Generalized hashing and parent-identifying codes

  • Authors:

  • Noga Alon;Gérard Cohen;Michael Krivelevich;Simon Litsyn

  • Affiliations:

  • Schools of Mathematical Science and of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel;Département Informatique et Réseaux, ENST, 46 rue Barrault, 75013, Paris, France;Department of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv, Israel

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2003

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Abstract

Let C be a code of length n over an alphabet of q letters. For a pair of integers 2 ≤ t u, C is (t,u)-hashing if for any two subsets T, U ⊂C, satisfying T ⊂ U, |T| = t, |U| = u, there is a coordinate 1 ≤ i ≤ n such that for any x ∈ T, y ∈ U - x, x and y differ in the ith coordinate. This definition, generalizing the standard notion of a t-hashing family, is motivated by an application in designing the so-called parent identifying codes, used in digital fingerprinting. In this paper, we provide lower and upper bounds on the best possible rate of (t, u)-hashing families for fixed t, u and growing n. We also describe an explicit construction of (t, u)-hashing families. The obtained lower bound on the rate of (t, u)-hashing families is applied to get a new lower bound on the rate of t-parent identifying codes.