Exploiting randomness on continuous sets

In this paper, we introduce a new type of field—continuous sets, where we can exploit randomness in the non-repeating decimal expansions of irrationals for cryptographical purposes, and present two specific sets, a real interval [0, 1) and a functional space F[0, 1). On [0, 1), we propose ideal irrational random number generator (IIRNG) which generates non-repeating random number sequence as a truly RNG by computing the decimal expansion of an randomly chosen irrational. On F[0, 1), we propose integral encryption scheme (IES) with which we can encrypt an infinite message and obtain perfect security in one-time encryption by computing the integration of the message on a randomly chosen function. Either the seeds of IIRNG or the keys of IES are sufficiently safe and immune to exhaustive key search. Both IIRNG and IES require the assumption that an element of [0, 1) or F[0, 1) can be uniformly randomly chosen. Though the assumption cannot be achieved in classical finite machine, we present the discretization of the assumption, i.e., randomly choosing an element of U or V (the set of all possible methods of generating irrationals or functions). The immunity of seeds or keys to exhaustive key search still exists, since any finite search for a random element of U or V is inefficient. This is the basic idea of implementing IIRNG and IES in finite machine. Two corresponding examples IRNG and IBC are also presented, whose securities are guaranteed by the randomly chosen elements of U or V.