Thompson’s Group and Public Key Cryptography

Abstract

Abstract. 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 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.

Cryptographic analysis of the generalized ElGamal's cipher over GL(8,F251)

“…Схема формирования ключа в протоколе Хехта является частным случаем схемы из протокола Шпильрайна Ушакова [2].…”

Post-Quantum Cryptography: generalized ElGamal cipher over GF(251^8)

“…A whole branch of new protocols was developed which do not rely on extended arithmetic's precision and instead exploit internal asymmetry of abstract algebraic structures like partial grupoids, categories, magmas, monoids, quasigroups, groups, rings, loops or neofields [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. The new developed one-way trapdoor functions (OWTF) include conjugator search (CSP), decomposition (DP), commutative subgroup search (CSSP), symmetric decomposition (SDP) and generalized symmetric decomposition (GSDP) [9,15,17,25,26].…”

Cryptanalysis of Group-Based Key Agreement Protocols Using Subgroup Distance Functions

“…The protocol below was suggested by Shpilrain and Ushakov in [12]. The authors suggested to use Thompson's group F for its implementation.…”

“…The best known example of such a protocol is the Diffie-Hellman protocol, which uses a (commutative) cyclic group. Over the last few years, there was a lot of interest in key agreement protocols based on noncommutative groups, and much research was dedicated to analyzing these proposals and suggesting alternative ones (see, e.g., [1,4,5,6,7,8,10,11,12], and references therein).…”

“…We construct the theoretical framework for distance-based attacks and demonstrate its applicability using the Shpilrain-Ushakov protocol in Thompson's group F [12] as an example. Although it has recently been shown by Matucci [8] that the implementation of the proposed protocol in F can be broken deterministically using a specialized attack based on the structural properties of the group, it is still an interesting test case for more generic attacks, such as the one proposed here.…”