  The quantum dense coding (DC) protocol, which has no security feature, deals
with the transmission of classical information encoded in a quantum state by
using shared entanglement between a single sender and a single receiver. Its
appropriate variant has been established as a quantum key distribution (QKD)
scheme for shared two-qubit maximally entangled states, with the security proof
utilizing the uncertainty relation of complementary observables and the
Shor-Preskill entanglement purification scheme. We present the DC-based QKD
protocol for higher dimensional systems and report the lower bounds on secret
key rate, when the shared state is a two-qudit maximally entangled state, and
mixtures of maximally entangled states with different ranks. The analysis also
includes the impact of noisy channels on the secure key rates, before and after
encoding. In both the noiseless and the noisy scenarios, we demonstrate that
the key rate as well as the robustness of the protocol against noise increases
with the dimension. Further, we prove that the set of useless states in the
DC-based QKD protocol is convex and compact.
