Skip to main content
SpringerLink
Log in

LCD codes and almost optimally extendable codes from self-orthogonal codes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

LCD codes and (almost) optimally extendable codes can be used to safeguard against fault injection attacks (FIA) and side-channel attacks (SCA) in the implementations of block ciphers. The first objective of this paper is to use a family of binary self-orthogonal codes given by Ding and Tang (Cryptogr Commun 12:1011–1033, 2020) to construct a family of binary LCD codes with new parameters. The parameters of the binary LCD codes and their duals are explicitly determined. It turns out that the codes by Ding and Tang are almost optimally extendable codes. The second objective is to prove that two families of known q-ary linear codes given by Heng et al. (IEEE Trans Inf Theory 66(11):6872–6883, 2020) are self-orthogonal. Using these two families of self-orthogonal codes, we construct another two families of q-ary LCD codes. The parameters of the LCD codes are determined and many optimal codes are produced. Besides, the two known families of q-ary linear codes are also proved to be almost optimally extendable codes.

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

Not applicable.

References

  1. Alahmadi A., Altassan A., AlKenani A., Çalkavur S., Shoaib H., Solé P.: A multisecret-sharing scheme based on LCD codes. Mathematics 8(2), 272 (2020).

    Article  Google Scholar 

  2. Bringer J., Carlet C., Chabanne H., Guilley S., Maghrebi H.: Orthogonal direct sum masking a smartcard friendly computation paradigm in a code, with builtin protection against side-channel and fault attacks. In: Naccache D., Sauveron D. (eds.) WISTP, Heraklion, LNCS, vol. 8501, pp. 40–56. Springer, Heidelberg (2014).

    Google Scholar 

  3. Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. Adv. Math. Commun. 10(1), 131–150 (2016).

    Article  MathSciNet  Google Scholar 

  4. Carlet C., Guilley S.: Complementary dual codes for counter-measures to side-channel attacks. In: Pinto E.R., et al. (eds.) Coding Theory and Applications, CIM Series in Mathematical Sciences, pp. 97–105. Springer Verlag (2014).

  5. Carlet C., Güneri C., Mesnager S., Ózbudak F.: Construction of some codes suitable for both side channel and fault injection attacks. In: Budaghyan L., Rodrguez-Henrquez F. (eds.) Arithmetic of Finite Fields, WAIFI 2018, LNCS, vol. 11321, pp. 95–107. Springer, Cham (2018).

    Google Scholar 

  6. Carlet C., Li C., Mesnager S.: Some (almost) optimally extendable linear codes. Des. Codes Cryptogr. 87, 2813–2834 (2019).

    Article  MathSciNet  Google Scholar 

  7. Carlet C., Mesnager S., Tang C., Qi Y., Pellikaan R.: Linear codes over \({\mathbb{F} }_{q}\) are equivalent to LCD codes for \(q>3\). IEEE Trans. Inf. Theory 64(4), 3010–3017 (2018).

    Article  Google Scholar 

  8. Dillon J.F.: Elementary Hadamard difference sets. In: Hoffman F., et al. (eds.) Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 237–249. Congressus Numerantium XIV, Winnipeg Utilitas Math (1975).

  9. Ding C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60, 3265–3275 (2015).

    Article  MathSciNet  Google Scholar 

  10. Ding C., Tang C.: Designs from Linear Codes, vol. 2nd. World Scientific, Singapore (2022).

    Book  Google Scholar 

  11. Ding C., Tang C.: Combinatorial \(t\)-designs from special functions. Cryptogr. Commun. 12, 1011–1033 (2020).

    Article  MathSciNet  Google Scholar 

  12. Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).

    Book  Google Scholar 

  13. Heng Z., Wang Q., Ding C.: Two families of optimal linear codes and their subfield codes. IEEE Trans. Inf. Theory 66(11), 6872–6883 (2020).

    Article  MathSciNet  Google Scholar 

  14. Huang X., Yue Q., Wu Y., Shi X.: Ternary primitive LCD BCH codes. Adv. Math. Commun. 17(3), 644–659 (2023).

    Article  MathSciNet  Google Scholar 

  15. Huang X., Yue Q., Wu Y., Shi X., Michel J.: Binary primitive LCD BCH codes. Des. Codes Cryptogr. 88, 2453–2473 (2020).

    Article  MathSciNet  Google Scholar 

  16. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).

    Google Scholar 

  17. Li C., Ding C., Li S.: LCD cyclic codes over finite fields. IEEE Trans. Inf. Theory 63(7), 4344–4356 (2017).

    Article  MathSciNet  Google Scholar 

  18. Liu Y., Kai X., Zhu S.: On binary LCD BCH codes of length \(\frac{{{2^ m}+ 1}}{3} \). Adv. Math. Commun. (2022). https://doi.org/10.3934/amc.2022053.

    Article  Google Scholar 

  19. Liu H., Liu S.: Construction of MDS twisted Reed-Solomon codes and LCD MDS codes. Des. Codes Cryptogr. 89(9), 2051–2065 (2021).

    Article  MathSciNet  Google Scholar 

  20. Mann H.B.: Addition Theorems. Wiley, New York (1965).

    Google Scholar 

  21. Mesnager S.: Bent Functions: Fundamentals and Results, pp. 1–544. Springer, New York (2016).

    Book  Google Scholar 

  22. Massey J.L.: Linear codes with complementary duals. Discret. Math. 106(107), 337-C342 (1992).

    Article  MathSciNet  Google Scholar 

  23. Massey J.L.: Orthogonal, antiorthogonal and self-orthogonal matrices and their codes. In: Communications and Coding, pp. 3–9. Research Studies Press, Somerset (1998).

  24. Quan X., Yue Q., Hu L.: Some constructions of (almost) optimally extendable linear codes. Adv. Math. Commun. (2022). https://doi.org/10.3934/amc.2022027.

    Article  Google Scholar 

  25. Sendrier N.: Linear codes with complementary duals meet the Gilbert-Varshamov bound. Discret. Math. 285(1–3), 345–347 (2004).

    Article  MathSciNet  Google Scholar 

  26. Wan Z.: A characteristic property of self-orthogonal codes and its application to lattices. Bull. Belg. Math. Soc. 5, 477–482 (1998).

    MathSciNet  Google Scholar 

  27. Wu Y., Lee Y.: Binary LCD codes and self-orthogonal codes via simplicial complexes. IEEE Commun. Lett. 24(6), 1159–1162 (2020).

    Article  Google Scholar 

  28. Wu Y., Hyun J.Y., Lee Y.: New LCD MDS codes of non-Reed-Solomon type. IEEE Trans. Inf. Theory 67(8), 5069–5078 (2021).

    Article  MathSciNet  Google Scholar 

  29. Yang X., Massey J.L.: The condition for a cyclic code to have a complementary dual. J. Discret. Math. 126, 391–393 (1994).

    Article  MathSciNet  Google Scholar 

  30. Zhou Z., Li X., Tang C., Ding C.: Binary LCD codes and self-orthogonal codes from a generic construction. IEEE Trans. Inf. Theory 65(1), 16–27 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the reviewers and the editor for their constructive suggestions which greatly improve the quality of this paper. The authors also thank Prof. Cunsheng Ding for many valuable discussions.

Funding

Z. Heng’s research was supported in part by the National Natural Science Foundation of China under Grant 12271059 and 11901049, in part by the open research fund of National Mobile Communications Research Laboratory of Southeast University under Grant 2024D10, and in part by the Fundamental Research Funds for the Central Universities, CHD, under Grant 300102122202. F. Li’s research was supported in part by the National Natural Science Foundation of China under Grant 12171420, and in part by the Natural Science Foundation of Shandong Province under Grant ZR2021MA046. Q. Yue’s research was supported by the National Natural Science Foundation of China (No. 62172219).

Author information

Authors and Affiliations

  1. School of Science, Chang’an University, Xi’an, 710064, China

    Xinran Wang & Ziling Heng

  2. National Mobile Communications Research Laboratory, Southeast University, Nanjing, 211111, China

    Xinran Wang & Ziling Heng

  3. College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Fengwei Li

  4. School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

    Qin Yue

Authors
  1. Xinran Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ziling Heng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fengwei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qin Yue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ziling Heng.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest relevant to the content of this article.

Additional information

Communicated by J. Jedwab.

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

Wang, X., Heng, Z., Li, F. et al. LCD codes and almost optimally extendable codes from self-orthogonal codes. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-024-01420-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10623-024-01420-y

Keywords

Mathematics Subject Classification

Search

Navigation