Problems in theory of cryptanalytical invertibility of finite automata

The paper continues an investigation of the cryptanalytical invertibility concept of finite automata with a finite delay introduced by the author in his previous papers where he also gave a constructive set theory test for an automaton $A$ to be cryptanalytically invertible, that is, to have a recovering function $f$ which allows to calculate a prefix of a length $m$ in an input sequence of the automaton $A$ by using its output sequence of a length $m+\tau$ and some additional information about $A$ known to cryptanalysts, defining a type of its invertibility and of its recovering functon. Here, we expound a test for that of another kind, namely some logical necessary and sufficient conditions for an automaton $A$ to have or not a recovering function $f$ of a certain type. Results related to specific types of automata invertibility (invertibility tests, inversion algorithms, synthesis of inverse automata and others) are subjects of further researching and publications.