Improved Lower Bounds for Secure Codes and Related Structures

Secure codes are widely-studied combinatorial structures which were introduced for traitor tracing in broadcast encryption. To determine the maximum size of such structures is the main research objective. In this paper, we investigate the lower bounds for secure codes and their related structures. First, we give some improved lower bounds for the rates of 2-frameproof codes and 2-separable codes for slightly large alphabet size. Then we improve the lower bounds for the rate of some related structures, i.e., strongly 2-separable matrices and 2-cancellative set families. Finally, we give a general method to derive new lower bounds for strongly t-separable matrices and t-cancellative set families for t ≥ 3.

[1]  James B. Shearer,et al.  New Bounds for Union-free Families of Sets , 1998, Electron. J. Comb..

[2]  D. Du,et al.  Pooling Designs And Nonadaptive Group Testing: Important Tools For Dna Sequencing , 2006 .

[3]  James B. Shearer,et al.  A New Construction for Cancellative Families of Sets , 1996, Electron. J. Comb..

[4]  Chaoping Xing.  Asymptotic bounds on frameproof codes , 2002, IEEE Trans. Inf. Theory.

[5]  Gennian Ge,et al.  New Lower Bounds for Secure Codes and Related Hash Families: A Hypergraph Theoretical Approach , 2017, IEEE Transactions on Information Theory.

[6]  Minquan Cheng,et al.  Strongly separable codes , 2016, Des. Codes Cryptogr..

[7]  D. Du,et al.  Combinatorial Group Testing and Its Applications , 1993 .

[8]  Hugues Randriambololona,et al.  (2, 1)-Separating systems beyond the probabilistic bound , 2010, ArXiv.

[9]  V. V. Rykov,et al.  Superimposed distance codes , 1989 .

[10]  Yujie Gu,et al.  Improved Bounds for Separable Codes and $B_2$ Codes , 2020, IEEE Communications Letters.

[11]  H. Niederreiter,et al.  Rational Points on Curves Over Finite Fields: Theory and Applications , 2001 .

[12]  Gyula O. H. Katona,et al.  Extremal Problems for Hypergraphs , 1975 .

[13]  Maiko Shigeno,et al.  Strongly separable matrices for nonadaptive combinatorial group testing , 2021, Discret. Appl. Math..

[14]  Ludo M. G. M. Tolhuizen,et al.  New rate pairs in the zero-error capacity region of the binary multiplying channel without feedback , 2000, IEEE Trans. Inf. Theory.

[15]  Gennian Ge,et al.  New Bounds on Separable Codes for Multimedia Fingerprinting , 2014, IEEE Transactions on Information Theory.

[16]  Zoltán Füredi.  2-Cancellative Hypergraphs and Codes , 2012, Comb. Probab. Comput..

[17]  Zoltán Füredi,et al.  Union-free Hypergraphs and Probability Theory , 1984, Eur. J. Comb..

[18]  János Körner,et al.  On Cancellative Set Families , 2007, Comb. Probab. Comput..

[19]  Simon R. Blackburn.  Probabilistic Existence Results for Separable Codes , 2015, IEEE Transactions on Information Theory.

[20]  Zoltán Füredi,et al.  Families of Finite Sets in Which No Set Is Covered by the Union of Two Others , 1982, J. Comb. Theory, Ser. A.

[21]  Douglas R. Stinson,et al.  On generalized separating hash families , 2008, J. Comb. Theory, Ser. A.

[22]  Dan Boneh,et al.  Collusion-Secure Fingerprinting for Digital Data , 1998, IEEE Trans. Inf. Theory.

[23]  Minquan Cheng,et al.  On Anti-Collusion Codes and Detection Algorithms for Multimedia Fingerprinting , 2011, IEEE Transactions on Information Theory.

[24]  S. Wei.  Secure Frameproof Codes, Key Distribution Patterns, Group Testing Algorithms and Related Structures , 1997 .