An authentication scheme based on the twisted conjugacy problem
  The conjugacy search problem in a group $G$ is the problem of recovering an $x \in G$ from given $g \in G$ 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 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) phi, \psi of a group G and a pair of elements w, t \in G, find an element s \in G such that t = \psi(s^{-1}) w \phi(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 2x2 matrices over truncated one-variable polynomials over F_2, the field of two elements, with transposition used instead of inversion in the equality above. 
Vladimir Shpilrain and Alexander Ushakov