Ultra-high energy (UHE) photons play an important role as an independent probe of the photo-pion production mechanism by UHE cosmic rays. Their observation, or non-observation, may constrain astrophysical scenarios for the origin of UHECRs and help to understand the nature of the flux suppression observed by several experiments at energies above $10^{19.5}$ eV. Whereas the interaction length of UHE photons above $10^{17}$ eV ranges from a few hundred kpc up to tenths of Mpc, photons can interact with the extragalactic background radiation initiating the development of electromagnetic cascades which affect the fluxes of photons observed at Earth. The interpretation of the current experimental results rely on the simulations of the UHE photon propagation. In this paper, we present the novel Monte Carlo code EleCa to simulate the $Ele$ctromagnetic $Ca$scading initiated by high-energy photons and electrons. We provide an estimation of the surviving probability for photons inducing electromagnetic cascades as a function of their distance from the observer and we calculate the distances within which we expect to observe UHE photons with energy between $10^{17}$ and $10^{19}$ eV. Furthermore, the flux of GZK photons at Earth is investigated in several astrophysical scenarios where we vary both injection spectrum and composition at the source and the intensity of the intervening extragalactic magnetic field. Although the photon propagation depends on several astrophysical factors, our numerical predictions combined with future experimental observations (or non-observations) of UHE photons -- in the energy range between $10^{17.5}$ eV and $10^{20}$ eV -- can help to constrain these scenarios.
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