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
Ahmadi H., Safavi-Naini R.: Detection of algebraic manipulation in the presence of leakage, ICITS 2013. Lect. Notes Comput. Sci. 8317, 238–258 (2013).
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).
Buratti M., Yan J., Wang C.: From a \(1\)-rotational RBIBD to a partitioned difference family. Electron. J. Comb. 17, R139 (2010).
Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs, vol. 42. Chapman & Hall/CRC, London (2006).
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).
Cramer R., Fehr S., Padró C.: Algebraic manipulation codes. Sci. China Math. 56(7), 1349–1358 (2013).
Cramer R., Padró C., Xing C.: Optimal algebraic manipulation detection codes in the constant-error model. Lect. Notes Comput. Sci. 9014, 481–501 (2015).
Ding C.: Optimal constant composition codes from zero-difference balanced functions. IEEE Trans. Inf. Theory 54(12), 5766–5770 (2008).
Ding C.: Optimal and perfect difference systems of sets. J. Comb. Theory A 116(1), 109–119 (2009).
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).
Fujiwara Y., Tonchev V.D.: High-rate self-synchronizing codes. IEEE Trans. Inf. Theory 59(4), 2328–2335 (2012).
Huczynska S., Paterson M.B.: Existence and non-existence results for strong external difference families. Discret. Math. 341(1), 87–95 (2018).
Huczynska S., Paterson M.B.: Weighted external difference families and \(R\)-optimal AMD codes. Discret. Math. 342(3), 855–867 (2019).
Huczynska S., Paterson M.B.: Characterising bimodal collections of sets in finite groups. Arch. Math. 113(6), 571–580 (2019).
Jedwab J., Li S.: Construction and nonexistence of strong external difference families. J. Algebraic Comb. 49(1), 21–48 (2019).
Levenshtein V.I.: Combinatorial problems motivated by comma-free codes. J. Comb. Des. 12(3), 184–196 (2004).
Lu X., Niu X., Cao H.: Some results on generalized strong external difference families. Des. Codes Cryptogr. 86(12), 2857–2868 (2018).
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).
Ng S.L., Paterson M.B.: Disjoint difference families and their applications. Des. Codes Cryptogr. 78(1), 103–127 (2016).
Paterson M.B., Stinson D.R.: Combinatorial characterizations of algebraic manipulation detection codes involving generalized difference families. Discret. Math. 339(12), 2891–2906 (2016).
Wen, J., Yang, M., Feng, K.: The \((n,m,k,\lambda )\)-strong external difference family with \(m\ge 5\) exists. arXiv:1612.09495v1 (2016)
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).
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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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