Noisy Feedback and Loss Unlimited Private Communication

Cryptographic protocols often involve the assistance of public side channels to which all parties have perfectly noiseless access. For instance, in the BB84 quantum key distribution protocol, the side channel is used to share the bases in which Alice and Bob encoded or measured their qubits. In this paper, we find that in the case of continuous variable communication, by slightly altering this model such that Eve's copy of the initial round of feedback is corrupted by an iota of noise while keeping Alice's copies noiseless, the capacity can be increased dramatically. Specifically, it is known that the private capacity with noiseless feedback for a pure-loss bosonic channel is at most -\log(1-\eta) bits per mode, where \eta is the transmissivity, in the limit of infinite input photon number. This is a very pessimistic result as there is a finite rate limit even with an arbitrarily large number of input photons. We refer to this as a loss limited rate. However, in our altered model we find that we can achieve a rate of (1/2) \log(1+4\eta N-{S}) bits per mode with weak security, where N-{S} is the input photon number. This rate diverges with N-{S}, in sharp contrast to the result for the original model. This suggests that physical considerations behind the eavesdropping model should be taken more seriously, as they can create strong dependencies of the achievable rates on the model. For by a seemingly inconsequential weakening of Eve, we obtain a loss-unlimited rate. Our protocol also works verbatim for arbitrary i.i \mathrm{d}, noise (not even necessarily Gaussian) injected by Eve in every round, and even if Eve is given access to copies of the initial transmission and noise. The error probability of the protocol decays super-exponentially with the blocklength.