Identifiable Parent Property Codes Research Articles

Bounds and Constructions of Parent Identifying Schemes via the Algorithmic Version of the Lovász Local Lemma

The usefulness of Identifiable Parent Property (IPP) schemes in diverse scenarios has led to several distinct but related concepts. This work focuses on three of these concepts: “classical” IPP codes, Multimedia IPP codes, and IPP set systems. Although several existence bounds for all of the above schemes are known, constructions are scarce. In this paper, we present explicit constructions of all mentioned IPP notions, in the form of combinatorial objects. Our discussion follows a systematic procedure. First, we use the Lovász Local Lemma (LLL) to obtain existence bounds for the object to be constructed. The bounds derived essentially match the previously best-known ones. Additionally, our proof strategy enables for further development. It allows us to use the Moser-Tardos algorithmic version of the LLL in order to construct, with polynomial complexity, the actual objects. Moreover, we extend the results of Giotis et al. to precisely establish the computational complexity of the proposed algorithms.

Multimedia IPP codes with efficient tracing

Binary multimedia identifiable parent property codes (binary $t$-MIPPCs) are used in multimedia fingerprinting schemes where the identification of users taking part in the averaging collusion attack to illegally redistribute content is required. In this paper, we first introduce a binary strong multimedia identifiable parent property code (binary $t$-SMIPPC) whose tracing algorithm is more efficient than that of a binary $t$-MIPPC. Then a composition construction for binary $t$-SMIPPCs from $q$-ary $t$-SMIPPCs is provided. Several infinite series of optimal $q$-ary $t$-SMIPPCs of length $2$ with $t = 2, 3$ are derived from the relationships among $t$-SMIPPCs and other fingerprinting codes, such as $\overline{t}$-separable codes and $t$-MIPPCs. Finally, combinatorial properties of $q$-ary $2$-SMIPPCs of length $3$ are investigated, and optimal $q$-ary $2$-SMIPPCs of length $3$ with $q \equiv 0, 1, 2, 5 \pmod 6$ are constructed.

Codes with the identifiable parent property for multimedia fingerprinting

Let ${\cal C}$ be a $q$-ary code of length $n$ and size $M$, and ${\cal C}(i) = \{{\bf c}(i) \ | \ {\bf c}=({\bf c}(1), {\bf c}(2), \ldots, {\bf c}(n))^{T} \in {\cal C}\}$ be the set of $i$th coordinates of ${\cal C}$. The descendant code of a sub-code ${\cal C}^{'} \subseteq {\cal C}$ is defined to be ${\cal C}^{'}(1) \times {\cal C}^{'}(2) \times \cdots \times {\cal C}^{'}(n)$. In this paper, we introduce a multimedia analogue of codes with the identifiable parent property (IPP), called multimedia IPP codes or $t$-MIPPC$(n, M, q)$, so that given the descendant code of any sub-code ${\cal C}^{'}$ of a multimedia $t$-IPP code ${\cal C}$, one can always identify, as IPP codes do in the generic digital scenario, at least one codeword in ${\cal C}^{'}$. We first derive a general upper bound on the size $M$ of a multimedia $t$-IPP code, and then investigate multimedia $3$-IPP codes in more detail. We characterize a multimedia $3$-IPP code of length $2$ in terms of a bipartite graph and a generalized packing, respectively. By means of these combinatorial characterizations, we further derive a tight upper bound on the size of a multimedia $3$-IPP code of length $2$, and construct several infinite families of (asymptotically) optimal multimedia $3$-IPP codes of length $2$.

Improved bounds for separating hash families

An $${(N;n,m,\{w_1,\ldots, w_t\})}$$ -separating hash family is a set $${\mathcal{H}}$$ of N functions $${h: \; X \longrightarrow Y}$$ with $${|X|=n, |Y|=m, t \geq 2}$$ having the following property. For any pairwise disjoint subsets $${C_1, \ldots, C_t \subseteq X}$$ with $${|C_i|=w_i, i=1, \ldots, t}$$ , there exists at least one function $${h \in \mathcal{H}}$$ such that $${h(C_1), h(C_2), \ldots, h(C_t)}$$ are pairwise disjoint. Separating hash families generalize many known combinatorial structures such as perfect hash families, frameproof codes, secure frameproof codes, identifiable parent property codes. In this paper we present new upper bounds on n which improve many previously known bounds. Further we include constructions showing that some of these bounds are tight.

On the relationship between the traceability properties of Reed-Solomon codes

Fingerprinting codes are used to prevent dishonest users (traitors) from redistributing digital contents. In this context, codes with the traceability (TA) property and codes with the identifiable parent property (IPP) allow the unambiguous identification of traitors. The existence conditions for IPP codes are less strict than those for TA codes. In contrast, IPP codes do not have an efficient decoding algorithm in the general case. Other codes that have been widely studied but possess weaker identification capabilities are separating codes. It is a well-known result that a TA code is an IPP code, and an IPP code is a separating code. The converse is in general false. However, it has been conjectured that for Reed-Solomon codes all three properties are equivalent. In this paper we investigate this equivalence, providing a positive answer when the number of traitors divides the size of the ground field.

Identifying Traitors Using the Koetter–Vardy Algorithm

This paper deals with the use of the Koetter-Vardy soft-decision decoding algorithm to perform identification of guilty users in traitor tracing and fingerprinting schemes. In these schemes, each user is assigned a copy of an object with an embedded codeword. Placing different codewords in different copies makes each copy unique and at the same time allows unique identification of each user. The weakness of these schemes comes in the form of a collusion attack, where a group of dishonest users get together and, by comparing their copies, they create a pirate copy that tries to hide their identities. The concern of the paper is restricted to traitor tracing and fingerprinting schemes based on Reed-Solomon codes. By using the Koetter-Vardy soft-decision decoding algorithm as the core part of the tracing process, three different settings are approached: tracing in traceability codes, tracing in identifiable parent property codes and tracing in binary concatenated fingerprinting codes. It is also discussed how by a careful setting of a reliability matrix all possibly identifiable users can be found.

Maximum IPP Codes of Length 3

Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a1, . . . , an) and b = (b1, . . . , bn) in C, let \({D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}\). Let \({C^* = {{\bigcup}_{\rm {a,\,b}\in{C}}}D({\bf a, b})}\). The code C is said to have the identifiable parent property (IPP) if, for any \({{\rm {\bf x}} \in C^*}\), \({{\bigcap}_{{\rm x}{\in}D({\rm a,\,b})}\{{\bf a, b}\}\neq \emptyset} \). Codes with the IPP were introduced by Hollmann et al [J. Combin. Theory Ser. A 82 (1998) 21–133]. Let F(n, q) = max{|C|: C is a q-ary code of length n with the IPP}.To and Safavi-Naini [SIAM J. Discrete Math. 17 (2004) 548–570] showed that \({3q + 6 - 6 \lceil\sqrt{q+1}\rceil \leq F(3, q) \leq 3q + 6 - \lceil 6 \sqrt{q+1}\rceil}\), and determined F (3, q) precisely when q ≤ 48 or when q can be expressed as r2 + 2r or r2 + 3r +2 for r ≥ 2. In this paper, we establish a precise formula of F(3, q) for q ≥ 24. Moreover, we construct IPP codes of size F(3, q) for q ≥ 24 and show that, for any such code C and any \({{\rm {\bf x}} \in C^*}\), one can find, in constant time, \({{\rm {\bf a}} \in C}\) such that if \({{\rm {\bf x}} \in D ({\bf c, d})}\) then \({{\rm {\bf a}} \in \{{\rm {\bf c, d}}\}}\).

Prolific Codes with the Identifiable Parent Property

Let $\cal C$ be a code of length n over an alphabet of size q. A word $\mathbf{d}$ is a descendant of a pair of codewords $\mathbf{x},\mathbf{y} \in \cal C$ if $d_i \in \{x_i ,y_i \}$ for $1 \leq i \leq n$. A code $\cal C$ is an identifiable parent property (IPP) code if the following property holds. Whenever we are given $\cal C$ and a descendant $\mathbf{d}$ of a pair of codewords in $\cal C$, it is possible to determine at least one of these codewords. The paper introduces the notion of a prolific IPP code. An IPP code is prolific if all $q^n$ words are descendants. It is shown that linear prolific IPP codes fall into three infinite (“trivial”) families, together with a single sporadic example which is ternary of length 4. There are no known examples of prolific IPP codes which are not equivalent to a linear example: the paper shows that for most parameters there are no prolific IPP codes, leaving a relatively small number of parameters unsolved. In the process the paper obtains upper bounds on the size of a (not necessarily prolific) IPP code which are better than previously known bounds.

New Constructions for IPP Codes

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.