Abstract
An \({(N;n,m,\{w_1,\ldots, w_t\})}\) -separating hash family is a set \({\mathcal{H}}\) of N functions \({h: \; X \longrightarrow Y}\) with \({|X|=n, |Y|=m, t \geq 2}\) having the following property. For any pairwise disjoint subsets \({C_1, \ldots, C_t \subseteq X}\) with \({|C_i|=w_i, i=1, \ldots, t}\) , there exists at least one function \({h \in \mathcal{H}}\) such that \({h(C_1), h(C_2), \ldots, h(C_t)}\) are pairwise disjoint. Separating hash families generalize many known combinatorial structures such as perfect hash families, frameproof codes, secure frameproof codes, identifiable parent property codes. In this paper we present new upper bounds on n which improve many previously known bounds. Further we include constructions showing that some of these bounds are tight.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Atici M., Magliveras S.S., Stinson D.R., Wei W.-D.: Some recursive constructions for perfect hash families. J. Comb. Des. 4, 353–363 (1996)
Bazrafshan M., van Trung T.: Bounds for separating hash families. J. Comb. Theory Ser. A 118, 1129–1135 (2011)
Bazrafshan M.: Separating hash families. PhD thesis, University of Duisburg-Essen (2011).
Blackburn S.R., Etzion T., Stinson D.R., Zaverucha G.M.: A bound on the size of separating hash families. J. Comb. Theory Ser. A 115, 1246–1256 (2008)
Blackburn S.R.: Perfect hash families: probabilistic methods and explicit constructions. J. Comb. Theory Ser. A 92, 54–60 (2000)
Blackburn S.R.: Perfect hash families with few functions. Unpublished manuscript (2000).
Blackburn S.R.: Frameproof codes. SIAM J. Discret. Math. 16, 499–510 (2003)
Blackburn S.R., Wild P.R.: Optimal linear perfect hash families. J. Comb. Theory Ser. A 83, 1897–1905 (1998)
Boneh D., Shaw J.: Collusion-free fingerprinting for digital data. IEEE Trans. Inf. Theory 44, 1897–1905 (1998)
Bush K.A.: A generalization of a theorem due to MacNeish. Ann. Math. Stat. 23, 293–295 (1952)
Bush K.A.: Orthogonal arrays of index unity. Ann. Math. Stat. 23, 426–434 (1952)
Cohen G.D., Encheva S.B., Schaathun H.G.: On separating codes, Technical report 2001D003, TELECOM ParisTech—Ecole Nationale Superieure des Telecommunications (2001).
Dinitz, J.H, Colbourn, C.J. (eds): The CRC Handbook of Combinatorial Designs, 2nd edn. Chapman and Hall/CRC, Boca Raton, FL (2007)
Hollmann H.D.L., van Lint J.H., Linnartz J.-P., Tolhuizen L.M.G.M.: On codes with the identifiable parent property. J. Comb. Theory Ser. A 82, 121–133 (1998)
Li P.C., Wei R., van Rees G.H.J.: Constructions of 2-cover-free families and related separating hash families. J. Comb. Des. 14, 423–440 (2006)
Martirosyan S.S., van Trung T.: Explicit constructions for perfect hash families. Des. Codes Cryptogr. 46, 97–112 (2008)
Mehlhorn K.: Data Structures and Algorithms 1: Sorting and Searching. Springer, Berlin (1984)
Staddon J.N., Stinson D.R., Wei R.: Combinatorial properties of frameproof and traceability codes. IEEE Trans. Inf. Theory 47, 1042–1049 (2001)
Stinson D.R., Wei R., Zhu L.: New constructions for perfect hash families and related structures using combinatorial designs and codes. J. Comb. Des. 8, 189–200 (2000)
Stinson D.R., van Trung T., Wei R.: Secure frameproof codes, key distribution patterns, group testing algorithms and related structures. J. Stat. Plan. Inference 86, 595–617 (2000)
Stinson D.R., Zaverucha G.M.: Some improved bounds for secure frameproof codes and related separating hash families. IEEE Trans. Inf. Theory 54, 2508–2514 (2008)
Stinson D.R., Wei R., Chen K.: On generalized separating hash families. J. Comb. Theory Ser. A 115, 105–120 (2008)
van Trung T., Martirosyan S.S.: New constructions for IPP codes. Des. Codes Cryptogr. 35, 227–239 (2005)
Walker R.A. II, Colbourn C.J.: Perfect hash families: constructions and existence. J. Math. Cryptogr. 1, 125–150 (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by S. D. Galbraith.
Rights and permissions
About this article
Cite this article
Bazrafshan, M., van Trung, T. Improved bounds for separating hash families. Des. Codes Cryptogr. 69, 369–382 (2013). https://doi.org/10.1007/s10623-012-9673-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9673-7