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**11 - 16**of**16**### The Conjugacy Problem in Amalgamated Products I: Regular Elements and Black Holes

, 2009

"... We discuss the time complexity of the word and conjugacy search 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 ..."

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We discuss the time complexity of the word and conjugacy search 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.

### Generic complexity of . . .

, 2009

"... We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller’s groups. We show that for “almost all”, in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may h ..."

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We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller’s groups. We show that for “almost all”, in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may have be undecidable. Our results show that “hard” instances of the problem comprise a negligibly small part of the group.

### 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.

### Length-Based Cryptanalysis: The Case Of

, 2006

"... The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make them applicable to Thompson's group with significan ..."

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The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make them applicable to Thompson's group with significant success rates. In particular, this shows that the Shpilrain-Ushakov public key cryptosystem based on Thompson's group is insecure, and suggests that no practical public key cryptosystem based on this group can be secure.