Effects of Quantization on the Multiple-Round Secret-Key Capacity

We consider the strong secret key (SK) agreement problem for the satellite communication setting, where a remote source (a satellite) chooses a common binary phase shift keying (BPSK) modulated input for three statistically independent additive white Gaussian noise (AWGN) channels whose outputs are observed by, respectively, two legitimate receivers (Alice and Bob) and an eavesdropper (Eve). Legitimate receivers have access to an authenticated, noiseless, two-way, and public communication link, so they can exchange multiple rounds of public messages to agree on a SK hidden from Eve. Without loss of essential generality, the noise variances for Alice's and Bob's measurement channels are both fixed to a value $Q>1$, whereas the noise over Eve's measurement channel has a unit variance, so $Q$ represents a channel quality ratio. The significant and not necessarily expected effect of quantizations at all receivers on the scaling of the SK capacity with respect to a sufficiently large and finite channel quality ratio $Q$ is illustrated by showing 1) the achievability of a constant SK for any finite BPSK modulated satellite output by proposing a thresholding algorithm as an advantage distillation protocol for AWGN channels and 2) the converse (i.e., unachievability) bound for the case when all receivers apply a one-bit uniform quantizer to their noisy observations before SK agreement, for which the SK capacity is shown to decrease quadratically in $Q$. Our results prove that soft information increases not only the reliability and the achieved SK rate but also the scaling of the SK capacity at least quadratically in $Q$ as compared to hard information.