Impact of channel sparsity and correlated eavesdropping on secret key generation from multipath channel randomness

We study secret key generation from reciprocal multipath wireless channels modeled as multiple parallel fading channels. We consider two channel characteristics that heavily impact secret key capacity: channel sparsity and correlation between main and eavesdropper´s channels. We propose a joint model to capture the channel sparsity and correlated eavesdropping. For the case without eavesdroppers, we show that at each transmitter SNR γ there is an optimal sparsity 0 ≤ ρ_opt ≤ 1, which specifies the fraction of non-zero channel coefficients, that yields the maximum secret key capacity. The reduction in secret key capacity due to eavesdropping is due to two sources. First is the overlap between the main and eavesdropper channels, i.e., the pattern of non-zero subchannels common to both. The second is the correlation between the channel coefficients of the overlapping channels. We show that when the power of the training signal is uniformly distributed over the non-zero channels, there is a cutoff SNR γ_c below which the secret key rate is zero, but non-zero (and increasing in γ) when γ > γ_c. We also show that in the low SNR regime, the optimal input signal is peaky (a non-uniform training signal) by which the secret key capacity is non-zero at all γ > 0.