Based on the assumption that the Pomeron is a sum of Regge multipoles, each of which corresponds to a finite gluon ladder, we suggest a way to search the Odderon by studying the diffractive processes of -pp and pp elastic scattering. From our theoretical calculations and fits to the corresponding experimental data at high energies and all available momentum transfer, we find that the Odderon plays a significant role in the description of differential cross section dσ / dt in the whole experimental domain. Accordingly, we can claim that the -pp and pp diffractive processes at high energies s and moderate momentum transfer t could be a good tool to search for Odderon and to test QCD since the processes are purely non-perturbative QCD phenomena. Therefore, the experimental measurements are urgently demanded.