Pseudorandom sequence generators based on shift registers over finite chain rings

This article is based on a report made at the conference CTCrypt'2018. The paper contains an overview of the author's results related to the synthesis of pseudorandom sequence generators. For arbitrary $m$ and for Galois ring $R$ the maximum length $L_m(R)$ of cycles of bijective polynomial transformations of module $R^m$ is calculated. An algorithm is proposed that constructs polynomial transformations with a cycle of length $L_m(R)$. Some estimates of the periods and ranks of the output sequences of self-controlled $2$-dimensional linear shift registers ($2$-LFSR) are obtained. The frequencies of occurrence of signs on the cycles of the output sequences of $2$-LFSR are investigated. A new result is announced in the article, consisting of the fact that over Galois ring $R$ there are polynomial shift registers of length $m$, the state transition graph of which contains a cycle of length $L_m(R)$.