Real-Time Prime Generators Implemented on Small-State Cellular Automata

Abstract

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 the prime generation problem on cellular automata consisting of infinitely many cells each with finite state memory and present two implementations 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 a 1-bit inter-cell communication cellular automaton with 25-states, which is an improvement over a 34-state implementation given 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 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 developed by Korec [1998]. Those two implementations on cellular automata with different communication models are the smallest realizations in the number of states, at present.