Abstract
We describe the construction of a piecewise polynomial generator over a Galois ring and prove a transitivity criterion for it. We give an estimate for the discrepancy of the output sequences of such a generator. We show that the obtained estimate is asymptotically equivalent to known estimates for special cases of a piecewise polynomial generator, and in some cases it is asymptotically sharper.
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Ermilov, D.M. and Kozlitin, O.A., Cyclic Structure of a Polynomial Generator over the Galois Ring, Mat. Vopr. Kriptogr., 2013, vol. 4, no. 1, pp. 27–57.
Solé, P. and Zinoviev, D., Inversive Pseudorandom Numbers over Galois Rings, European J. Combin., 2009, vol. 30, no. 2, pp. 458–467.
Vernigora, E.V., Inversive Congruential Generator over a Galois Ring of Characteristic pl, Ukr. Mat. Visnik, 2011, vol. 8, no. 4, pp. 607–618 [J. Math. Sci. (N.Y.) (Engl. Transl.), 2012, vol. 182, no. 1, pp. 108–116.].
Nechaev, A.A., Kerdock Code in a Cyclic Form, Diskret. Mat., 1989, vol. 1, no. 4, pp. 123–139 [Discrete Math. Appl. (Engl. Transl.), 1991, vol. 1, no. 4, pp. 365–384].
McDonald, B.R., Finite Rings with Identity, New York: Dekker, 1974.
Helleseth, T., Kumar, P.V., and Shanbhag, A.G., Exponential Sums over Galois Rings and Their Applications, Finite Fields and Applications (Proc. 3rd Int. Conf., Glasgow, Scotland, July 11–14, 1995), Cohen, S. and Niederreiter, H., Eds., Lond. Math. Soc. Lecture Notes Ser., vol. 233, Cambridge: Cambridge Univ. Press, 1996, pp. 109–128.
Glukhov, M.M., Elizarov, V.P., and Nechaev, A.A., Algebra, Moscow: Gelios ARV, 2003, Part 1.
Niederreiter, H. and Shparlinski, I.E., On the Distribution and Lattice Structure of Nonlinear Congruential Pseudorandom Numbers, Finite Fields Appl., 1999, vol. 5, no. 3, pp. 246–253.
Vasin, A.R., Bounds on the Discrepancy of Linear Recurring Sequences over Galois Rings, Diskret. Mat., 2019, vol. 31, no. 3, pp. 17–25.
Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, New York: Wiley, 1974. Translated under the title Ravnomernoe raspredelenie posledovatel’nostei, Moscow: Nauka, 1985.
Chou, W.-S., The Period Lengths of Inversive Pseudorandom Vector Generations, Finite Fields Appl., 1995, vol. 1, no. 1, pp. 126–132.
El-Mahassni, E., Exponential Sums for Nonlinear Recurring Sequences in Residue Rings, Albanian J. Math., 2010, vol. 4, no. 1, pp. 3–13.
Topuzoğlu, A. and Winterhof, A., Pseudorandom Sequences, Topics in Geometry, Coding Theory and Cryptography, García, A. and Stichtenoth, H., Eds., Dordrecht: Springer, 2007, pp. 135–166.
Niederreiter, H. and Winterhof, A., Exponential Sums for Nonlinear Recurring Sequences, Finite Fields Appl., 2008, vol. 14, no. 1, pp. 59–64.
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Russian Text © The Author(s), 2020, published in Problemy Peredachi Informatsii, 2020, Vol. 56, No. 1, pp. 99–111.
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Vasin, A.R. Piecewise Polynomial Sequences over the Galois Ring. Probl Inf Transm 56, 91–102 (2020). https://doi.org/10.1134/S0032946020010081
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DOI: https://doi.org/10.1134/S0032946020010081