Geodesic
Algorithm for constructing the system of representatives of maximal length cycles of polynomial substitution over the Galois ring
Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 64-67

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There are no polynomials with full cycle over the Galois ring. The maximal length of cycle of polynomial mapping over the Galois ring equals $q(q-1)p^{n-2},$ where $q^n$ – cardinality of ring and $p^n$ – its characteristic. In this work, an algorithm is presented for constructing the system of representatives of all maximal length cycles of a polynomial substitution over the Galois ring. Let an elementary operation be the production in the Galois ring, then the complexity of the algorithm equals $\mathrm O(lq^{n-1})$ elementary operations as $n$ tends to infinity, where $l$ is the degree of the polynomial.
Keywords: nonlinear recurrent sequences, Galois ring. 
D. M. Ermilov. Algorithm for constructing the system of representatives of maximal length cycles of polynomial substitution over the Galois ring. Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 64-67. http://geodesic.mathdoc.fr/item/PDMA_2014_7_a27/
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     title = {Algorithm for constructing the system of representatives of maximal length cycles of polynomial substitution over the {Galois} ring},
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     url = {http://geodesic.mathdoc.fr/item/PDMA_2014_7_a27/}
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[1] Ermilov D. M., Kozlitin O. A., “Tsiklovaya struktura polinomialnogo generatora nad koltsom Galua”, Matematicheskie voprosy kriptografii, 4:1 (2013), 27–57

[2] Ermilov D. M., “O tsiklovoi strukture polinomialnykh preobrazovanii kolets Galua maksimalnogo perioda”, Obozrenie prikl. i promyshl. matem., 20:3 (2013)