Generic-case complexity, decision problems in group theory, and random walks

Kapovich, Ilya; Myasnikov, Alexei; Schupp, Paul E.; Shpilrain, Vladimir

doi:10.1016/s0021-8693(03)00167-4

Cited by 170 publications

(211 citation statements)

References 39 publications

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“…Informally, generic case complexity is the complexity on "almost all" of the inputs, see [7,8]. We prove that the claim is not true and, in fact, the conjugacy search problem is trivial in most of the cases (see Theorem 3).…”

Section: )mentioning

confidence: 90%

Cryptanalysis of matrix conjugation schemes

Myasnikov

Ushakov

2014

Journal of Mathematical Cryptology

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Abstract. In this paper we cryptanalyze two protocols: GrigorievShpilrain authentication protocol and Wang et al. public key encryption protocols that use computational hardness of some variations of the conjugacy search problem in noncommutative monoids. We devise a practical heuristic algorithm solving those problems. As a conclusion we claim that these protocols are insecure for the proposed parameter values.

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Section: )mentioning

confidence: 90%

Cryptanalysis of matrix conjugation schemes

Myasnikov

Ushakov

2014

Journal of Mathematical Cryptology

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“…We give a counterexample in Example 15. Still, the mistake did not affect the correctness of the proof in [23] as only a special case of the statement is used.…”

Section: Definition 7 a Graph Is Called Non-amenable If It Satisfiesmentioning

confidence: 99%

“…38], see [23,Prop 6.2]. However, the proposition does not hold in full generality as stated in [5], see Example 15 below.…”

Section: Definition 7 a Graph Is Called Non-amenable If It Satisfiesmentioning

confidence: 99%

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

Diekert

Myasnikov

Wei

2017

Journal of Symbolic Computation

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In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. Thus, the question arises "how many" such elements there are. This question can be formalized using the notion of strongly generic sets and lower bounds can be proven by applying the theory of amenable graphs:In this work we examine Schreier graphs of amalgamated products and HNN extensions. As application of these characterizations we show that the conjugacy problem in fundamental groups of finite graphs of groups with finitely generated free abelian vertex groups can be solved in polynomial time on a strongly generic set. Furthermore, the conjugacy problem in groups with more than one end can be solved with a strongly generic algorithm which has essentially the same time complexity as the word problem. These are rather striking results as the word problem might be easy, but the conjugacy problem might be even undecidable. Finally, our results yield a new proof that the set where the conjugacy problem of the Baumslag group G 1,2 is decidable in polynomial time is also strongly generic.

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“…В работе [1] развита теория генерической сложности вычислений. В рамках этого подхода алгоритмическая проблема рассматривается не на всем множестве входов, а на некотором подмножестве почти всех входов.…”

Section: а н рыбаловunclassified

On generic complexity of the quadratic resi-duosity problem

Rybalov¹

2015

Applied Discrete Mathematics. Supplement

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Generic-case complexity, decision problems in group theory, and random walks

Cited by 170 publications

References 39 publications

Cryptanalysis of matrix conjugation schemes

Cryptanalysis of matrix conjugation schemes

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

On generic complexity of the quadratic resi-duosity problem

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