In this paper, we propose a ÿnite-automaton public-key cryptosystem. The compound ÿnite automaton of the public key of this cryptosystem consists of n + 1 ÿnite automata, where n is an arbitrary positive integer. This cryptosystem contains FAPKC3 and FAPKC4 as special cases, and can be used for encryption and for digital signature. We deal with the problem of generating safe keys. Some results on generating ÿnite automata with desired properties on invertibility are brie y mentioned. Relations of R a R b transformation sequences with linear Ra are characterized. We prove that for inversion of ÿnite automata by linear Ra R b transformation method, in a sense, only one linear Ra R b transformation sequence is enough. We give a feasible inversion algorithm for compound ÿnite automaton, of which one component belongs to quasi-linear ÿnite automata and the other component belongs to a kind of weakly invertible or weak inverse ÿnite automata generated by linear Ra R b inverse transformation. Some applications to the key-generator of FAPKCs are discussed and several linear attacks are considered.