Probabilistic Existence Results for Parent-Identifying Schemes

Abstract

Parent-identifying schemes provide a way to identify causes from effects for some information systems such as digital fingerprinting and group testing. In this paper, we consider combinatorial structures for parent-identifying schemes. First, we establish an equivalent relationship between parent-identifying schemes and forbidden configurations. Based on this relationship, we derive probabilistic existence lower bounds for two related combinatorial structures, that is, t-parent-identifying set systems (t-IPPS) and t-multimedia parent-identifying codes (t-MIPPC), which are used in broadcast encryption and multimedia fingerprinting respectively. The probabilistic lower bound for the maximum size of a t-IPPS has the asymptotically optimal order of magnitude in many cases, and that for t-MIPPC provides the asymptotically optimal code rate when t = 2 and the best known asymptotic code rate when t ≥ 3. Furthermore, we analyze the structure of 2-IPPS and prove some bounds for certain cases. Index TermsParent-identifying scheme, forbidden configuration, probabilistic construction, graph removal lemma. I. INTRODUCTIONThe elucidation of cause-effect relationships among variables or events is the central aim of many studies, see preface of Pearl's book [24]. Parent-identifying schemes provide a way to identify causes from an effect for some information systems such as digital fingerprinting and group testing.We use the terminology of f -channel to represent a kind of cause-effect relationship, which has at most t inputs (causes) and a collection of possible outputs (effects). Let Q be a finite alphabet of cardinality |Q| = q, in which elements represent possible causes. Let Q n = {c = (c(1), . . . , c(n)) : c(i) ∈ Q}, and 2 Q denote the power set of Q. We define f -channel as a function f from subsets of some ground set, which will be Q n or 2 Q in our settings, to subsets of the same ground set. Two most interesting particular choices of f will be described below.Definition 1: A subset C of Q n or 2 Q is a t-parent-identifying scheme under f -channel if for any C ⊆ C with |C | ≤ t and any d ∈ f (C ), we have P⊆C: |P|≤t, d∈f (P) P = ∅.(1)The above definition exhibits the essential idea of cause clarification algorithms in parent-identifying schemes, that is, once an effect d is observed then at least one cause can be found which is common to all subsets P ⊆ C with size at most t capable of generating d. If d can be generated by P, then we call d a descendant of P and call P a possible parent set of d.A set C ⊆ Q n is called a q-ary code of length n and each c ∈ C is called a codeword. In Definition 1, if C ⊆ Q n , then it can be regarded as a class of fingerprinting codes used in digital copyright protection. Well-known examples include codes with the t-identifiable parent property (t-IPP codes) [3]-[6], [22], [27], [29], where f (P) = desc(P) = P(1) × P(2) × · · · × P(n),and P(i) = {c(i) : c ∈ P} is the ith projection of the set P.Y. Gu is with the

Universally Sparse Hypergraphs with Applications to Coding Theory

“…Recently, Gu, Cheng, Kabatiansky and Miao [22] showed that for fixed integers t ≥ 2, r ≥ 3, there exists a t-IPPS(r, m, n) with m = Ω(n r ⌊t 2 /4⌋+t ), which implies that for ⌊t 2 /4⌋ + t | r the upper bound in [23] is tight up to a constant factor. We slightly improve the lower bound of [22] for some pairs of r, t. Proposition 9 is proved by establishing a connection between IPPSs and sparse hypergraphs, as stated below. Note that a similar observation with different phrasing was obtained independently in [22].…”

“…We slightly improve the lower bound of [22] for some pairs of r, t. Proposition 9 is proved by establishing a connection between IPPSs and sparse hypergraphs, as stated below. Note that a similar observation with different phrasing was obtained independently in [22].…”

On $2$-parent-identifying set systems of block size $4$

“…In this paper we shall investigate the value I 2 (n, 4) for 2-IPPS(n, 4), especially, when n is sufficiently large. Notice that it is reasonable to consider large n and relatively small k. In fact, the dealer needs a large set of base keys to accommodate amounts of authorized users, however, each authorized user is usually assigned with a limited number of base keys which are used as the user's inputs to the decryption devices [6].…”

On non-binary traceability set systems

“…Another known scheme, based on arbitrary (w, n)threshold SSS, was proposed in [25,7] and is known under the name of set systems with the identifiable parent property (IPP set systems). The most recent results can be found in [11,16,10,15,12]. The generalization of these two schemes was proposed in [9].…”