# American Institute of Mathematical Sciences

February  2021, 15(1): 155-165. doi: 10.3934/amc.2020050

## Some properties of the cycle decomposition of WG-NLFSR

 a. Science and Technology on Information Assurance Laboratory, Beijing, China b. Data Communication Science and Technology Research Institute, Beijing, China

Received  December 2018 Revised  September 2019 Published  February 2021 Early access  January 2020

In this paper, we give some properties of the cycle decomposition of a nonlinear feedback shift register called WG-NLFSR which was presented by Mandal and Gong recently. First we give the parity of the state transition transformation of WG-NLFSR and then by the relation of the parity of a permutation and its number of cycles given in Theorem 2 in Section 1, we show that the number of cycles in the cycle decomposition of WG-NLFSR is even. Second we study the properties of the cycle decomposition of WG-NLFSR when the coefficients of the characteristic polynomial belong to the proper subfields of the finite field on which the WG-NLFSR is defined. Finally, we give some properties of the cycle decomposition of the filtering WG7-NLFSR.

Citation: Yujuan Li, Huaifu Wang, Peipei Zhou, Guoshuang Zhang. Some properties of the cycle decomposition of WG-NLFSR. Advances in Mathematics of Communications, 2021, 15 (1) : 155-165. doi: 10.3934/amc.2020050
##### References:
 [1] P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra, 2nd Edition, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9781139174237. [2] U. Cheng, On the cycle structure of certain classes of nonlinear shift registers, Journal of Combinatorial Theory, 37 (1984), 61-68.  doi: 10.1016/0097-3165(84)90019-0. [3] H. Dobbertin, Kasami power functions, permutation polynomials and cyclic difference sets, Difference Sets, Sequences and Their Correlation Properties (Bad Windsheim, 1998), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 542 (1999), 133-158. [4] S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1967. [5] G. Gong and A. M. Youssef, Cryptographic properties of the Welch-Gong transformation sequence generators, IEEE Transactions on Information Theory, 48 (2002), 2837-2846.  doi: 10.1109/TIT.2002.804043. [6] T. Helleseth, Nonlinear shift registers - A survey and challenges, Algebraic Curves and Finite Fields, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 16 (2014), 121-144. [7] K. Kjeldsen, On the cycle structure of a set of nonlinear shift registers with symmetric feedback functions, Journal of Combinatorial Theory Ser. A, 20 (1976), 154-169.  doi: 10.1016/0097-3165(76)90013-3. [8] K. Mandal and G. Gong, Filtering nonlinear feedback shift registers using Welch-Gong transformations for securing RFID applications, ICST Trans. Security Safety, 3 (2016), e3. [9] J. Mykkeltveit, M. K. Siu and P. Tong, On the cycle structure of some nonlinear shift register sequences, Information and Control, 43 (1979), 202-215.  doi: 10.1016/S0019-9958(79)90708-3. [10] Z. X. Wan and Z. D. Dai, Nonlinear Feedback Shift Registers, Science Press, Beijing, 1975.

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##### References:
 [1] P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra, 2nd Edition, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9781139174237. [2] U. Cheng, On the cycle structure of certain classes of nonlinear shift registers, Journal of Combinatorial Theory, 37 (1984), 61-68.  doi: 10.1016/0097-3165(84)90019-0. [3] H. Dobbertin, Kasami power functions, permutation polynomials and cyclic difference sets, Difference Sets, Sequences and Their Correlation Properties (Bad Windsheim, 1998), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 542 (1999), 133-158. [4] S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1967. [5] G. Gong and A. M. Youssef, Cryptographic properties of the Welch-Gong transformation sequence generators, IEEE Transactions on Information Theory, 48 (2002), 2837-2846.  doi: 10.1109/TIT.2002.804043. [6] T. Helleseth, Nonlinear shift registers - A survey and challenges, Algebraic Curves and Finite Fields, Radon Ser. Comput. Appl. Math., De Gruyter, Berlin, 16 (2014), 121-144. [7] K. Kjeldsen, On the cycle structure of a set of nonlinear shift registers with symmetric feedback functions, Journal of Combinatorial Theory Ser. A, 20 (1976), 154-169.  doi: 10.1016/0097-3165(76)90013-3. [8] K. Mandal and G. Gong, Filtering nonlinear feedback shift registers using Welch-Gong transformations for securing RFID applications, ICST Trans. Security Safety, 3 (2016), e3. [9] J. Mykkeltveit, M. K. Siu and P. Tong, On the cycle structure of some nonlinear shift register sequences, Information and Control, 43 (1979), 202-215.  doi: 10.1016/S0019-9958(79)90708-3. [10] Z. X. Wan and Z. D. Dai, Nonlinear Feedback Shift Registers, Science Press, Beijing, 1975.
An Architecture of the WG-NLFSR
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