Relations Between Transformations

Abstract

In application to public key cryptosystems, for finite automata in public keys no feasible inversion algorithm had been found. In Sect. 3.1 of the preceding chapter, an inversion method by R _a R _b transformation was given implicitly. In this chapter, a relation between two R _a R _b transformation sequences beginning at the same equation is derived. It means that in the inversion process it is enough to choose any one of the linear R _a R _b transformation sequences. Then, by exploring properties of “composition” of two R _a R _b transformation sequences, it is shown that the inversion method by R _a R _b transformation works for some special compound finite automata.

Other two inversion methods are by reduced echelon matrices, and by canonical diagonal matrix polynomials. Results in the last two sections show that the two inversion methods are “equivalent” to the inversion method by R _a R _b transformation.

This chapter provides a foundation for assertions on the weak key of the public key cryptosystem based on finite automata in Sect. 9.4.