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- Vladimir Shpilrain, Gabriel Zapata
- Applicable Algebra in Engineering, Communication…
- 2004

After some excitement generated by recently suggested public key exchange protocols due to Anshel–Anshel–Goldfeld and Ko–Lee et al., it is a prevalent opinion now that the conjugacy search problem is unlikely to provide sufficient level of security if a braid group is used as the platform. In this paper we address the following questions: (1) whether… (More)

- Vladimir Shpilrain, Alexander Ushakov
- ACNS
- 2005

Recently, several public key exchange protocols based on symbolic computation in non-commutative (semi)groups were proposed as a more efficient alternative to well established protocols based on numeric computation. Notably, the protocols due to Anshel-Anshel-Goldfeld and Ko-Lee et al. exploited the conjugacy search problem in groups, which is a… (More)

- Vladimir Shpilrain, Alexander Ushakov
- IACR Cryptology ePrint Archive
- 2005

In this paper we present a new key establishment protocol based on the decomposition problem in non-commutative groups which is: given two elements w, w1 of the platform group G and two subgroups A, B ⊆ G (not necessarily distinct), find elements a ∈ A, b ∈ B such that w1 = awb. Here we introduce two new ideas that improve the security of key establishment… (More)

- Vladimir Shpilrain
- IACR Cryptology ePrint Archive
- 2003

One of the possible generalizations of the discrete logarithm problem to arbitrary groups is the so-called conjugacy search problem (sometimes erroneously called just the conjugacy problem) : given two elements a, b of a group G and the information that ax = b for some x ∈ G, find at least one particular element x like that. Here ax stands for xax. The… (More)

We prove that Whitehead’s algorithm for solving the automorphism problem in a fixed free group Fk has strongly linear time generic-case complexity. This is done by showing that the “hard” part of the algorithm terminates in linear time on an exponentially generic set of input pairs. We then apply these results to one-relator groups. We obtain a Mostow-type… (More)

We are now witnessing a rapid growth of a new part of group theory which has become known as “statistical group theory”. A typical result in this area would say something like “a random element (or a tuple of elements) of a group G has a property P with probability p”. The validity of a statement like that does, of course, heavily depend on how one defines… (More)

- Vladimir Shpilrain
- CSR
- 2008

We offer cryptanalysis of a key exchange scheme due to Stickel [11], which was inspired by the well-known Diffie-Hellman protocol. We show that Stickel’s choice of platform (the group of invertible matrices over a finite field) makes the scheme vulnerable to linear algebra attacks with very high success rate in recovering the shared secret key (100% in our… (More)

- Dima Grigoriev, Vladimir Shpilrain
- Groups Complexity Cryptology
- 2009

We propose an authentication scheme where forgery (a.k.a. impersonation) seems infeasible without finding the prover’s long-term private key. The latter would follow from solving the conjugacy search problem in the platform (noncommutative) semigroup, i.e., to recovering X from XAX and A. The platform semigroup that we suggest here is the semigroup of n×n… (More)

- Vladimir Shpilrain, Alexander Ushakov
- ACNS
- 2008

The conjugacy search problem in a group G is the problem of recovering an x ∈ G from given g ∈ G and h = xgx. The alleged computational hardness of this problem in some groups was used in several recently suggested public key exchange protocols, including the one due to Anshel, Anshel, and Goldfeld, and the one due to Ko, Lee et al. Sibert, Dehornoy, and… (More)

This is a collection of open problems in combinatorial group theory, which is based on a similar list available on-line at our web site http://zebra.sci.ccny.cuny.edu/web/ In selecting the problems, our choices have been, in part, determined by our own tastes. In view of this, we have welcomed suggestions from other members of the community. We want to… (More)