Linear Autonomous Finite Automata

Abstract

Autonomous finite automata are regarded as sequence generators. For the general case, the set of output sequences of an autonomous finite automaton consists of ultimately periodic sequences and is closed under translation operation. From a mathematical viewpoint, such sets have been clearly characterized, although such a characterization is not very useful to cryptology. On the other hand, nonlinear autonomous finite automata can be linearized. So we confine ourself to the linear case in this chapter. Notice that each linear autonomous finite automaton with output dimension 1 is equivalent to a linear shift register and that linear shift registers as a special case of linear autonomous finite automata have been so intensively and extensively studied. In this chapter, we focus on the case of arbitrary output dimension. After reviewing some preliminary results of combinatory theory, we deal with representation, translation, period, and linearization for output sequences of linear autonomous finite automata. A result of decimation of linear shift register sequences is also presented.