Efficient Algebraic Method for Testing the Invertibility of Finite State Machines

The emergence of new embedded system technologies, such as IoT, requires the design of new lightweight cryptosystems to meet different hardware restrictions. In this context, the concept of Finite State Machines (FSMs) can offer a robust solution when using cryptosystems based on finite automata, known as FAPKC (Finite Automaton Public Key Cryptosystems), introduced by Renji Tao. These cryptosystems have been proposed as alternatives to traditional public key cryptosystems, such as RSA. They are based on composing two private keys, which are two FSMs M1 and M2 with the property of invertibility with finite delay to obtain the composed FSM M=M1oM2, which is the public key. The invert process (factorizing) is hard to compute. Unfortunately, these cryptosystems have not really been adopted in real-world applications, and this is mainly due to the lack of profound studies on the FAPKC key space and a random generator program. In this paper, we first introduce an efficient algebraic method based on the notion of a testing table to compute the delay of invertibility of an FSM. Then, we carry out a statistical study on the number of invertible FSMs with finite delay by varying the number of states as well as the number of output symbols. This allows us to estimate the landscape of the space of invertible FSMs, which is considered a first step toward the design of a random generator.