Length-based attacks in polycyclic groups

Abstract

Abstract The Anshel–Anshel–Goldfeld (AAG) key-exchange protocol was implemented and studied with the braid groups as its underlying platform. The length-based attack, introduced by Hughes and Tannenbaum, has been used to cryptanalyze the AAG protocol in this setting. Eick and Kahrobaei suggest to use the polycyclic groups as a possible platform for the AAG protocol. In this paper, we apply several known variants of the length-based attack against the AAG protocol with the polycyclic group as the underlying platform. The experimental results show that, in these groups, the implemented variants of the length-based attack are unsuccessful in the case of polycyclic groups having high Hirsch length. This suggests that the length-based attack is insufficient to cryptanalyze the AAG protocol when implemented over this type of polycyclic groups. This implies that polycyclic groups could be a potential platform for some cryptosystems based on conjugacy search problem, such as non-commutative Diffie–Hellman, El Gamal and Cramer–Shoup key-exchange protocols. Moreover, we compare for the first time the success rates of the different variants of the length-based attack. These experiments show that, in these groups, the memory length-based attack introduced by Garber, Kaplan, Teicher, Tsaban and Vishne does better than the other variants proposed thus far in this context.