Explicit constructions of separating hash families from algebraic curves over finite fields |
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Authors: | Lihua Liu Hao Shen |
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Institution: | (1) Department of Information and Computation Science, Shanghai Maritime University, Shanghai, China;(2) Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China |
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Abstract: | 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
of N functions from X to Y such that for any
with
, |X
1| = w
1 and |X
2| = w
2, there exists an element
such that
. 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
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
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
, which substantially improve the results in
8, 15, 17] give an answer to the fifth open problem presented in 11]. |
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Keywords: | Algebraic curve Separating hash family Strong separating hash family Frameproof (FP) code Secure frameproof (SFP) code Identifiable parent property (IPP) code |
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