On the Nth maximum order complexity and the expansion complexity of a Rudin-Shapiro-like sequence


Based on the parity of the number of occurrences of a pattern 10 as a scattered subsequence in the binary representation of integers, a Rudin-Shapiro-like sequence is defined by Lafrance, Rampersad and Yee. The N th maximum order complexity and the expansion complexity of this Rudin-Shapiro-like sequence are calculated in this paper.

This is a preview of subscription content, log in to check access.


  1. 1.

    Allouche, J.P., sequences, J. Shallit.: Automatic Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  2. 2.

    Allouche, J.P., Liardet, P.: Generalized Rudin-Shapiro sequences. Acta Arithmet LX.1, 1–27 (1991)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Allouche, J.P.: On a Golay-Shapiro-like sequence. Unif. Distrib. Theory 11(2), 205–210 (2016)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Brillhart, J., Morton, P.: A case study in mathematical research: the Golay-Rudin-Shapiro sequence. Am. Math. Mon. 103(10), 854–869 (1996)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chan, L., Grimm, U.: Spectrum of a Rudin-Shapiro-like sequence. Adv. Appl. Math. 87, 16–23 (2017)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Diem, C.: On the use of expansion series for stream ciphers. LMS J. Comput. Math. 15, 326–340 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Jansen, C.J.A.: Investigations on Nonlinear Streamcipher Systems: Construction and Evaluation Methods. Ph.D.dissertation, Technical University of Delft, Delft (1989)

  8. 8.

    Jansen, C.J.A.: The Maximum Order Complexity of Sequence Ensembles. In: Davies, D.W. (ed.) Advances in Cryptology - EUROCRYPT ’91, Lect. Notes Comput. Sci., vol. 547, pp 153–159. Springer, Berlin (1991)

  9. 9.

    Lafrance, P., Rampersad, N., Yee, R.: Some properties of a Rudin-Shapiro-like sequence. Adv. Appl. Math. 63, 19–40 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Mauduit, C., Sárközy, A.: On finite pseudorandom binary sequences II. The Champernowne, Rudin-Shapiro, and Thue-Morse sequences, a further construction. J. Number Theory 73(2), 256–276 (1998)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Mérai, L., Niederreiter, H., Winterhof, A.: Expansion complexity and linear complexity of sequences over finite fields. Cryptogr. Commun. 9, 501–509 (2017)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Mérai, L., Winterhof, A.: On the N th linear complexity of automatic sequences. J. Number Theory 187, 415–429 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Müllner, C.: The Rudin-Shapiro sequence and similar sequences are normal along squares. Can. J. Math. 70(5), 1096–1129 (2018)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Sun, Z., Winterhof, A.: On the maximum order complexity of the Thue-Morse sequence and the Rudin-Shapiro sequence. Preprint (2017)

  15. 15.

    Sun, Z., Winterhof, A.: On the maximum order complexity of subsequences of the Thue-Morse sequence and the Rudin-Shapiro sequence along squares. Int. J. Comput. Math. Comput. Syst. Theory 4(1), 30–36 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Xing, C.P., Lam, K.Y.: Sequence with almost perfect linear complexity profiles and curves over finite fields. IEEE Trans. Inf. Theory 45(4), 1267–1270 (1999)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Xiangyong Zeng.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are supported by the National Natural Science Foundation of China Grant 61472120. The first author is also supported by China Scholarship Council.

This article is part of the Topical Collection on Special Issue on Sequences and Their Applications

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sun, Z., Zeng, X. & Lin, D. On the Nth maximum order complexity and the expansion complexity of a Rudin-Shapiro-like sequence. Cryptogr. Commun. 12, 415–426 (2020). https://doi.org/10.1007/s12095-019-00396-0

Download citation


  • Rudin-Shapiro-like sequence
  • Maximum order complexity
  • Expansion complexity

Mathematics Subject Classification (2010)

  • 11B50
  • 11B85
  • 11K45