We investigate the cosmological evolution of the hard X-ray luminosity function (HXLF) of Active Galactic Nuclei (AGN) in the 2-10 keV luminosity range of 10^{41.5} - 10^{46.5} erg s^-1 as a function of redshift up to 3. From a combination of surveys conducted at photon energies above 2 keV with HEAO1, ASCA, and Chandra, we construct a highly complete (>96%) sample consisting of 247 AGNs over the wide flux range of 10^{-10} - 3.8*10^{-15} erg cm^-2 s^-1 (2-10 keV). For our purpose, we develop an extensive method of calculating the intrinsic (before-absorption) HXLF and the absorption (N_H) function. This utilizes the maximum likelihood method fully correcting for observational biases with consideration of the X-ray spectrum of each source. We find that (i) the fraction of X-ray absorbed AGNs decreases with the intrinsic luminosity and (ii) the evolution of the HXLF of all AGNs (including both type-I and type-II AGNs) is best described with a luminosity dependent density evolution (LDDE) where the cutoff redshift increases with the luminosity. Our results directly constrain the evolution of AGNs that produce a major part of the hard X-ray background, thus solving its origin quantitatively. A combination of the HXLF and the NH function enables us to construct a purely "observation based" population synthesis model. We present basic consequences of this model, and discuss the contribution of Compton-thick AGNs to the rest of the hard X-ray background.

62 pages, 26 figures (13 colour figures included). Accepted for publication in ApJ. Minor corrections. References are updated