The computation of the scattering properties of cirrus cloud ice crystals by the ray-tracing approach is described. A light beam is represented by its Stokes quadrivector I, which describes intensity as well as polarization, and the scattering properties of particles (molecules, droplets, or ice crystals) are introduced by means of a 4 x 4 transformation matrix M known as the Mueller matrix, or M matrix. Obtaining such a matrix for each kind of particle gives access to a complete description of the scattering medium. Most computations of the M matrices of cirrus ice crystals have introduced several simplifying hypotheses, by using basic shapes, by assuming a random orientation of the particles, or both. The present study focuses on the calculation of the complete M matrix for a specific shape of particles (i.e., hexagonally based crystals) either with optional oscillation about the horizontal plane or with random orientation. The validity of the computation code is checked against specific well-known cases for randomly oriented particles. For horizontally oscillating particles the computation of this matrix is a new result. Sensitivity of the M matrix to the following variables is studied: refractive index, amplitude of oscillation, particle shape and size, and angle of incidence.