Volume 30 Issue 3
May  2021
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WANG Hongyu, ZHENG Qunxiong, WANG Zhongxiao, QI Wenfeng. Binary Sequences Derived from Dickson Permutation Polynomials over Binary Extension Field[J]. Chinese Journal of Electronics, 2021, 30(3): 523-533. doi: 10.1049/cje.2021.04.006
 Citation: WANG Hongyu, ZHENG Qunxiong, WANG Zhongxiao, QI Wenfeng. Binary Sequences Derived from Dickson Permutation Polynomials over Binary Extension Field[J]. Chinese Journal of Electronics, 2021, 30(3): 523-533.

# Binary Sequences Derived from Dickson Permutation Polynomials over Binary Extension Field

##### doi: 10.1049/cje.2021.04.006
Funds:

This work is supported by the National Natural Science Foundation of China(No.61872383), Young Elite Scientists Sponsorship Program by CAST (No.2016QNRC001), National Postdoctoral Program for Innovative Talents (No.BX201600188), and China Postdoctoral Science Foundation (No.2017M611035).

• In this paper, based on a result of Lidl and Mullen (Mathematical Journal of Okayama University, 1991), the maximum length and the second maximum length that can be attained by cycles of Dickson permutation polynomial (of the first kind) with parameter 1 are studied. Necessary and sufficient conditions for these two lengths to be attained are given, which are connected with Fermat primes and Mersenne primes, respectively. Furthermore, a class of coordinate sequences that maintains a large period is obtained, which is shown to be the coordinate sequences derived from cycles of the second maximum length. Explicit formulas for their periodicity and shift-equivalences are also presented.
•  R. Lidl and H. Niederreiter, Finite Fields, in: Encyclopedia of Mathematics and its Applications, Vol.20, Cambridge University Press, Cambridge, 1997. G.L. Mullen and D. Panario, Handbook of Finite Fields, Taylor & Francis, Boca Raton, 2013. X.D. Hou, “Permutation polynomials over finite fields – A survey of recent advances”, Finite Fields Their Appl., Vol.32, pp.82–119, 2015. Z.R. Tu and X.Y. Zeng, “A class of permutation trinomials over finite fields of odd characteristic”, Cryptogr. Commun., Vol.11, No.4, pp.563–583, 2019. Z.R. Tu, X.Y. Zeng and Y.P. Jiang, “Two classes of permutation polynomials having the form (x2m +x+δ)s +x”, Finite Fields Their Appl., Vol.53, pp.99–112, 2018. L.B. Wang and B.F. Wu, “General constructions of permutation polynomials of the form (x2m +x+δ)i(2m-1)+1+ x over F22m”, Finite Fields Their Appl., Vol.52, pp.137–155, 2018. X.T. Feng, D.D. Lin, L.P. Wang, et al., “Further results on complete permutation monomials over finite fields”, Finite Fields Their Appl., Vol.57, pp.47–59, 2019. X.F. Xu, X.T. Feng and X.Y. Zeng, “Complete permutation polynomials with the form (xpm - x +δ)s + axpm + bx over Fpn”, Finite Fields Their Appl., Vol.57, pp.309–343, 2019. B.F. Wu and D.D. Lin, “On constructing complete permutation polynomials over finite fields of even characteristic”, Discret. Appl. Math., Vol.184, pp.213–222, 2015. R. Lidl, G.L. Mullen and G. Turnwald, Dickson Polynomials, in: Pitmen Monographs and Surveys in Pure and Applied Mathematics, Vol.65, Longmen Scientific and Technical, Essex, England, 1993. S. Ahmad, “Cycle structure of automorphisms of finite cyclic groups”, Journal of Combinatorial Theory, Series A, Vol.6, No.4, pp.370–374, 1969. Q.X. Zheng, Y.P. Jiang, D.D. Lin, et al., “Binary sequences derived form monomial permutation polynomials over GF(2p)”, private communication. I. Rubio and C. Corrada, “Cyclic decomposition of permutations of finite fields obtained using monomials”, in: 7th Int. Conf. on Finite Fields and Their Applications, Springer-Verlag, pp.254–261, 2003. R. Lidl and G.L. Mullen, “Cycle structure of Dickson permutation polynomials”, Mathematical Journal of Okayama University, Vol.33, No.1, pp.1–11, 1991. W.S. Chou, “The period lengths of inversive congruential recursions”, Acta Arithmetica, Vol.73, No.4, pp.325–341, 1995. A. Çeşmelioğlu, W. Meidl and A. Topuzoğlu, “On the cycle structure of permutation polynomials”, Finite Fields Their Appl., Vol.14, No.3, pp.593–614, 2008. A. Sakzad, M.R. Sadeghi and D. Panario, “Cycle structure of permutation functions over finite fields and their applications”, Adv. Math. Commun., Vol.6, No.3, pp.347–361, 2012. H. Niederreiter, “Pseudorandom vector generation by the inversive method”, ACM Trans. Model. Comput. Simul., Vol.4, No.2, pp.191–212, 1994. S.L. Anderson, “Random number generators on vector supercomputers and other advanced architectures”, SIAM Rev., Vol.32, pp.221–251, 1990. V.C. Bhavsar and J.R. Isaac, “Design and analysis of parallel Monte Carlo algorithms”, SIAM J. Sci. Stat. Comput., Vol.8, pp.s73–s95, 1987. W.F. Eddy, “Random number generators for parallel processors”, J. Comput. Appl. Math., Vol.31, pp.63–71, 1986.

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