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doi: 10.3934/amc.2020050

Some properties of the cycle decomposition of WG-NLFSR

Yujuan Li a,, , Huaifu Wang a, , Peipei Zhou b, and  Guoshuang Zhang a,

a. 

Science and Technology on Information Assurance Laboratory, Beijing, China
b. 

Data Communication Science and Technology Research Institute, Beijing, China

* Corresponding author

Received  December 2018 Revised  September 2019 Published  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.

Keywords: Permutation, WG-NLFSR, cycle decomposition.
Mathematics Subject Classification: 94A55, 94A60.
Citation: Yujuan Li, Huaifu Wang, Peipei Zhou, Guoshuang Zhang. Some properties of the cycle decomposition of WG-NLFSR. Advances in Mathematics of Communications, 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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[4]

S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1967.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10] Z. X. Wan and Z. D. Dai, Nonlinear Feedback Shift Registers, Science Press, Beijing, 1975.   Google Scholar

show all references

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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[4]

S. W. Golomb, Shift Register Sequences, Holden-Day, San Francisco, Calif.-Cambridge-Amsterdam, 1967.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10] Z. X. Wan and Z. D. Dai, Nonlinear Feedback Shift Registers, Science Press, Beijing, 1975.   Google Scholar
Figure 1.  An Architecture of the WG-NLFSR
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