A Polynomial Time Algorithm for the Braid Diffie-Hellman Conjugacy Problem

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(n 14.4 l 3.2) or O(n 4τ + 2e l 2e ) bit operations for τ = log2 7 ≈ 2.8 and e> log2 3 ≈ 1.57.

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