Finite-length analysis for secret random number generation and coding theorems

Shannon theory had been considered as an unpractical engineering theory because the obtained optimizations focused on the asymptotic analysis, which are far from a practical setting. However, recent progresses are resolving this defect from achieving finite-length analysis on several information theoretical topics based on the traditional asymptotic results. These progresses enable us to evaluate the performances of realizable systems. The main factors of this research stream are the reflection for the traditional stream and the progress of the method of information spectrum, which has mainly been developed in Japan. However, we cannot ignore the effect by the min entropy developed in cryptography community and the practical demands required by the quantum cryptography system. As results, the finite-length theory has been developed in an interdisciplinary area across information theory community, the communication theory, cryptography theory, and quantum information theory. In this talk, we review the finite-length analysis for the secure random number generation as the most successful case, and briefly explain its extension to source coding, channel coding, and the quantum extension.