Skip to main content
SpringerLink
Log in

A construction for optimal c-splitting authentication and secrecy codes

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

Abstract

Authentication and secrecy codes which provide both secrecy and authentication have been intensively studied in the case where there is no splitting; however the results concerning the case where there is splitting are far fewer. In this paper, we focus on the case with c-splitting, and obtain a bound on the number of encoding rules required in order to obtain maximum levels of security. A c-splitting authentication and secrecy code is called optimal if it obtains maximum levels of security and has the minimum number of encoding rules. We define a new design, called an authentication perpendicular multi-array, and prove that the existence of authentication perpendicular multi-arrays implies the existence of optimal c-splitting authentication and secrecy codes. Further, we study the constructions and existence of authentication perpendicular multi-arrays, and then obtain two new infinite classes of optimal c-splitting authentication and secrecy 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

References

  1. Blundo C., De Santis A., Kurosawa K., Ogata W.: On a fallacious bound for authentication codes. J. Cryptol. 12(3), 155–159 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  2. Casse L.R.A., Martin K.M., Wild P.R.: Bounds and characterizations of authentication/secrecy schemes. Des. Codes Cryptogr. 13(2), 107–129 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang Y., Ji L.: Optimal \((4up, 5, 1)\) optical orthogonal codes. J. Combin. Des. 12(5), 346–361 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  4. Chee Y.M., Zhang X., Zhang H.: Infinite families of optimal authentication codes secure against spoofing attacks of highter order. Adv. Math. Commun. 5(1), 59–68 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  5. Colbourn C.J., Hoffman D.G., Rees R.: A new class of group divisible designs with block size three. J. Combin. Theory Ser A. 59(1), 73–89 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Ding C., Salomaa A., Solé P., Tian X.: Three constructions of authenticaton/secrecy codes. Journal of Pure and Applied Algebra. 196(2–3), 149–168 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  8. Du B.: Splitting balanced incomplete block designs with block size \(3\times 2\). J. Combin. Des. 12(6), 404–420 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  9. Du B.: Splitting balanced incomplete block designs. Australas. J. Combin. 31, 287–298 (2005).

    MathSciNet  MATH  Google Scholar 

  10. Ge G., Zhu L.: Authentication perpendicular arrays \(APA_{1}(2,5, v)\). J. Combin. Des. 4(5), 365–375 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  11. Ge G., Zhu L.: Authentication perpendicular arrays \(APA_{1}(2,7, v)\). J. Combin. Des. 5(2), 111–124 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  12. Ge G., Miao Y., Wang L.: Combinatorial constructions for optimal splitting authentication codes. SIAM J. Discret. Math. 18(4), 663–678 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  13. Huber M.: Information theoretic authentication and secrecy codes in the splitting model. arXiv:1112.0038.

  14. Huber M.: Combinatorial bounds and characterizations of splitting authentication codes. Cryptogr. Commun. 2(2), 173–185 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  15. Liang M., Du B.: A new class of splitting \(3\)-designs. Des. Codes Cryptogr. 60(3), 283–290 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  16. Liang M., Ji L., Zhang J.: Some new classes of \(2\)-fold optimal or perfect splitting authentication codes. Cryptogr. Commun. 9(3), 407–430 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  17. Lidl R., Niederreiter H.: Finite Fields, pp. 225–226. Cambridge University Press, Cambridge (1997).

    MATH  Google Scholar 

  18. Massey J.L.: Cryptographya selective survey. In: Biglier E., Prati G. (eds.) Digital Communications, pp. 3–21. North-Holland, Amsterdam, New York, Oxford (1986).

    Google Scholar 

  19. Ogata W., Kurosawa K., Stinson D.R., Saido H.: New combinatorial designs and their applications to authentication codes and secret sharing schemes. Discret. Math. 279(1–3), 383–405 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  20. Simmons G.J.: A game theory model of digital message authentication. Congr. Numer. 34, 413–424 (1982).

    MATH  Google Scholar 

  21. Simmons G.J.: Message authentication: a game on hypergraphs. Congr. Numer. 45, 161–192 (1984).

    MathSciNet  MATH  Google Scholar 

  22. Simmons G.J.: Authentication theorey/coding theory. In: Blakley G.R., Chaum D. (eds.) Advances in Cryptology-CRYPTO 1984, vol. 196, pp. 411–431. Lecture Notes in Computer ScienceSpringer, Berlin, Heidelberg, New York (1985).

    Google Scholar 

  23. Simmons G.J.: A survey of information authentication. In: Simmons G.J. (ed.) Contemporary Cryptology: The Science of Information Integrity, pp. 379–419. IEEE Press, Piscataway (1992).

    Google Scholar 

  24. Stinson D.R.: Some constructions and bounds for authentication codes. J. Cryptol. 1(1), 37–51 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  25. Stinson D.R.: A construction for authentication/secrecy codes from certain combinatorial designs. J. Cryptol. 1(2), 119–127 (1988).

    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. Stinson D.R., Teirlink L.: A construction for authentication/secrecy codes from \(3\)-homogeneous permutation groups. Eur. J. Combin. 11(1), 73–79 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  28. Tran V.T.: On the construction of authentication and secrecy codes. Des. Codes Cryptogr. 5(3), 269–280 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang J.: A new class of optimal 3-splitting authentication codes. Des. Codes Cryptogr. 38(3), 373–381 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang J., Su R.: Further results on the existence of splitting BIBDs and application to authentication codes. Acta Appl. Math. 109(3), 791–803 (2010).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The research of Mingchao Li was supported by the National Natural Science Foundation of China under Grant No. 11501161 and the Natural Science Foundation of Hebei Province under Grant No. A2016402164. The research of Miao Liang was 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. The research of Beiliang Du was supported by the National Natural Science Foundation of China under Grant No. 11571251.

Author information

Authors and Affiliations

  1. Department of Mathematics, Hebei University of Engineering, Handan, 056038, People’s Republic of China

    Mingchao Li

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

    Miao Liang

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

    Beiliang Du & Jingyuan Chen

Authors
  1. Mingchao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miao Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Beiliang Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jingyuan Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miao Liang.

Additional information

Communicated by C. Blundo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, M., Liang, M., Du, B. et al. A construction for optimal c-splitting authentication and secrecy codes. Des. Codes Cryptogr. 86, 1739–1755 (2018). https://doi.org/10.1007/s10623-017-0421-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-017-0421-x

Keywords

Mathematics Subject Classification

Search

Navigation