Identification over the Gaussian Channel in the Presence of Feedback

We analyze message identification via Gaussian channels with noiseless feedback, which is part of the Post Shannon theory. The consideration of communication systems beyond Shannon's approach is useful to increase the efficiency of information transmission for certain applications. We consider the Gaussian channel with feedback. If the noise variance is positive, we propose a coding scheme that generates infinite common randomness between the sender and the receiver and show that any rate for identification via the Gaussian channel with noiseless feedback can be achieved. The remarkable result is that this applies to both rate definitions $\frac 1n \log M$ (as Shannon defined it for the transmission) and $\frac 1n \log\log M$ (as defined by Ahlswede and Dueck for identification). We can even show that our result holds regardless of the selected scaling for the rate.