Quantum encoding based on optical phase states and mutually unbiased bases in the presence of several eavesdroppers

We investigate quantum key distribution where information is carried by a d-level optical quantum system. The sender uses the mutually unbiased bases generated from the phase states of the quantum system. We consider the effects of several intercept and resend attacks. We derive explicitly the mutual information between the legitimate parties as well as the mutual information between the sender and the different eavesdroppers in the communication channel. We discuss the security of the cryptographic scheme and present some numerical analysis.