Skip to main content
SpringerLink
Log in

On (t, L)-fold perfect authentication and secrecy codes with arbitration

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

Abstract

An authentication code with arbitration is t-fold perfect if the numbers of decoding rules and encoding rules meet the information-theoretic lower bounds with equality. A code has perfect L-fold secrecy if observing a set of \(L'\le L\) messages in the channel gives no information to the opponent regarding the \(L'\) source states. In this paper, we investigate (t, L)-fold perfect authentication and secrecy codes with arbitration which provide both t-fold perfect and perfect L-fold secrecy. We define a new design, L-secrecy perfect ordered restricted strong partially balanced t-design, which is used to construct a (t, L)-fold perfect authentication and secrecy code with arbitration. We also obtain some infinite classes of (t, 1)-fold perfect authentication and secrecy codes with arbitration, especially for \(t>2\).

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

References

  1. Bao J., Ji L.: The completion determination of optimal \((3,4)\)-packings. Des. Codes Cryptogr. 77(1), 217–229 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  2. Bush K.A.: Orthogonal arrays of index unity. Ann. Math. Stat. 23(3), 293–295 (1952).

    Article  MathSciNet  MATH  Google Scholar 

  3. De Soete M.: New bounds and constructions for Authentication/secrecy codes with splitting. J. Cryptol. 3(3), 173–186 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  4. Du B.: Existence of optimal strong partially balanced designs. Appl. Math. J. Chin. Univ. Ser. B 22(2), 169–173 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  5. Du B.: Existence of optimal strong partially balanced designs with block size five. Discret. Math. 279(1–3), 173–190 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  6. Du B.: The spectrum of optimal strong partially balanced designs with block size five. Discret. Math. 288(1–3), 19–28 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  7. Ge G., Ling A.C.H.: Some more \(5\)-GDDs and optimal (\(v,5,1\))-packings. J. Comb. Des. 12(2), 132–141 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  8. Ge G., Miao Y., Zhu L.: GOB designs for authentication codes with arbitration. Des. Codes Cryptogr. 40(3), 303–317 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  9. Hanani H.: Balanced incomplete block designs and related designs. Discret. Math. 11(3), 255–369 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  10. Johansson T.: Lower bounds on the probability of deception in authentication with arbitration. IEEE Trans. Inf. Theory. 40(5), 1573–1585 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  11. Ji L.: Asymptotic determination of the last packing number of quadruples. Des. Codes Cryptogr. 38(1), 83–95 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  12. Ji L., Dong Z.: Existence of optimal strong partially balanced \(3\)-designs with block size four. Des. Codes Cryptogr. 79(1), 19–36 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  13. Ji L., Yin J.: Constructions of new orthogonal arrays and covering arrays of strength three. J. Comb. Theory Ser. A. 117(3), 236–347 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  14. Li M., Liang M., Du B., Chen J.: A construction for optimal c-splitting authentication and secrecy codes. Des. Codes Cryptogr. 86(8), 1739–1755 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  15. Lu X., Jimbo M.: Affine-invariant strictly cyclic Steiner quadruple systems. Des. Codes Cryptogr. 83(1), 33–69 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  16. Liang M., Du B.: A new class of \(3\)-fold perfect splitting authentication codes. Des. Codes Cryptogr. 62(1), 109–119 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  17. Liang M., Li M., Du B.: A construction for \(t\)-fold perfect authentication codes with arbitration. Des. Codes Cryptogr. 73(3), 781–790 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  18. Mills W.H.: On the existence of H designs. Congr. Numer. 79, 129–141 (1990).

    MathSciNet  MATH  Google Scholar 

  19. Mills W.H., Mullin R.C.: Coverings and packings. In: Dinitz J.H., Stinson D.R. (eds.) Contemporary Design Theory, pp. 371–399. Wiley, New York (1992).

    Google Scholar 

  20. Obana S., Kurosawa K.: Combinatorial classification of optimal authentication codes with arbitration. Des. Codes Cryptogr. 20(3), 281–305 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  21. Pei D., Li Y., Wang Y., Safavi-Naini R.: Characterization of authentication codes with arbitration. Lecture Notes in Computer Science, vol. 1587, pp. 303–313. Springer, Berlin (1999).

  22. Pei D.: Authentication Codes and Combinatorial Designs. Chapman Hall/CRC, Boca Raton (2006).

    MATH  Google Scholar 

  23. Pei D.: Message authentication codes (in Chinese). Press: USCT, Hefei (2009).

    Google Scholar 

  24. Simmons G.J.: Message authentication with arbitration of transmitter/receiver disputes. In: Blakley, G.R., Chaum, D. (eds.) Advances in Cryptology-EUROCRYPT 1987, Lecture Notes in Computer Science, vol. 304, pp. 151–165. Springer, Berlin (1998).

  25. Stinson D.R.: A construction for authentication/secrecy codes from certain combinatorial designs. J. Cryptol. 1(2), 119–127 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  26. Stinson D.R.: The combinatorics of authentication and secrecy codes. J. Cryptol. 2(1), 23–49 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang Y., Safavi-Naini R., Pei D.: Combinatorial characterisation of \(l\)-optimal authentication codes with arbitration. J. Comb. Math. Comb. Comput. 37, 205–224 (2001).

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of Miao Liang is supported by the National Natural Science Foundation of China under Grant Nos. 11301370 and 11571251, the China Postdoctoral Science Foundation under Grant No. 2016M601873, and sponsored by Qing Lan Project of Jiangsu Province and Suzhou Vocational University. The research of Lijun Ji is supported by the National Natural Science Foundation of China under Grant Nos. 11871363 and 11431003.

Author information

Authors and Affiliations

  1. Department of Mathematics, Soochow University, Suzhou, 215006, People’s Republic of China

    Miao Liang & Lijun Ji

  2. Department of Mathematics and Physics, Suzhou Vocational University, Suzhou, 215104, People’s Republic of China

    Miao Liang

Authors
  1. Miao Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lijun Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miao Liang.

Additional information

Communicated by M. Paterson.

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, M., Ji, L. On (t, L)-fold perfect authentication and secrecy codes with arbitration. Des. Codes Cryptogr. 87, 2003–2026 (2019). https://doi.org/10.1007/s10623-018-00602-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-018-00602-9

Keywords

Mathematics Subject Classification

Search

Navigation