Minimum Energy Per Bit for Secret Key Acquisition Over Multipath Wireless Channels

We study fundamental limits on the generation of secret keys based on the randomness inherent to reciprocal wireless multipath channels. Estimates of the common channel at the two ends of a link are jointly Gaussian sources from which secret keys can be generated. The key generation problem is cast as an equivalent communication problem to characterize the secret key capacity. We analyze the low-SNR regime to quantify the minimum energy per secret key bit required for reliable key acquisition. Our results show that, in contrast to the low SNR behavior of conventional channel capacity, there is a non-zero SNR gamma* that achieves the minimum energy per key bit. A time-sharing scheme is proposed to achieve the minimum energy per key bit at any SNR below gamma*. We also investigate the reliability of secret key generation via error exponent analysis. In particular, our results yield a tight upper bound on the minimum energy required to generate a finite-length key with a specified probability of error in key acquisition.