  The Ekert quantum key distribution protocol uses pairs of entangled qubits
and performs checks based on a Bell inequality to detect eavesdropping. The
N-DEB protocol uses instead pairs of entangled qudits to achieve better noise
resistance than the Ekert protocol. It performs checks based on the Bell CGLMP
inequality for qudits. In this paper, we present the generalization for qudits
of our protocol h3DEB (for qutrits). This protocol also uses pairs of entangled
qudits, but achieves even better noise resistance than N-DEB and is showed to
be secure against the same family of cloning attacks than N-DEB. This gain of
performance is obtained by using another inequality called here hCHSH-$d$. For
each party, the hCHSH-$d$ inequality involves $2d$ observables. We explain how
the parties can measure these observables and thus are able to check the
violation of hCHSH-$d$. In the presence of noise, this violation allows the
parties to ensure the secrecy of the key because it guarantees the absence of a
local Trojan horse attack. The advantage of our proposed scheme is that it
results in an increased resistance to noise while remaining secure against
individual attacks.
