Abstract
Nonlinear feedback shift registers (NFSRs) have been used in many recent stream ciphers. They are generally classified as Fibonacci NFSRs and Galois NFSRs in terms of their implementation configurations. Two NFSRs are said to be isomorphic if their state diagrams are isomorphic, and two NFSRs are equivalent if their sets of output sequences are equal. Equivalent NFSRs must be isomorphic NFSRs, but not the vice versa. Previous work has been done on the isomorphism and equivalence of Fibonacci NFSRs. This paper continues this research for Galois NFSRs. It first gives some characterizations for several kinds of isomorphic Galois NFSRs, which improves and generalizes the previous corresponding results for Fibonacci NFSRs. It then presents some characterizations for two kinds of equivalent Galois NFSRs, helpful to the design of NFSR-based stream ciphers.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Hell, M., Johansson, T., Maximov, A., Meier, W.: The grain family of stream ciphers. In: Robshaw, M., Billet, O. (eds.) New Stream Cipher Designs. LNCS, vol. 4986, pp. 179–190. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68351-3_14
De Cannière, C., Preneel, B.: Trivium. In: Robshaw, M., Billet, O. (eds.) New Stream Cipher Designs. LNCS, vol. 4986, pp. 244–266. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-68351-3_18
Dubrova, E.: A transformation from the Fibonacci to the Galois NLFSRs. IEEE Trans. Inf. Theory 55(11), 5263–5271 (2009)
Cheng D., Qi H., Li Z.: Analysis and Control of Boolean Networks. Springer, London (2011) https://doi.org/10.1007/978-0-85729-097-7
Cheng D., Qi H., Zhao Y.: An Introduction To Semi-Tensor Product of Matrices And Its Applications. World Scientific Publishing Company, Singapore (2012)
Zhao, D., Peng, H., Li, L., Hui, S., Yang, Y.: Novel way to research nonlinear feedback shift register. Sci. China Inf. Sci. 57(9), 1–14 (2014)
Zhong, J., Lin, D.: Driven stability of nonlinear feedback shift registers. IEEE Trans. Commun. 64(6), 2274–2284 (2016)
Zhong, J., Lin, D.: On minimum period of nonlinear feedback shift registers in Grainlike structure. IEEE Trans. Inf. Theory 64(9), 6429–6442 (2018)
Wan Z., Dai Z., Liu M. et al.: Nonlinear Shift Register (in Chinese), Science Press, Beijing, China (1978)
Zhong, J., Lin, D.: A new linearization method of nonlinear feedback shift registers. J. Comput. Syst. Sci. 81(4), 783–796 (2015)
Zhao, X.-X., Zheng, Q.-X., Wang, Z.-X., Qi, W.-F.: On a class of isomorphic NFSRs. Des. Codes Cryptogr. 88(6), 1205–1226 (2020)
Dubrova, E.: Finding matching initial states for equivalent NLFSRs in the Fibonacci and the Galois configurations. IEEE Trans. Inf. Theory 56(6), 2961–2966 (2010)
Lin Z.: The transformation from the Galois NLFSR to the Fibonacci Configuration. In: EI-DWT 2013, USA, NJ, Piscataway: IEEE Press, pp. 335–339 (2013)
Mykkeltveit, J., Siu, M.-K., Ton, P.: On the cylcle structure of some nonlinear shift register sequences. Inf. Control 43(2), 202–215 (1979)
Zhong, J., Pan, Y., Lin, D.: On Galois NFSRs equivalent to Fibonacci ones. In: Wu, Y., Yung, M. (eds.) Inscrypt 2020. LNCS, vol. 12612, pp. 433–449. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-71852-7_29
Pan Y., Zhong J. and Lin D.: On Galois NFSRs with terminal bits. In: 2021 IEEE International Symposium on Information Theory (ISIT 2021), to appear
Roger A.H., Johnson C.R.: Topics in Matrix Analysis. Cambridge University Press, UK (1991)
Qi, H., Cheng, D.: Logic and logic-based control. J. Contr. Theory Appl. 6(1), 123–133 (2008)
Barbier, M., Cheballah, H., Le Bars, J.-M.: On the computation of the Mobius transform. Theor. Comput. Sci. 809, 171–188 (2020)
Golomb S. W.: Shift Register Sequences. Holden-Day, Laguna Hills, CA, USA (1967)
Zhong J. and Lin D.: Decomposition of nonlinear feedback shift registers based on Boolean networks. Sci. China Inf. Sci. 62(3), 39110:1–39110:3 (2019)
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61772029 and 61872359.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Kong, W., Zhong, J., Lin, D. (2021). Isomorphism and Equivalence of Galois Nonlinear Feedback Shift Registers. In: Yu, Y., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2021. Lecture Notes in Computer Science(), vol 13007. Springer, Cham. https://doi.org/10.1007/978-3-030-88323-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-88323-2_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-88322-5
Online ISBN: 978-3-030-88323-2
eBook Packages: Computer ScienceComputer Science (R0)