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.

On the invertibility of finite linear transducers

“…The invertibility theory of transducers used in the FAPKCs relies heavily on invertible linear transducers. The first study on this subject was published by Tao [7], based on the work of Massey and Slain [4] on inverses of linear sequential circuits. The notion of linear finite transducer used in that paper is a slight variant of the one Nerode studies in his seminal article [5].…”

“…Later on, Tao published a paper with a study on the invertibility of linear finite transducers defined over a ring [8]. Other studies were pursued some years later [2,[15][16][17], but as in the first paper on this subject [7], none of them presents an algorithm to invert finite linear transducers. Some authors refer to a book written in Chinese on the invertibiliy of automata, where supposedly one finds an algorithm to invert linear transducers.…”