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A construction of smallest real-time prime generators on cellular automata

For a long time there was little use of prime numbers in practical applications. But nowadays, it has been known that large scale prime numbers play a very important role in encryption in computer security networks. In this paper, we explore a real-time prime generation problem on cellular automata consisting of infinitely many agents each with finite state memory and present two constructions of real-time prime generators on cellular automata having smallest number of internal states, known at present. It is shown that there exists a real-time prime generator on 1-bit inter-cell communication cellular automaton with 25-states, which is an improvement over a 34-state implementation in Umeo and Kamikawa [2003]. In addition, we show that an infinite prime sequence can be generated in real-time by an eight-state cellular automaton with constant-bit local communications. Both the algorithms presented are based on the classical sieve of Eratosthenes, and our eight-state implementation is an improvement over a nine-state prime generator presented in Korec [1998]. Those two constructions on cellular automata with different communication models are the smallest realizations in the number of states.