  Using the cosmic ray (CR) data available in the energy interval $(10 - 2
\times 10^{7})$ GeV/particle, we have calculated the profile of the primary
$\gamma$-ray spectrum produced by the interaction of these CR with thermal
nuclei of the ISM. Normalized to the EGRET measurements, this allows an
estimate of the galactic diffuse $\gamma$-ray background due to intermediate
and high energy CR at TeV energies. On the other hand, over the last few years,
several particles with energies above $10^{20}$ eV (beyond the
Greisen-Zatsepin-Kuzmin cut-off) have been detected. These particles are very
likely extragalactic protons originated at distances not greater than $30 - 50$
Mpc [e.g., 1]. The propagation of these ultra-high energy protons (UHEP)
through the intergalactic medium leads to the development of $\gamma$-ray
cascades and an ultimate signature at TeV energies. To assess the statistical
significance of this $\gamma$-ray signature by the UHEP, we have also simulated
the development of electromagnetic cascades triggered by the decay of a
$10^{19}$ eV $\pi^{o}$ in the intergalactic medium after an UHEP collision with
a cosmic microwave background photon.
