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The graph structure of Chebyshev polynomials over finite fields and applications

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Abstract

We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials.

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Acknowledgements

Claudio Qureshi was supported by Fapesp grant 201526420-1 and the second author by NSERC of Canada. Most of this work was done while the Claudio Qureshi was visiting Carleton University. This author wishes to thank the Fields Institute for funding this visit.

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Correspondence to Daniel Panario.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Qureshi, C., Panario, D. The graph structure of Chebyshev polynomials over finite fields and applications. Des. Codes Cryptogr. 87, 393–416 (2019). https://doi.org/10.1007/s10623-018-0545-7

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