# American Institute of Mathematical Sciences

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

* 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.

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:

show all references

##### References:
 [1] Amin Sakzad, Mohammad-Reza Sadeghi, Daniel Panario. Cycle structure of permutation functions over finite fields and their applications. Advances in Mathematics of Communications, 2012, 6 (3) : 347-361. doi: 10.3934/amc.2012.6.347 [2] Nataša Djurdjevac Conrad, Ralf Banisch, Christof Schütte. Modularity of directed networks: Cycle decomposition approach. Journal of Computational Dynamics, 2015, 2 (1) : 1-24. doi: 10.3934/jcd.2015.2.1 [3] Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505 [4] Dmitry N. Kozlov. Cobounding odd cycle colorings. Electronic Research Announcements, 2006, 12: 53-55. [5] Ethel Mokotoff. Algorithms for bicriteria minimization in the permutation flow shop scheduling problem. Journal of Industrial & Management Optimization, 2011, 7 (1) : 253-282. doi: 10.3934/jimo.2011.7.253 [6] Ricardo P. Beausoleil, Rodolfo A. Montejo. A study with neighborhood searches to deal with multiobjective unconstrained permutation problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 193-216. doi: 10.3934/jimo.2009.5.193 [7] Nian Li, Qiaoyu Hu. A conjecture on permutation trinomials over finite fields of characteristic two. Advances in Mathematics of Communications, 2019, 13 (3) : 505-512. doi: 10.3934/amc.2019031 [8] Claudio Qureshi, Daniel Panario, Rodrigo Martins. Cycle structure of iterating Redei functions. Advances in Mathematics of Communications, 2017, 11 (2) : 397-407. doi: 10.3934/amc.2017034 [9] Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems & Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035 [10] Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549 [11] Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439 [12] Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447 [13] Yu-Hsien Chang, Guo-Chin Jau. The behavior of the solution for a mathematical model for analysis of the cell cycle. Communications on Pure & Applied Analysis, 2006, 5 (4) : 779-792. doi: 10.3934/cpaa.2006.5.779 [14] Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009 [15] Domingo Gomez-Perez, Ana-Isabel Gomez, Andrew Tirkel. Arrays composed from the extended rational cycle. Advances in Mathematics of Communications, 2017, 11 (2) : 313-327. doi: 10.3934/amc.2017024 [16] Katarzyna Pichór, Ryszard Rudnicki. Applications of stochastic semigroups to cell cycle models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2365-2381. doi: 10.3934/dcdsb.2019099 [17] Qigang Yuan, Yutong Sun, Jingli Ren. How interest rate influences a business cycle model. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020190 [18] Peter Müller, Gábor P. Nagy. On the non-existence of sharply transitive sets of permutations in certain finite permutation groups. Advances in Mathematics of Communications, 2011, 5 (2) : 303-308. doi: 10.3934/amc.2011.5.303 [19] Tim Gutjahr, Karsten Keller. Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4207-4224. doi: 10.3934/dcds.2019170 [20] Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391