We considered Compton scattering of photons of two elliptical polarizations by electrons in plasma in a strong magnetic field. The frequency redistribution functions of the photons were derived, and characteristics such as the average photon momentum after scattering, average energy, and dispersions of these quantities were obtained numerically. These were averaged over an ensemble of electrons with a relativistic Maxwell distribution along the field. Multiple scattering in the medium was simulated and the above characteristics were obtained in this case.