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On optimal weak algebraic manipulation detection codes and weighted external difference families

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Abstract

This paper provides a combinatorial characterization of weak algebraic manipulation detection (AMD) codes via a kind of generalized external difference families called bounded standard weighted external difference families (BSWEDFs). By means of this characterization, we improve a known lower bound on the maximum probability of successful tampering for the adversary’s all possible strategies in weak AMD codes. We clarify the relationship between weak AMD codes and BSWEDFs with various properties. We also propose several explicit constructions for BSWEDFs, some of which can generate new optimal weak AMD codes.

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References

  1. Ahmadi H., Safavi-Naini R.: Detection of algebraic manipulation in the presence of leakage, ICITS 2013. Lect. Notes Comput. Sci. 8317, 238–258 (2013).

    Article  Google Scholar 

  2. Bao J., Ji L., Wei R., Zhang Y.: New existence and nonexistence results for strong external difference families. Discret. Math. 341(6), 1798–1805 (2018).

    Article  MathSciNet  Google Scholar 

  3. Buratti M., Yan J., Wang C.: From a \(1\)-rotational RBIBD to a partitioned difference family. Electron. J. Comb. 17, R139 (2010).

    Article  MathSciNet  Google Scholar 

  4. Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs, vol. 42. Chapman & Hall/CRC, London (2006).

    Book  Google Scholar 

  5. Cramer R., Dodis Y., Fehr S., Padró C., Wichs D.: Detection of algebraic manipulation with applications to robust secret sharing and fuzzy extractors. EUROCRYPT 4965, 471–488 (2008).

    MathSciNet  MATH  Google Scholar 

  6. Cramer R., Fehr S., Padró C.: Algebraic manipulation codes. Sci. China Math. 56(7), 1349–1358 (2013).

    Article  MathSciNet  Google Scholar 

  7. Cramer R., Padró C., Xing C.: Optimal algebraic manipulation detection codes in the constant-error model. Lect. Notes Comput. Sci. 9014, 481–501 (2015).

    Article  MathSciNet  Google Scholar 

  8. Ding C.: Optimal constant composition codes from zero-difference balanced functions. IEEE Trans. Inf. Theory 54(12), 5766–5770 (2008).

    Article  MathSciNet  Google Scholar 

  9. Ding C.: Optimal and perfect difference systems of sets. J. Comb. Theory A 116(1), 109–119 (2009).

    Article  MathSciNet  Google Scholar 

  10. Fan C., Ge G.: A unified approach to Whiteman’s and Ding-Helleseth’s generalized cyclotomy over residue class rings. IEEE Trans. Inf. Theory 60(2), 1326–1336 (2014).

    Article  MathSciNet  Google Scholar 

  11. Fujiwara Y., Tonchev V.D.: High-rate self-synchronizing codes. IEEE Trans. Inf. Theory 59(4), 2328–2335 (2012).

    Article  MathSciNet  Google Scholar 

  12. Huczynska S., Paterson M.B.: Existence and non-existence results for strong external difference families. Discret. Math. 341(1), 87–95 (2018).

    Article  MathSciNet  Google Scholar 

  13. Huczynska S., Paterson M.B.: Weighted external difference families and \(R\)-optimal AMD codes. Discret. Math. 342(3), 855–867 (2019).

    Article  MathSciNet  Google Scholar 

  14. Huczynska S., Paterson M.B.: Characterising bimodal collections of sets in finite groups. Arch. Math. 113(6), 571–580 (2019).

    Article  MathSciNet  Google Scholar 

  15. Jedwab J., Li S.: Construction and nonexistence of strong external difference families. J. Algebraic Comb. 49(1), 21–48 (2019).

    Article  MathSciNet  Google Scholar 

  16. Levenshtein V.I.: Combinatorial problems motivated by comma-free codes. J. Comb. Des. 12(3), 184–196 (2004).

    Article  MathSciNet  Google Scholar 

  17. Lu X., Niu X., Cao H.: Some results on generalized strong external difference families. Des. Codes Cryptogr. 86(12), 2857–2868 (2018).

    Article  MathSciNet  Google Scholar 

  18. Martin W.J., Stinson D.R.: Some nonexistence results for strong external difference families using character theory. Bull. Inst. Comb. Appl. 80, 79–92 (2017).

    MathSciNet  MATH  Google Scholar 

  19. Ng S.L., Paterson M.B.: Disjoint difference families and their applications. Des. Codes Cryptogr. 78(1), 103–127 (2016).

    Article  MathSciNet  Google Scholar 

  20. Paterson M.B., Stinson D.R.: Combinatorial characterizations of algebraic manipulation detection codes involving generalized difference families. Discret. Math. 339(12), 2891–2906 (2016).

    Article  MathSciNet  Google Scholar 

  21. Wen, J., Yang, M., Feng, K.: The \((n,m,k,\lambda )\)-strong external difference family with \(m\ge 5\) exists. arXiv:1612.09495v1 (2016)

  22. Wen J., Yang M., Fu F., Feng K.: Cyclotomic construction of strong external difference families in finite fields. Des. Codes Cryptogr. 86(5), 1149–1159 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Prof. Marco Buratti for the helpful discussion about difference families. The authors are also indebted to the anonymous referees for their very detailed comments and suggestions in revising this paper, which greatly improved the presentation of the ideas, notions, and results contained in this paper. This research is supported by JSPS Grant-in-Aid for Scientific Research (B) under Grant No. 18H01133.

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  1. Graduate School of Systems and Information Engineering, University of Tsukuba, Tennodai 1-1-1, Tsukuba, 305-8573, Japan

    Minfeng Shao & Ying Miao

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  1. Minfeng Shao
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  2. Ying Miao
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Correspondence to Minfeng Shao.

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Communicated by M. Paterson.

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Shao, M., Miao, Y. On optimal weak algebraic manipulation detection codes and weighted external difference families. Des. Codes Cryptogr. 88, 1349–1369 (2020). https://doi.org/10.1007/s10623-020-00754-7

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  • DOI: https://doi.org/10.1007/s10623-020-00754-7

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