The origin of wide band cosmic X-and γ-ray background has not been well understood, but recently it has been considered that the cosmic X-ray background in the energy range below several 100 keV is well reproduced by the superposition of AGN X-ray spectra with a hard bump (Morisawa et al. 1990; Fabian et al. 1990; Rogers & Field 1991; Terasawa 1991). It should be noted that this interpretation is based on the recent Ginga observations of the X-ray spectrum of AGN (Morisawa et al. 1990; Matsuoka et al. 1990). On the other hand, the cosmic γ-ray background above 0.5 MeV which shows a broad bump around a few MeV seems to require an additional component to the superposition spectrum of AGN X-ray radiation. In order to explain this cosmic γ-ray background we investigate the model in which π 0 γ-rays are produced by p-p and/or α-p interaction of cosmological cosmic rays. The cosmological cosmic rays could be provided from explosions of Population III objects. It is known that this basic theory has been proposed by Stecker (1971).