Skip to main content
SpringerLink
Log in

Sequences with ideal auto-correlation derived from group actions

  • Research
  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Bent functions have a number of practical applications in cryptography, coding theory, and other fields. Fourier transform is a key tool to study bent functions on finite abelian groups. Using Fourier transforms, in this paper, we first present two necessary and sufficient conditions on the existence of bent functions via faithful actions of finite abelian groups and then show two constructions of sequences with ideal auto-correlation (SIACs). In addition, we construct a periodic complementary sequence set (PCSS) by rearranging a periodic multiple shift sequence (PMSS) corresponding to a bent function on a finite abelian group. Some concrete constructions of SIACs and PCSSs are provided to illustrate the efficiency of our methods.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

No datasets were generated or analysed during the current study.

References

  1. Mesnager, S.: Bent functions fundamentals and results. Springer, New York (2016)

    Book  Google Scholar 

  2. Rothaus, O.S.: On bent functions. J. Comb. Theory Ser. A. 20, 300–305 (1976)

    Article  Google Scholar 

  3. Logachev, O.A., Salnikov, A.A., Yashchenko, V.V.: Bent functions over a finite abelian group. Discret. Math. Appl. 7, 547–564 (1997)

    Article  Google Scholar 

  4. Poinsot, L.: Bent functions on a finite nonabelian group. J. Discret. Math. Sci. Cryptogr. 9, 349–364 (2006)

    Article  MathSciNet  Google Scholar 

  5. Solodovnikov, V.I.: Bent functions from a finite abelian group into a finite abelian group. Discret. Math. Appl. 12, 111–126 (2002)

    Article  MathSciNet  Google Scholar 

  6. Poinsot, L.: A new characterization of group action-based perfect nonlinearity. Discret. Appl. Math. 157, 1848–1857 (2009)

    Article  MathSciNet  Google Scholar 

  7. Poinsot, L., Harari, S.: Group actions based perfect nonlinearity. GESTS Int. Trans. Comput. Sci. Eng. 12, 1–14 (2005)

    Google Scholar 

  8. Fan, Y., Xu, B.: Fourier transforms and bent functions on faithful actions of finite abelian groups. Des. Codes Cryptogr. 82, 543–558 (2017)

    Article  MathSciNet  Google Scholar 

  9. Fan, Y., Xu, B.: Fourier transforms and bent functions on finite groups. Des. Codes Cryptogr. 86, 2091–2113 (2018)

    Article  MathSciNet  Google Scholar 

  10. Xu, B.: Absolute Maximum nonlinear functions on finite nonabelian groups. IEEE Trans. Inf. Theory. 66, 5167–5181 (2020)

    Article  MathSciNet  Google Scholar 

  11. Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complex. 20, 205–244 (2004)

    Article  MathSciNet  Google Scholar 

  12. Poinsot, L., Pott, A.: Non-boolean almost perfect nonlinear functions on non-abelian groups. Int. J. Found. Comput. Sci. 22, 1351–1367 (2011)

    Article  MathSciNet  Google Scholar 

  13. Xu, B.: Multidimensional Fourier transforms and nonlinear functions on finite groups. Linear Algebra Appl. 450, 89–105 (2014)

    Article  MathSciNet  Google Scholar 

  14. Xu, B.: Dual bent functions on finite groups and C-algebras. J. Pure Appl. Algebr. 220, 1055–1073 (2016)

    Article  MathSciNet  Google Scholar 

  15. Xu, B.: Bentness and nonlinearity of functions on finite groups. Des. Codes Cryptogr. 76, 409–430 (2015)

    Article  MathSciNet  Google Scholar 

  16. Golay, M.: Complementary series. IRE Trans. Inf. Theory. 7, 82–87 (1961)

    Article  MathSciNet  Google Scholar 

  17. Tseng, C.-C., Liu, C.: Complementary sets of sequences. IEEE Trans. Inf. Theory. 18, 644–652 (1972)

    Article  MathSciNet  Google Scholar 

  18. Bömer, L., Antweiler, M.: Periodic complementary binary sequences. IEEE Trans. Inf. Theory. 36, 1487–1494 (1990)

    Article  MathSciNet  Google Scholar 

  19. Feng, K., Jau-Shyong, P., Xiang, Q.: On aperiodic and periodic complementary binary sequences. IEEE Trans. Inf. Theory. 45, 296–303 (1999)

    Article  MathSciNet  Google Scholar 

  20. Chen, H.-H., Hank, D., Maganaz, M.E., Guizani, M.: Design of next-generation CDMA using orthogonal complementary codes and offset stacked spreading. IEEE Wirel. Commun. 14, 6–69 (2007)

    Article  Google Scholar 

  21. Davis, J.A., Jedwab, J.: Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Müller codes. IEEE Trans. Inf. Theory. 45, 2397–2417 (1999)

    Article  Google Scholar 

  22. Zeng, F.X., Zeng, X.P., Zhang, Z.Y., Xuan, G.X.: Perfect 16-QAM sequences and arrays. IEICE Trans. Fundamentals. 95, 1740–1748 (2012)

    Article  Google Scholar 

  23. Dokovic, D.Z.: Note on periodic complementary sets of binary sequences. Des. Codes Cryptogr. 13, 251–256 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The support of the National Natural Science Foundation of China (through grant no. 12171241) and Natural Science Foundation of Jiangsu Province (through grant no. BK20230867) are greatly acknowledged.

Author information

Authors and Affiliations

  1. College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, 211106, People’s Republic of China

    Hongyang Xiao & Xiwang Cao

  2. Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA),MIIT, College of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, People’s Republic of China

    Xiwang Cao

Authors
  1. Hongyang Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiwang Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Hongyang Xiao wrote the main manuscript and Xiwang Cao provided guidance for this manuscript. Both authors reviewed the manuscript.

Corresponding author

Correspondence to Xiwang Cao.

Ethics declarations

Competing interests

The authors declare no competing interests.

Human and animal rights

We would like to mention that this article does not contain any studies with animals and does not involve any studies over human being.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xiao, H., Cao, X. Sequences with ideal auto-correlation derived from group actions. Cryptogr. Commun. (2024). https://doi.org/10.1007/s12095-024-00710-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12095-024-00710-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation