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Nonsingularity of Grain-like cascade FSRs via semi-tensor product

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  • Special Focus on Analysis and Control of Finite-Valued Network Systems
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Abstract

In this paper, Grain-like cascade feedback shift registers (FSRs) are regarded as two Boolean networks (BNs), and the semi-tensor product (STP) of the matrices is used to convert the Grain-like cascade FSRs into an equivalent linear equation. Based on the STP, a novel method is proposed herein to investigate the nonsingularity of Grain-like cascade FSRs. First, we investigate the property of the state transition matrix of Grain-like cascade FSRs. We then propose their sufficient and necessary nonsingularity condition. Next, we regard the Grain-like cascade FSRs as Boolean control networks (BCNs) and further provide a sufficient condition of their nonsingularity. Finally, two examples are provided to illustrate the results obtained in this paper.

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References

  1. Goresky M, Klapper A. Algebraic Shift Register Sequences. Cambridge: Cambridge University Press, 2012

    MATH  Google Scholar 

  2. Golomb S W. Shift Register Sequences. Walnut Creek: Aegean Park Press, 1982

    MATH  Google Scholar 

  3. Goresky M, Klapper A. Pseudonoise sequences based on algebraic feedback shift registers. IEEE Trans Inf Theory, 2006, 52: 1649–1662

    Article  MATH  Google Scholar 

  4. Li C Y, Zeng X Y, Helleseth T, et al. The properties of a class of linear FSRs and their applications to the construction of nonlinear FSRs. IEEE Trans Inf Theory, 2014, 60: 3052–3061

    Article  MathSciNet  MATH  Google Scholar 

  5. Massey J. Shift-register synthesis and BCH decoding. IEEE Trans Inf Theory, 1969, 15: 122–127

    Article  MathSciNet  MATH  Google Scholar 

  6. Meier W, Staffelbach O. Fast correlation attacks on certain stream ciphers. J Cryptology, 1989, 1: 159–176

    Article  MathSciNet  MATH  Google Scholar 

  7. Hell M, Johansson T, Meier M. Grain: a stream cipher for constrained environments. Int J Wirel Mobile Comput, 2007, 2: 86–93

    Article  Google Scholar 

  8. Gammel B M, Gottfert R, Kniffler O. An NLFSR-based stream cipher. In: Proceedings of IEEE International Symposium on Circuits and Systems, Island of Kos, 2006

    Google Scholar 

  9. Chen K, Henricksen M, Millan W, et al. Dragon: a fast word based stream cipher. In: Proceedings of International Conference on Information Security and Cryptology. Berlin: Springer, 2004. 33–50

    Google Scholar 

  10. Gammel B, Göttfert R, Kniffler O. Achterbahn-128/80: design and analysis. ECRYPT Network of Excellence–SASC Workshop Record, 2007. https://www.cosic.esat.kuleuven.be/ecrypt/stream/papersdir/2007/020.pdf

    Google Scholar 

  11. Courtois N T, Meier W. Algebraic attacks on stream ciphers with linear feedback. In: Proceedings of International Conference on the Theory and Applications of Cryptographic Techniques. Berlin: Springer, 2003. 345–359

    Google Scholar 

  12. Robshaw M, Matsumoto M, Saito M, et al. New Stream Cipher Designs: the eSTREAM Finalists. Berlin: Springer, 2008

    Book  MATH  Google Scholar 

  13. Hell M, Johansson T, Maximov A. The grain family of stream ciphers. Lect Notes Comput Sci, 2008, 4986: 179–190

    Article  Google Scholar 

  14. Babbage S, Dodd M. The MICKEY Stream Ciphers. Berlin: Springer, 2008

    Book  Google Scholar 

  15. Maximov A. Cryptanalysis of the “Grain” family of stream ciphers. In: Proceedings of the 2006 ACM Symposium on Information, Computer and Communications Security, Taipei, 2006. 283–288

    Google Scholar 

  16. Berbain C, Gilbert H, Joux A. Algebraic and correlation attacks against linearly filtered non linear feedback shift registers. In: Proceedings of the 15th International Workshop on Selected Areas in Cryptography, Sackville, 2008. 184–198

    Google Scholar 

  17. Hu H G, Gong G. Periods on two kinds of nonlinear feedback shift registers with time varying feedback functions. Int J Found Comput Sci, 2011, 22: 1317–1329

    Article  MathSciNet  MATH  Google Scholar 

  18. Cheng D Z, Qi H S, Li Z Q. Analysis and Control of Boolean Networks. Berlin: Springer, 2011

    Book  MATH  Google Scholar 

  19. Li H T, Zhao G D, Meng M, et al. A survey on applications of semi-tensor product method in engineering. Sci China Inf Sci, 2018, 61: 010202

    Article  Google Scholar 

  20. Lu J Q, Li H T, Liu Y, et al. Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems. IET Control Theory Appl, 2017, 11: 2040–2047

    Article  Google Scholar 

  21. Lu J Q, Zhong J, Ho D W C, et al. On controllability of delayed Boolean control networks. SIAM J Control Optim, 2016, 54: 475–494

    Article  MathSciNet  MATH  Google Scholar 

  22. Liu Y, Chen H W, Wu B. Controllability of Boolean control networks with impulsive effects and forbidden states. Math Method Appl Sci, 2014, 37: 1–9

    Article  MATH  Google Scholar 

  23. Zhu Q X, Liu Y, Lu J Q, et al. Observability of Boolean control networks. Sci China Inf Sci, 2018, 61: 092201. doi: 10.1007/s11432-017-9135-4

    Article  MathSciNet  Google Scholar 

  24. Lu J Q, Zhong J, Huang C, et al. On pinning controllability of Boolean control networks. IEEE Trans Autom Control, 2016, 61: 1658–1663

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhong J, Lu J Q, Liu Y, et al. Synchronization in an array of output-coupled Boolean networks with time delay. IEEE Trans Neural Netw Learn Syst, 2014, 25: 2288–2294

    Article  Google Scholar 

  26. Liu Y, Li B W, Lu J Q, et al. Pinning control for the disturbance decoupling problem of Boolean networks. IEEE Trans Autom Control, 2017. doi:10.1109/TAC.2017.2715181

    Google Scholar 

  27. Liu Y, Sun L J, Lu J Q, et al. Feedback controller design for the synchronization of Boolean control networks. IEEE Trans Neural Netw Learn Syst, 2016, 27: 1991–1996

    Article  MathSciNet  Google Scholar 

  28. Li F F, Sun J T. Controllability of Boolean control networks with time delays in states. Automatica, 2011, 47: 603–607

    Article  MathSciNet  MATH  Google Scholar 

  29. Cheng D Z, Qi H S. Controllability and observability of Boolean control networks. Automatica, 2009, 45: 1659–1667

    Article  MathSciNet  MATH  Google Scholar 

  30. Laschov D, Margaliot M. Controllability of Boolean control networks via the perron-frobenius theory. Automatica, 2012, 48: 1218–1223

    Article  MathSciNet  MATH  Google Scholar 

  31. Li H T, Wang Y Z, Xie L H. Output tracking control of Boolean control networks via state feedback: constant reference signal case. Automatica, 2015, 59: 54–59

    Article  MathSciNet  MATH  Google Scholar 

  32. Cheng D Z, Qi H S, Li Z Q, et al. Stability and stabilization of Boolean networks. Int J Robust Nonlinear Control, 2011, 21: 134–156

    Article  MathSciNet  MATH  Google Scholar 

  33. Li H T, Wang Y Z. Controllability analysis and control design for switched Boolean networks with state and input constraints. SIAM J Control Optim, 2015, 53: 2955–2979

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhong J, Lu J Q, Huang T W, et al. Controllability and synchronization analysis of identical-hierarchy mixed-valued logical control networks. IEEE Trans Cybern, 2017, 47: 3482–3493

    Article  Google Scholar 

  35. Guo P L, Wang Y Z, Li H T. A semi-tensor product approach to finding Nash equilibria for static games. In: Proceedings of the 32nd Chinese Control Conference (CCC), Xi’an, 2013. 107–112

    Google Scholar 

  36. Li H T, Xie L H, Wang Y Z. On robust control invariance of Boolean control networks. Automatica, 2016, 68: 392–396

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhong J H, Lin D D. A new linearization method for nonlinear feedback shift registers. J Comput Syst Sci, 2014, 81: 783–796

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhong J H, Lin D D. Stability of nonlinear feedback shift registers. Sci China Inf Sci, 2016, 59: 012204

    Google Scholar 

  39. Zhong J H, Lin D D. Driven stability of nonlinear feedback shift registers with inputs. IEEE Trans Commun, 2016, 64: 2274–2284

    Article  Google Scholar 

  40. Zhong J, Ho D W C, Lu J Q, et al. Global robust stability and stabilization of Boolean network with disturbances. Automatica, 2017, 84: 142–148

    Article  MathSciNet  MATH  Google Scholar 

  41. Liu Y, Cao J D, Sun L J, et al. Sampled-data state feedback stabilization of Boolean control networks. Neural Comput, 2016, 28: 778–799

    Article  Google Scholar 

  42. Wu H J, Huang T, Nguyen P H, et al. Differential attacks against stream cipher ZUC. In: Proceedings of the 18th International Conference on the Theory and Application of Cryptology and Information Security, Beijing, 2012. 262–277

    Google Scholar 

  43. Lai X J. Condition for the nonsingularity of a feedback shift-register over a general finite field (corresp.). IEEE Trans Inf Theory, 1987, 33: 747–749

    Article  MATH  Google Scholar 

  44. Wang Q Y, Jin C H. Criteria for nonsingularity of Grain-like cascade feedback shift register (in Chinese). Comput Eng, 2014, 40: 519–523

    Google Scholar 

  45. Girard J Y. Linear logic. Theor Comput Sci, 1987, 50: 1–101

    Article  MathSciNet  MATH  Google Scholar 

  46. Liu Z B, Wang Y Z, Cheng D Z. Nonsingularity of feedback shift registers. Automatica, 2015, 55: 247–253

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 61573102, 11671361), Natural Science Foundation of Jiangsu Province of China (Grant No. BK20170019), Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (Grant No. BM2017002), China Postdoctoral Science Foundation (Grant Nos. 2014M560377, 2015T80483), Jiangsu Province Six Talent Peaks Project (Grant No. 2015-ZNDW-002), and Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, China

    Jianquan Lu & Meilin Li

  2. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, China

    Jianquan Lu & Yang Liu

  3. Department of Mathematics, Hong Kong, China

    Daniel W.C. Ho

  4. Potsdam Institute for Climate Impact Research, Potsdam, 14415, Germany

    Jürgen. Kurths

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  1. Jianquan Lu
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  2. Meilin Li
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  4. Daniel W.C. Ho
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  5. Jürgen. Kurths
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Correspondence to Jianquan Lu.

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Lu, J., Li, M., Liu, Y. et al. Nonsingularity of Grain-like cascade FSRs via semi-tensor product. Sci. China Inf. Sci. 61, 010204 (2018). https://doi.org/10.1007/s11432-017-9269-6

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