Abstract
Parent-identifying set systems and separable codes are useful combinatorial structures which were introduced, respectively, for traitor tracing in broadcast encryption and collusion-resistant fingerprints for copyright protection. Determining the maximum size of such structures is the main research objective. New upper bounds are presented in this paper. Specifically, for parent-identifying set systems, we determine the order of magnitude of \(I_2(4,v)\) and prove an exact bound when \(w\le \lfloor \frac{t^2}{4}\rfloor +t\). For q-ary separable codes, we give a new upper bound by estimating the distance distribution of such codes, improving the existing upper bound when q is relatively small.
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References
Aaltonen M.: A new upper bound on nonbinary block codes. Discret. Math. 83, 139–160 (1995).
Boneh D., Shaw J.: Collusion-secure fingerprinting for digital data. IEEE Trans. Inf. Theory 44, 1897–1905 (1998).
Bondy J.A., Simonovits M.: Cycles of even length in graphs. J. Comb. Theory Ser. B 16, 97–105 (1974).
Cheng M., Miao Y.: On anti-collusion codes and detection algorithms for multimedia fingerprinting. IEEE Trans. Inf. Theory 57, 4843–4851 (2011).
Chor B., Fiat A., Naor M.: Tracing traitors. In: Cryto’94, pp. 480–491. Springer, Berlin, Heigelberg, New York (1994).
Chor B., Fiat A., Naor M., Pinkas B.: Tracing traitors. IEEE Trans. Inf. Theory 46, 893–910 (2000).
Cohen G., Litsyn S., Zémor G.: Binary \(B_2\)-sequences: a new upper bound. J. Comb. Theory Ser. A 94, 152–155 (2001).
Collins M.J.: Upper bounds for parent-identifying set systems. Des. Codes Cryptogr. 51, 167–173 (2009).
Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 1–97 (1973).
Gu Y., Miao Y.: Bounds on traceability schemes. IEEE Trans. Inf. Theory 64, 3450–3460 (2018).
Gu Y., Satake S.: On 2-parent-identifying set systems of block size 4. arXiv:1908.03523v2.
Gu Y., Cheng M., Kabatiansky G., Miao Y.: Probabilistic existence results for parent-identifying schemes. IEEE Trans. Inf. Theory 65, 6160–6170 (2019).
Gu Y., Fan J., Miao Y.: Improved bounds for separable codes and \(B_2\) codes. IEEE Commun. Lett. 24, 15–19 (2020).
Levenshtein V.I.: Bounds for packings of metric spaces and some applications. Probl. Pered. Inf. 40, 43–110 (1983). (in Russian).
Lindström B.: Determination of two vectors from the sum. J. Comb. Theory 6, 402–407 (1969).
Lindström B.: On \(B_2\)-sequence of vectors. J. Number Theory 4, 261–265 (1972).
Litsyn S.: New upper bounds on error exponents. IEEE Trans. Inf. Theory 45, 385–398 (1999).
McEliece R.J., Rodemich E.R., Rumsey Jr. H., Welch L.R.: New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Trans. Inf. Theory 23, 157–166 (1977).
Sidon S.: Ein satz über trigonometrische polynome und seine anwendung in der theorie der Fourier-Reihen. Math. Ann. 106, 536–539 (1932).
Stinson D.R., Wei R.: Combinatorial properties and constructions of traceability schemes and frameproof code. SIAM J. Discret. Math. 11, 41–53 (1998).
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Communicated by C. Padro.
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Research supported by National Natural Science Foundation of China under Grant 11801392 and the Natural Science Foundation of Jiangsu Province under Grant BK20180833.
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Wang, X. Improved upper bounds for parent-identifying set systems and separable codes. Des. Codes Cryptogr. 89, 91–104 (2021). https://doi.org/10.1007/s10623-020-00809-9
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DOI: https://doi.org/10.1007/s10623-020-00809-9