Efficient methods to determine the reversibility of general 1D linear cellular automata in polynomial complexity

Abstract

In this paper, we study reversibility of one-dimensional(1D) linear cellular automata(LCA) under null boundary condition, whose core problems have been divided into two main parts: calculating the period of reversibility and verifying the reversibility in a period. With existing methods, the time and space complexity of these two parts are still too expensive to be employed. So the process soon becomes totally incalculable with a slightly big size, which greatly limits its application. In this paper, we set out to solve these two problems using two efficient algorithms, which make it possible to solve reversible LCA of very large size. Furthermore, we provide an interesting perspective to conversely generate 1D LCA from a given period of reversibility. Due to our methods' efficiency, we can calculate the reversible LCA with large size, which has much potential to enhance security in cryptography system.

On the Exploration of the Natural Sequence of Primes With Cellular Automata Targeting Enhanced Data Security and Privacy

“…In future, we plan to investigate further at higher automata to explore further. We already found in (Du et al, 2019) that the proposed additive CA rules-based primes generation at higher automata size may face the possible incalculability and hence we believe an investigation with reference to the computation of reversibility of 1-dimensional CA focused on the computation for the period of reversibility and verification of said reversibility (refer Du et al, 2019) may further explore the true application suitability of proposed approach towards its uses in enhanced data security and privacy.…”

“…A normalized method-based representation towards descriptive complexity for minimal time of CA was presented in (Grandjean et al, 2019). Further, reversibility of 1-dimensional CA focused on the compatibility of reversibility and verifiability of the reversibility in said period was presented in (Du et al, 2019) to deal with the problem of in-computability for CA at higher automata size; further a detailed discussion related to the group theoretical aspects for QCA (Quantum CA) towards its classification was presented in (Freedman et al, 2019).…”