Stealthy Secret Key Generation

In order to make a warden, Willie, unaware of the existence of meaningful communications, there have been different schemes proposed including covert and stealth communications. When legitimate users have no channel advantage over Willie, the legitimate users may need additional secret keys to confuse Willie, if the stealth or covert communication is still possible. However, secret key generation (SKG) may raise Willie's attention since it has a public discussion, which is observable by Willie. To prevent Willie's attention, we consider the source model for SKG under a strong secrecy constraint, which has further to fulfill a stealth constraint. Our first contribution is that, if the stochastic dependence between the observations at Alice and Bob fulfills the strict more capable criterion with respect to the stochastic dependence between the observations at Alice and Willie or between Bob and Willie, then a positive stealthy secret key rate is identical to the one without the stealth constraint. Our second contribution is that, if the random variables observed at Alice, Bob, and Willie induced by the common random source form a Markov chain, then the key capacity of the source model SKG with the strong secrecy constraint and the stealth constraint is equal to the key capacity with the strong secrecy constraint, but without the stealth constraint. For the case of fast fading models, a sufficient condition for the existence of an equivalent model, which is degraded, is provided, based on stochastic orders. Furthermore, we present an example to illustrate our results.