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19
Thompson’s group and public key cryptography
 In Third International Conference, ACNS 2005
, 2005
"... Abstract. Recently, several public key exchange protocols based on symbolic computation in noncommutative (semi)groups were proposed as a more efficient alternative to well established protocols based on numeric computation. Notably, the protocols due to AnshelAnshelGoldfeld and KoLee et al. exp ..."
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Cited by 25 (3 self)
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Abstract. Recently, several public key exchange protocols based on symbolic computation in noncommutative (semi)groups were proposed as a more efficient alternative to well established protocols based on numeric computation. Notably, the protocols due to AnshelAnshelGoldfeld and KoLee et al. exploited the conjugacy search problem in groups, which is a ramification of the discrete logarithm problem. However, it is a prevalent opinion now that the conjugacy search problem alone is unlikely to provide sufficient level of security no matter what particular group is chosen as a platform. In this paper we employ another problem (we call it the decomposition problem), which is more general than the conjugacy search problem, and we suggest to use R. Thompson’s group as a platform. This group is well known in many areas of mathematics, including algebra, geometry, and analysis. It also has several properties that make it fit for cryptographic purposes. In particular, we show here that the word problem in Thompson’s group is solvable in almost linear time. 1
Lengthbased conjugacy search in the braid group
"... Several key agreement protocols are based on the following Generalized Conjugacy Search Problem: Find, given elements b1,..., bn and xb1x −1,..., xbnx −1 in a nonabelian group G, the conjugator x. In the case of subgroups of the braid group BN, Hughes and Tannenbaum suggested a lengthbased approac ..."
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Cited by 17 (3 self)
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Several key agreement protocols are based on the following Generalized Conjugacy Search Problem: Find, given elements b1,..., bn and xb1x −1,..., xbnx −1 in a nonabelian group G, the conjugator x. In the case of subgroups of the braid group BN, Hughes and Tannenbaum suggested a lengthbased approach to finding x. Since the introduction of this approach, its effectiveness and successfulness were debated. We introduce several effective realizations of this approach. In particular, a length function is defined on BN which possesses significantly better properties than the natural length associated to the Garside normal form. We give experimental results concerning the success probability of this approach, which suggest that an unfeasible computational power is required for this method to successfully solve the Generalized Conjugacy Search Problem when its parameters are as in existing protocols.
A new key exchange protocol based on the decomposition problem
 Contemp. Math., Amer. Math. Soc
"... Abstract. In this paper we present a new key establishment protocol based on the decomposition problem in noncommutative 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 in ..."
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Cited by 12 (2 self)
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Abstract. In this paper we present a new key establishment protocol based on the decomposition problem in noncommutative 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 protocols based on the decomposition problem. In particular, we conceal (i.e., do not publish explicitly) one of the subgroups A, B, thus introducing an additional computationally hard problem for the adversary, namely, finding the centralizer of a given finitely generated subgroup. 1.
Key Agreement Protocol (KAP) Using Conjugacy and Discrete
 Logarithm Problems in Group Representation Level, Informatica
"... Abstract. The key agreement protocol based on infinite noncommutative group presentation and representation levels is proposed. Two simultaneous problems in group representation level are used: the conjugator search problem (CSP) and modified discrete logarithm problem (DLP). The modified DLP in o ..."
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Cited by 7 (2 self)
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Abstract. The key agreement protocol based on infinite noncommutative group presentation and representation levels is proposed. Two simultaneous problems in group representation level are used: the conjugator search problem (CSP) and modified discrete logarithm problem (DLP). The modified DLP in our approach is a matrix DLP and is different from that’s used in other publications. The algorithm construction does not allow to perform a cryptoanalysis by replacing the existing CSP solution to the decomposition problem (DP) solution. The group presentation level serves for two commuting subgroups and invertible group’s word image matrix construction. The group representation level allows reliable factors disguising in the initial word. The word equivalence problem (WEP) solution is transformed from the group presentation level to the group representation level. Hence there are not necessary to solve WEP in the group presentation level and hence there are no restrictions on the group complexity in this sense. The construction of irreducible representation of group is required. The presented protocol is a modernization of protocol declared in (Sakalauskas et al., 2005). Key words: key agreement protocol, conjugator search problem, discrete logarithm problem, group representation. 1.
Using decision problems in public key cryptography
, 2007
"... There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the info ..."
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Cited by 4 (3 self)
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There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the information that there are objects with the property P, find at least one particular object with the property P. So far, no cryptographic protocol based on a search problem in a noncommutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability “very close ” to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a “brute force attack”, the “1” or “0 ” bits in Bob’s binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her “brute force attack ” program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob’s sequence.
A K. Group Signature Scheme Using Braid Groups
"... Artin’s braid groups have been recently suggested as a new source for publickey cryptography. In this paper we propose the first group signature schemes based on the conjugacy problem, decomposition problem and root problem in the braid groups which are believed to be hard problems. ..."
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Artin’s braid groups have been recently suggested as a new source for publickey cryptography. In this paper we propose the first group signature schemes based on the conjugacy problem, decomposition problem and root problem in the braid groups which are believed to be hard problems.
Conjugacy in baumslag’s group, generic case complexity, and division in power circuits. arXiv preprint arXiv:1309.5314
, 2013
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On the conjugacy search problem and left conjugacy closed loops
 Appl. Algebra Engrg. Comm. Comput
, 2008
"... Abstract. The conjugacy search problem (CSP) is used as a primitive in several braid group based public key encryption schemes. It has been pointed out that, in braid groups, it unlikely provides adequate security. Therefore, new structures need to be found. In this paper, we give a formulation of ..."
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Abstract. The conjugacy search problem (CSP) is used as a primitive in several braid group based public key encryption schemes. It has been pointed out that, in braid groups, it unlikely provides adequate security. Therefore, new structures need to be found. In this paper, we give a formulation of the CSP for left conjugacy closed loops. In order to construct a generalization of the AnshelAnshelGoldfeld key establishment method, we also define a partial conjugacy search problem PCSP and show it to be equivalent to the CSP, if the underlying structure is a group. We also study closer the PCSP in a class of conjugacy closed loops of order p 2 , where p is a prime.