  Owing to recent advances in laser technology, it has become important to
investigate fundamental laser-assisted processes in very powerful laser fields.
In the present work and within the framework of laser-assisted quantum
electrodynamics (QED), electron-proton scattering was considered in the
presence of a strong electromagnetic field of circular polarization. First, we
present a study of the process where we only take into account the relativistic
dressing of the electron without the proton. Then, in order to explore the
effect of the proton dressing, we fully consider the relativistic dressing of
the electron and the proton together and describe them by using Dirac-Volkov
functions. The analytical expression for the differential cross section (DCS)
in both cases is derived at lowest-order of perturbation theory. As a result,
the DCS is notably reduced by the laser field. It is found that the effect of
proton dressing begins to appear at laser field strengths greater than or equal
to $10^{10}~\text{V/cm}$ and it therefore must be taken into account. The
influence of the laser field strength and frequency on the DCS is reported. A
comparison with the Mott scattering and the laser-free results is also
included.
