An Authentication Scheme Based on the Twisted Conjugacy Problem

The conjugacy search problem in a group G is the problem of recovering an x ∈ G from given g ∈ G and h = x-1gx. 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 Fiat-Shamir scheme involving repeating several times a three-pass challenge-response 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) θ, ψ of a group G and a pair of elements ω, t ∈ G, find an element s ∈ G such that t = ψ(s-1)ωθ(s) provided at least one such s exists. This problem appears to be very non-trivial even for free groups. We offer here another platform, namely, the semigroup of all 2 × 2 matrices over truncated one-variable polynomials over F2, the field of two elements, with transposition used instead of inversion in the equality above.

[1]  Alexander Ushakov,et al.  Length Based Attack and Braid Groups: Cryptanalysis of Anshel-Anshel-Goldfeld Key Exchange Protocol , 2007, Public Key Cryptography.

[2]  Adi Shamir,et al.  Cryptanalysis of Group-Based Key Agreement Protocols Using Subgroup Distance Functions , 2007, Public Key Cryptography.

[3]  D. Goldfeld,et al.  An algebraic method for public-key cryptography , 1999 .

[4]  V. Pan,et al.  Polynomial and Matrix Computations , 1994, Progress in Theoretical Computer Science.

[5]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[6]  Volker Gebhardt,et al.  Conjugacy in Garside groups I: cyclings, powers and rigidity , 2006, math/0605230.

[7]  Dennis Hofheinz,et al.  A Practical Attack on Some Braid Group Based Cryptographic Primitives , 2003, Public Key Cryptography.

[8]  Robert R. Alfano,et al.  Recent Advances in the Uses of Light in Physics, Chemistry, Engineering, and Medicine: 19-21 June 1991, the City College of New York , 1992 .

[9]  Vladimir Shpilrain.  Hashing with Polynomials , 2006, ICISC.

[10]  Patrick Dehornoy,et al.  Entity authentication schemes using braid word reduction , 2006, Discret. Appl. Math..

[11]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[12]  Jung Hee Cheon,et al.  New Public-Key Cryptosystem Using Braid Groups , 2000, CRYPTO.

[13]  Volker Gebhardt,et al.  Conjugacy in Garside groups II: structure of the ultra summit set , 2006 .

[14]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[15]  Oleg Bogopolski,et al.  The Conjugacy Problem is Solvable in Free‐By‐Cyclic Groups , 2006 .

[16]  R. Gregory Taylor,et al.  Modern computer algebra , 2002, SIGA.

[17]  Alexander Fel'shtyn,et al.  TWISTED CONJUGACY SEPARABLE GROUPS , 2006 .

[18]  A. Martino,et al.  Free-by-cyclic groups have solvable conjugacy problem , 2004 .

[19]  Michael Clausen,et al.  Algebraic Complexity Theory : With the Collaboration of Thomas Lickteig , 1997 .

[20]  David Garber,et al.  Probabilistic Solutions of Equations in the Braid Group , 2005, Adv. Appl. Math..

[21]  D. Garber,et al.  LENGTH-BASED CONJUGACY SEARCH IN THE BRAID GROUP , 2002, math/0209267.