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CRYPTANALYSIS OF STICKEL’S KEY EXCHANGE SCHEME
"... Abstract. We offer cryptanalysis of a key exchange scheme due to Stickel [11], which was inspired by the wellknown DiffieHellman 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 ver ..."
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Abstract. We offer cryptanalysis of a key exchange scheme due to Stickel [11], which was inspired by the wellknown DiffieHellman 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 experiments). We also show that obtaining the shared secret key in Stickel’s scheme is not harder for the adversary than solving the decomposition search problem in the platform (semi)group. 1.
AN AUTHENTICATION SCHEME BASED ON THE TWISTED CONJUGACY PROBLEM
"... Abstract. The conjugacy search problem in a group G is the problem of recovering an x ∈ G from given g ∈ G and h = x −1 gx. 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, ..."
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Abstract. The conjugacy search problem in a group G is the problem of recovering an x ∈ G from given g ∈ G and h = x −1 gx. 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 Girault used this problem in their authentication scheme, which was inspired by the FiatShamir scheme involving repeating several times a threepass challengeresponse step. In this paper, we offer an authentication scheme whose security is based on the apparent hardness of the twisted conjugacy search problem which is: given a pair of endomorphisms (i.e., homomorphisms into itself) ϕ, ψ of a group G and a pair of elements w, t ∈ G, find an element s ∈ G such that t = ψ(s −1)wϕ(s) provided at least one such s exists. This problem appears to be very nontrivial even for free groups. We offer here another platform, namely, the semigroup of all 2 × 2 matrices over truncated onevariable polynomials over F2, the field of two elements, with transposition used instead of inversion in the equality above. 1.
New Developments in Commutator Key Exchange
"... We study the algorithmic security of the AnshelAnshelGoldfeld (AAG) key exchange scheme and show that contrary to prevalent opinion, the computational hardness of AAG depends on the structure of the chosen subgroups, rather than on the conjugacy problem of the ambient braid group. Proper choice of ..."
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We study the algorithmic security of the AnshelAnshelGoldfeld (AAG) key exchange scheme and show that contrary to prevalent opinion, the computational hardness of AAG depends on the structure of the chosen subgroups, rather than on the conjugacy problem of the ambient braid group. Proper choice of these subgroups produces a key exchange scheme which is resistant to all known attacks on AAG.
Groups with two generators having unsolvable word problem and presentations of Mihailova subgroups
 SHENZHEN UNIVERSITY SHENZHEN CITY 518060, CHINA XUCHEN@TOM.COM
, 2014
"... A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F2 × F2 with finite number of generators are given. Where Mihailova subgroup of F2 × F2 enjoys the unsolvable subgroup membership problem. One then can use the ..."
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A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F2 × F2 with finite number of generators are given. Where Mihailova subgroup of F2 × F2 enjoys the unsolvable subgroup membership problem. One then can use the presentation to create entities’ private keys in a public key cryptsystem.
Polynomial time solutions of computational problems in noncommutativealgebraic cryptography
, 2013
"... By introducing extra shields on Shpilrain and Ushakov’s KoLeelike protocol based on the decomposition problem of group elements we propose two new key exchange schemes and then a number of public key cryptographic protocols. We show that these protocols are free of known attacks. Particularly, if ..."
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By introducing extra shields on Shpilrain and Ushakov’s KoLeelike protocol based on the decomposition problem of group elements we propose two new key exchange schemes and then a number of public key cryptographic protocols. We show that these protocols are free of known attacks. Particularly, if the entities taking part in our protocols create their private keys composed by the generators of the Mihailova subgroups of Bn, we show that the safety of our protocols are very highly guarantied by the insolvability of subgroup membership problem of the Mihailova subgroups.
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, 805
"... An authentication scheme based on the twisted conjugacy problem ..."
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Algorithms and . . . PIECEWISELINEAR HOMEOMORPHISMS
, 2008
"... The first part (Chapters 2 through 5) studies decision problems in Thompson’s groups F, T, V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson’s group F and suitable larger groups of piecewiselinear homeomorphisms of the unit interval. We describ ..."
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The first part (Chapters 2 through 5) studies decision problems in Thompson’s groups F, T, V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson’s group F and suitable larger groups of piecewiselinear homeomorphisms of the unit interval. We describe a conjugacy invariant both from the piecewiselinear point of view and a combinatorial one using strand diagrams. We determine algorithms to compute roots and centralizers in these groups and to detect periodic points and their behavior by looking at the closed strand diagram associated to an element. We conclude with a complete cryptanalysis of an encryption protocol based on the decomposition problem. In the second part (Chapters 6 and 7), we describe the structure of subgroups of the group of all homeomorphisms of the unit circle, with the additional requirement that they contain no nonabelian free subgroup. It is shown that in this setting the rotation number map is a group homomorphism. We give a classification of such subgroups as subgroups of certain wreath products and we show that such subgroups can exist by building examples. Similar techniques are then used to compute centralizers in these groups and to provide the base to generalize the techniques of the first part and to solve the simultaneous conjugacy problem.