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. 
@article{PDMA_2014_7_a27,
     author = {D. M. Ermilov},
     title = {Algorithm for constructing the system of representatives of maximal length cycles of polynomial substitution over the {Galois} ring},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {64--67},
     publisher = {mathdoc},
     number = {7},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2014_7_a27/}
}
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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/