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

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 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].

Constructions and bounds for separating hash families

“…Remark 1 : For the bounds of strong separating hash families, Sarkar and Stinson [22] proved that there exists an infinite class of SHF(N ; n, m, {1 w 1 , w 2 }) for which N is O((w 1 (w 1 + w 2 )) log * n log n). Liu and Shen [20] gave an infinite constructions of the SHF(N ; n, m, {1 w 1 , w 2 }) for which N is O(log n). Guo and Stinson [15] proved N ≥ n m−1 when w 1 ≥ m − 1 and w 1 + w 2 ≤ n ≤ 2(w 1 + w 2 ) − m. Now we compare our conclusion with the bound in [15].…”

Linear Time Constructions of Some $$d$$-Restriction Problems

“…For constant d, the (n, d)-universal set over Σ = {0, 1} constructed in [29] of size M = O(2 3d log n) (and in [30] of size M = 2 d+O(log 2 d) log n) is (n, (w, r))-CFF for any w and r of size O(log n). See also [23]. In [6], Bshouty gave the following locally explicit constructions of (n, (w, r))-CFF that can be constructed in (almost) linear time in their sizes (the third column in the table ).…”

“…In [23], Liu and Shen provide an explicit constructions of (M ; n, q, (d 1 , d 2 )) separating hash families using algebraic curves over finite fields. They show that for infinite sequence of integers n there is an explicit (M ; n, q, (d 1 , d 2 )) separating hash families of size O(log n) for fixed d 1 and d 2 .…”

Algorithms and Complexity

“…For constant d, the (n, d)-universal set over Σ = {0, 1} constructed in [57] of size M = O(2 3d log n) (and in [58] of size M = 2 d+O(log 2 d) log n) is (n, (w, r))-CFF for any w and r of size O(log n). See also [48]. In [13], Bshouty gave the following locally explicit constructions of (n, (w, r))-CFF that can be constructed in (almost) linear time in their sizes (the third column in the table).…”