  We present a detailed security analysis of a d-dimensional quantum key
distribution protocol based on two and three mutually unbiased bases (MUBs)
both in an asymptotic and finite key length scenario. The finite secret key
rates are calculated as a function of the length of the sifted key by (i)
generalizing the uncertainly relation-based insight from BB84 to any d-level
2-MUB QKD protocol and (ii) by adopting recent advances in the second-order
asymptotics for finite block length quantum coding (for both d-level 2- and
3-MUB QKD protocols). Since the finite and asymptotic secret key rates increase
with d and the number of MUBs (together with the tolerable threshold) such QKD
schemes could in principle offer an important advantage over BB84. We discuss
the possibility of an experimental realization of the 3-MUB QKD protocol with
the orbital angular momentum degrees of freedom of photons.
