Abstract
A random fault or a malicious attack can compromise the security of decryption systems. Using a stable and monotonous feedback shift register (FSR) as the main building block in a convolutional decoder can limit some error propagation. This work foucus on the synthesis of reliable (i.e., globally stable and monotonous) FSRs using the Boolean networks (BNs) method. First, we obtain an algebraic expression of the FSRs, based on which one necessary and sufficient condition for the monotonicity of the FSRs is given. Then, we obtain the upper bound of the number of cyclic attractors for monotonous FSRs. Furthermore, we propose one method of constructing n-stage reliable FSRs, and figure out that the number of reliable FSRs is \({{{2^{{2^{n - 4}}}}} \over {\phi (n)}}\) (n > 5) times of that constructed by the existing method, where ϕ denotes the Euler’s totient function. Finally, the proposed method and the obtained results are verified by some examples.
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References
Massey J, Liu R. Application of Lyapunov’s direct method to the error-propagation effect in convolutional codes. IEEE Trans Inform Theor, 1964, 10: 248–250
Wan Z X, Dai Z D, Liu M L, et al. Nonlinear Feedback Shift Registers (in Chinese). Beijing: Science Press, 1978
Ma Z, Qi W F, Tian T. On the decomposition of an NFSR into the cascade connection of an NFSR into an LFSR. J Complexity, 2013, 29: 173–181
Zhang J M, Qi W F, Tian T, et al. Further results on the decomposition of an NFSR into the cascade connection of an NFSR into an LFSR. IEEE Trans Inform Theor, 2015, 61: 645–654
Kauffman S A. Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol, 1969, 22: 437–467
Zhang Y, Liu Y. Nonlinear second-order multi-agent systems subject to antagonistic interactions without velocity constraints. Appl Math Comput, 2020, 364: 124667
Li N, Cao J D. Synchronization criteria for multiple memristor-based neural networks with time delay and inertial term. Sci China Tech Sci, 2018, 61: 612–622
Cheng D Z, Qi H S, Li Z Q. Analysis and Control of Boolean Networks: A Semi-tensor Product Approach. Berlin: Springer, 2010
Cheng D Z, Qi H S. Controllability and observability of Boolean control networks. Automatica, 2009, 45: 1659–1667
Liang S, Li H T, Wang S L. Structural controllability of Boolean control networks with an unknown function structure. Sci China Inf Sci, 2020, 63: 219203
Hou B Y, Li X, Chen G R. Structural controllability of temporally switching networks. IEEE Trans Circ Syst I, 2016, 63: 1771–1781
Zhu S Y, Liu Y, Lou Y J, et al. Stabilization of logical control networks: an event-triggered control approach. Sci China Inf Sci, 2020, 63: 112203
Xu M X, Liu Y, Lou J G, et al. Set stabilization of probabilistic Boolean control networks: a sampled-data control approach. IEEE Trans Cybern, 2020, 50: 3816–3823
Lin L, Cao J D, Rutkowski L. Robust event-triggered control invariance of probabilistic Boolean control networks. IEEE Trans Neural Netw Learn Syst, 2020, 31: 1060–1065
Zhong J, Li B W, Liu Y, et al. Output feedback stabilizer design of Boolean networks based on network structure. Front Inform Technol Electron Eng, 2020, 21: 247–259
Liu A X, Li H T. On feedback invariant subspace of Boolean control networks. Sci China Inf Sci, 2020, 63: 229201
Zhu Q X, Liu Y, Lu J Q, et al. Observability of Boolean control networks. Sci China Inf Sci, 2018, 61: 092201
Liu H C, Liu Y, Li Y Y, et al. Observability of Boolean networks via STP and graph methods. IET Control Theor Appl, 2018, 13: 1031–1037
Chen H W, Liang J L, Huang T W, et al. Synchronization of arbitrarily switched Boolean networks. IEEE Trans Neural Netw Learn Syst, 2017, 28: 612–619
Zhao J T, Chen Z Q, Liu Z X. Modeling and analysis of colored petri net based on the semi-tensor product of matrices. Sci China Inf Sci, 2018, 61: 010205
Li H T, Wang Y Z. Boolean derivative calculation with application to fault detection of combinational circuits via the semitensor product method. Automatica, 2012, 48: 688–693
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
Liu Z B, Wang Y Z, Cheng D Z. Nonsingularity of feedback shift registers. Automatica, 2015, 55: 247–253
Zhong J H, Lin D D. Stability of nonlinear feedback shift registers. Sci China Inf Sci, 2016, 59: 012204
Li B W, Lu J Q. Boolean-network-based approach for construction of filter generators. Sci China Inf Sci, 2020, 63: 212206
Lu J Q, Li M L, Liu Y, et al. Nonsingularity of Grain-like cascade FSRs via semi-tensor product. Sci China Inf Sci, 2018, 61: 010204
Dubrova E. Finding matching initial states for equivalent NLFSRs in the Fibonacci and the Galois configurations. IEEE Trans Inform Theor, 2010, 56: 2961–2966
Lu J Q, Li M L, Huang T W, et al. The transformation between the Galois NLFSRs and the Fibonacci NLFSRs via semi-tensor product of matrices. Automatica, 2018, 96: 393–397
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
Zhong J H, Lin D D. On minimum period of nonlinear feedback shift registers in Grain-like structure. IEEE Trans Inform Theor, 2018, 64: 6429–6442
Golomb S. Shift Register Sequences. Laguna Hills: Aegean Park Press, 1982
Mowle F J. An algorithm for generating stable feedback shift registers of order n. J ACM, 1967, 14: 529–542
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 61973078, 61903339). Natural Science Foundation of Jiangsu Province of China (Grant No. BK20170019), Jiangsu Province Six Talent Peaks Project (Grant No. 2015-ZNDW-002), Fundamental Research Funds for the Central Universities (Grant No. 2242019k1G013), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX19_0111).
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Lu, J., Li, B. & Zhong, J. A novel synthesis method for reliable feedback shift registers via Boolean networks. Sci. China Inf. Sci. 64, 152207 (2021). https://doi.org/10.1007/s11432-020-2981-4
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DOI: https://doi.org/10.1007/s11432-020-2981-4