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New Developments in Commutator Key Exchange

We study the algorithmic security of the Anshel-Anshel-Goldfeld (AAG) key exchange scheme and show that contrary to prevalent opinion, the computational hardness of AAG depends on the structure of the chosen subgroups, rather than on the conjugacy problem of the ambient braid group. Proper choice of these subgroups produces a key exchange scheme which is resistant to all known attacks on AAG.