  We give a proof that entanglement purification, even with noisy apparatus, is
sufficient to disentangle an eavesdropper (Eve) from the communication channel.
In the security regime, the purification process factorises the overall initial
state into a tensor-product state of Alice and Bob, on one side, and Eve on the
other side, thus establishing a completely private, albeit noisy, quantum
communication channel between Alice and Bob. The security regime is found to
coincide for all practical purposes with the purification regime of a two-way
recurrence protocol. This makes two-way entanglement purification protocols,
which constitute an important element in the quantum repeater, an efficient
tool for secure long-distance quantum cryptography.
