The origin of the Cosmic X-ray Background (CXB), whose intensity dominates in the hard X-ray band above 2 keV, has been a key issue of the X-ray astronomy. In order to study the nature of the hard X-ray source population and thus provide a clear picture of the composition of the CXB, we have carried out the first systematic wide-area survey around a galactic pole area observed with the ASCA satellite in the 0.7-10 keV energy band (Large Sky Survey; LSS). The area observed amounts to more than 6 deg2. The excellent sensitivity of ASCA, higher than any previous missions by two orders of magnitudes in the 2-10 keV band, has enabled us to investigate statistical properties of X-ray sources at a flux level down to ~10-13 erg cm-2 s-1 (2-10 keV). To search for faint X-ray sources and determine their fluxes and spectra, we have developed a new source-detection and image fitting method where the complicated detector responses are fully taken into account. Applying this method to the data obtained from the entire LSS field, independently in three energy bands (0.7-2 keV, 2-10 keV, and 0.7-7 keV), we detected 119 sources in total. Particularly, in the hard X-ray band of 2-10 keV, we detected 47 sources with the GIS instrument, whose fluxes distribute in a wide range from 1 × 10-13 to 2 × 10-12 erg cm-2 s-1 (2-10 keV). Derived Log N - Log S relation in this energy band lies on the extrapolation from the previous results obtained with Ginga and HEAO1 with the Euclidean slope of -3/2. The average spectrum of the sources with fluxes less than 2.5 × 10-13 erg cm-2 s-1 (2-10 keV) shows a photon index of 1.5 ± 0.2 in the 2-10 keV band, which is harder than the typical spectrum of bright Active Galactic Nuclei (AGN). This suggests that the "spectral paradox" is beginning to be solved. Our results imply that a population of sources with an average photon index similar to the CXB begins to dominate at the energy above 2 keV when the sensitivity goes up to 10-13 erg cm-2 s-1 (2-10 keV). Highly absorbed sources, such as type II AGN, could be the candidate for such sources.