Measuring the radiation environment at the surface of Mars is the primary goal of the Radiation Assessment Detector on the NASA Mars Science Laboratory's Curiosity rover. One of the conditions that Curiosity will likely encounter is a dust storm. The objective of this paper is to compute the cosmic ray ionization in different conditions, including dust storms, as these various conditions are likely to be encountered by Curiosity at some point. In the present work, the Nowcast of Atmospheric Ionizing Radiation for Aviation Safety model, recently modified for Mars, was used along with the Badhwar & O'Neill 2010 galactic cosmic ray model. In addition to galactic cosmic rays, five different solar energetic particle event spectra were considered. For all input radiation environments, radiation dose throughout the atmosphere and at the surface was investigated as a function of atmospheric dust loading. It is demonstrated that for galactic cosmic rays, the ionization depends strongly on the atmosphere profile. Moreover, it is shown that solar energetic particle events strongly increase the ionization throughout the atmosphere, including ground level, and can account for the radio blackout conditions observed by the Mars Advanced Radar for Subsurface and Ionospheric Sounding instrument on the Mars Express spacecraft. These results demonstrate that the cosmic rays' influence on the Martian surface chemistry is strongly dependent on solar and atmospheric conditions that should be taken into account for future studies.