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By Trappe, Wu, Liu

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By De Morgan’s Law, this corresponds to uniqueness of ∪j∈J Aj for all sets J with cardinality less than or equal to k − 1. Suppose we have a set of k − 1 blocks A1 , A2 , · · · , Ak−1 , we must show that there does not exist another set of blocks whose union produces the same set. There are two cases to consider. First, assume there is another set of blocks {Ai }i∈I with ∪j∈J Aj = ∪i∈I Ai such that I ∩ J = ∅ and |I| ≤ k − 1. Suppose we take a block Ai0 for i0 ∈ I. Then Ai0 must share at most one element with each Aj , otherwise it would violate the λ = 1 assumption of the BIBD.

If we choose i0 ∈ I\(I ∩ J) and look at Ai0 , then 29 again we have that Ai0 can share at most 1 element with each Aj for j ∈ J, and thus Ai0 would have fewer than k elements, contradicting the fact that Ai0 belongs to a (v, k, 1)-BIBD. Thus, ∪j∈J Aj is unique. References [1] N. Heintze, “Scalable document fingerprinting,” in 1996 USENIX Workshop on Electronic Commerce, November 1996. [2] I. Cox, J. Bloom, and M. Miller, Digital Watermarking: Principles & Practice, Morgan Kaufmann, 2001. [3] B. Chor, A.

Sun, and S. W. Cheng, “Modifications of competitive group testing,” SIAM Journal of Computing, vol. 23, pp. 82–96, Feb. 1994. [32] D. Z. Du and H. Park, “On competitive group testing,” SIAM Journal of Computing, vol. 23, pp. 1019–1025, Oct. 1994. [33] J. Dittmann, P. Schmitt, E. Saar, J. Schwenk, and J. Ueberberg, “Combining digital watermarks and collusion secure fingerprints for digital images,” SPIE Journal of Electronic Imaging, vol. 9, pp. 456–467, 2000. [34] T. W. Hungerford, Algebra, Springer-Verlag, 1974.

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