  We present a new implementation of the BB84 quantum key distribution protocol
that employs a $d$-dimensional Hilbert space spanned by spatial modes of the
propagating beam that have a definite value of orbital angular momentum. Each
photon carries $\log d$ bits of information, increasing the key generation rate
of the protocol. The states used in the transmission part of the protocol are
invariant under rotations about the propagation direction, making this
implementation independent of the alignment between the reference frames of the
sender and receiver. The protocol still works when these reference frames
rotate with respect to each other.
