. Infinite groups have been used for cryptography since about twenty years ago. However, it has not been so fruitful as using finite groups. An important reason seems the lack of research on building a solid mathematical foundation for the use of infinite groups in cryptography. As a first step for this line of research, this paper pays attention to a property, the socalled rightinvariance, which makes finite groups so convenient in cryptography, and gives a mathematical framework for correct, appropriate use of it in infinite groups. 1
A PROPERTY FOR CRYPTOGRAPHY BASED ON INFINITE GROUPS
"... Abstract. Cryptography using infinite groups has been studied since about twenty years ago. However, it has not been so fruitful as using finite groups. An important reason is the absence of research on probability in this area. Indeed, a number of cryptographic tools concerning probability are play ..."
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Abstract. Cryptography using infinite groups has been studied since about twenty years ago. However, it has not been so fruitful as using finite groups. An important reason is the absence of research on probability in this area. Indeed, a number of cryptographic tools concerning probability are playing significant roles in analyses in the case of finite groups. Our purpose is twofold—to deal with not a particular finite subset (as before) of an infinite group but the whole group itself, and to make cryptographic tools developed in finite groups still useful in infinite groups. As a first step to serve this purpose, we study a probabilitytheoretic property, the socalled rightinvariance, that has been widely used in cryptography. Like the uniform distribution over finite sets, rightinvariance property simplifies many complex situations. However, it can be unused or misused since it is not known when this property can be used. We propose a method of deciding whether or not we can use this property in a given situation, and prove that there is no rightinvariant probability distribution on most infinite groups which can be universally used. Therefore, we discuss weaker, yet practical alternatives with concrete examples. 1.
Contents
"... Abstract. We discuss the time complexity of the word and conjugacy problems for free products G = A ⋆C B of groups A and B with amalgamation over a subgroup C. We stratify the set of elements of G with respect to the complexity of the word and conjugacy problems and show that for the generic stratum ..."
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Abstract. We discuss the time complexity of the word and conjugacy problems for free products G = A ⋆C B of groups A and B with amalgamation over a subgroup C. We stratify the set of elements of G with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable assumptions about groups A, B, C. Moreover, the decision algorithm is fast on the generic stratum.