  In this paper, we consider the problem of secret key generation for multiple
parties. Multi-user networks usually require a trusted party to efficiently
distribute keys to the legitimate users and this process is a weakness against
eavesdroppers. With the help of the physical layer security techniques, users
can securely decide on a secret key without a trusted party by exploiting the
unique properties of the channel. In this context, we develop a physical layer
group key generation scheme that is also based on the ideas of the analog
function computation studies. We firstly consider the key generation as a
function to be computed over the wireless channel and propose two novel methods
depending on the users transmission capability (i.e. half-duplex and
full-duplex transmissions). Secondly, we exploit the uniqueness of the prime
integers in order to enable the simultaneous transmission of the users for key
generation. As a result, our approach contributes to the scalability of the
existing physical layer key generation algorithms since all users transmit
simultaneously rather than using pairwise communications. We prove that our
half-duplex network model reduces the required number of communications for
group key generation down to a linear scale. Furthermore, the full-duplex
network model reduces to a constant scale.
