A hypergraph approach to the identifying parent property: the case of multiple parents

Let X be a set of order n and Y be a set of order m. An (n, m, {w 1 , w 2 })-separating hash family is a set F of N functions from X to Y such that for any X 1 , X 2 ⊆ X with X 1 ∩ X 2 = ∅, |X 1 | = w 1 and |X 2 | = w 2 , there exists an element f ∈ F such that f (X 1 ) ∩ f (X 2 ) = ∅. In this paper, we provide explicit constructions of separating hash families using algebraic curves over finite fields. In particular, applying the Garcia-Stichtenoth curves, we obtain an infinite class of explicitly constructed (n, m, {w 1 , w 2 })-separating hash families with N = O(log n) for fixed m, w 1 , and w 2 . Similar results for strong separating hash families are also obtained. As consequences of our main results, we present explicit constructions of infinite classes of frameproof codes, secure frameproof codes and identifiable parent property codes with length N = O(log n) where n is the size of the codes. In fact, all the above explicit constructions of hash families and codes provide the best asymptotic behavior achieving the bound N = O(log n), which substantially improve the results in [8,15,17] and give an answer to the fifth open problem presented in [11].

Section: Theorem 34mentioning

confidence: 99%

Explicit constructions of separating hash families from algebraic curves over finite fields

Liu

Shen

2006

“…In fact, the construction works for any starter code. For instance, for given M, q, w ≥ 2, the probabilistic method in [2] shows the existence of (n , M, q, w)-I P P codes with q > w and some n . Thus, if we take this (n , M, q, w)-I P P code as a starter code and carry out the same recursive construction, then we get a more general result as follows.…”

Section: An Infinite Class Of W-ipp Codes With Efficient Traitor Tracmentioning

confidence: 99%

“…Note that the vector space F i+1 q is an (n i (2), q i+1 , q, 2)-PHF, where n i (2) = i + 1. Thus C 2 i exists for all i ≥ 1.…”

Section: A Recursive Construction Of Perfect Hash Familiesmentioning

confidence: 99%

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New Constructions for IPP Codes

Trung

Martirosyan

2005

Identifiable parent property (IPP) codes are introduced to provide protection against illegal producing of copyrighted digital material. In this paper we consider explicit construction methods for IPP codes by means of recursion techniques. The first method directly constructs IPP codes, whereas the second constructs perfect hash families that are then used to derive IPP codes. In fact, the first construction provides an infinite class of IPP codes having the best known asymptotic behavior. We also prove that this class has a traitor tracing algorithm with a runtime of O(M) in general, where M is the number of codewords.

“…Hollman et al [5] study upper bounds on the size of codes with the identifiable parent property. Barg et al [1] show that if the number of codewords used to create the word in the descendant set is strictly less than the size of the alphabet over which the code is defined, then there exist sequences of IPP-codes with asymptotically non-vanishing rates.…”

Section: Related Workmentioning

confidence: 99%

Binary Fingerprinting Codes

Löfvenberg

2005

Three binary fingerprinting code classes with properties similar to codes with the identifiable parent property are proposed. In order to compare such codes a new combinatorial quality measure is introduced. In the case of two cooperating pirates the measure is derived for the proposed codes, upper and lower bounds are constructed and the results of computer searches for good codes in the sense of the quality measure are presented. Some properties of the quality measure are also derived.