A polynomial time algorithm for the braid Diffie-Hellman conjugacy problem

Abstract

We propose the first polynomial time algorithm for the braid Diffie-Hellman conjugacy problem (DHCP) on which the braid key exchange scheme and the braid encryption scheme are based [9]. We show the proposed method solves the DHCP for the image of braids under the Lawrence-Krammer representation and the solutions play the equivalent role of the original key for the DHCP of braids. Given a braid index n and a canonical length ℓ, the complexity is about O(n14.4ℓ3.2) or O(n4τ+2εℓ2ε) bit operations for τ = log2 7 ≈ 2.8 and ε > log2 3 ≈ 1.57. © International Association for Cryptologic Research 2003.