GENERALIZED TRACEABILITY CODES

Transcription

1 U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 2, 2016 ISSN GENERALIZED TRACEABILITY CODES Majid MAZROOEI 1, Ali ZAGHIAN 2 In this paper, we introduce generalizations of frameproof, secure frameproof, traceability and identifiable parent property codes and will study some of their basic combinatorial properties. Keywords: Frameproof, Identifiable parent property, Hash family, Traceability. 1. Introduction Throughout this paper, is an alphabet of size and is a q array code of length. If =, then we call an (,, ) code. The elements of are called code words and each code word will have the form,...,, where, 1. The matrix representation of the code, denoted H(), is an matrix on symbols where each row of the matrix corresponds to one of the code words. Codes providing some forms of traceability (TA, for short) to protect copyrighted digital data against piracy have been extensively studied in the recent years. The weak forms of such codes are frameproof codes introduced by Boneh and Shaw [2], and secure frameproof codes. The more strong form of these codes are identifiable parent property (IPP, for short) codes which have been introduced by Hollmann, Van Lint, Linnartz and Tolhuizen [4] and defined as follows. Let be a -array code. For any subset of code words, the set of descendants of, denoted desc( ), is defined by desc,1. Thus desc( ) consists of all tuples that could be produced by a coalition holding the code words in. If desc( ), then we say that produces. Let be an integer. Define the descendant code, denoted desc, as follows: desc desc, Thus desc consists of all tuples that could be produced by some coalition of size at most. Now the code is called an,,, identifiable parent property code ( IPP) provided that, for all desc, it holds that 1 Assistant Prof., Dept.of Mathematical Sciences, University of Kashan, Isfahan, Iran, P. O. Box: , m.mazrooei@kashanu.ac.ir 2 Assistant Prof., Dept. of Mathematics, Malek Ashtar University of Technology, Isfahan, Iran, P. O. Box: , a_zaghian@mut-es.ac.ir

2 206 Majid Mazrooei, Ali Zaghian : desc, Another strong versions of such codes are TA schemes and TA codes introduced by Chor, Fiat and Naor in [3]. In fact, TA codes turn out to be a subclass of IPP codes and are defined as follows. Let define, for any,. Suppose is an (,, ) code and 2 is an integer. is called a traceability code ( TA) provided that, for all and all desc, and, there is at least one code word such that,, for any. Combinatorial properties of IPP codes and TA codes have been studied by Staddon, Stinson and Wei [9], Sarkar and Stinson [7], Barg, et al. [1], and also in [11]. The question of traitor tracing algorithms for IPP and TA codes is treated in [8], e.g. certain classes of TA codes are shown to have a faster tracing algorithm than their initially known linear runtime by using the list decoding techniques. New results on bounds of frame-proof codes and TA schemes can be found in [6]. Recently, a major theoretical challenge is to derive more codes which have efficient tracing algorithms. To do this, some authors have given some generalizations of IPP and TA codes. Sarkar and Stinson [7] defined an (,,,, ) IPP code as a code with the property that for every coalitions (1 ),..., of, each of size at most, desc implies. They have studied this generalization in part. For example, they proved that a code is an (,,, 2) IPP code if and only if it is an (,,, 2, 3) IPP code. Another generalization is given by Wu and Sattar [12] for TA codes. They have defined TA codes relative to any well-defined metric and an efficient tracing algorithm is presented for Lee-metric TA codes. In this paper, we introduce new generalizations of frame-proof, secure frame-proof, identifiable parent property and traceability codes and will study them. 2. Generalized Traceability Codes From now,,..., : are maps and α = {,..., }. For any and for any, let,..., and. For any subset of code words, we define the set of descendants of, denoted desc ( ), by desc, 1. Now, for any integer 2 define the (, ) descendant code, denoted desc,, as follows: desc, desc,

3 Generalized traceability codes 207 Definition 2.1 Let be an (,, ) code and 2 be an integer. We say that: 1. (, ) Frameproof code: is an (,,,, ) frameproof code ((, ) FP), if for any of size at most, desc implies. 2. (, ) Secure frameproof code: is an (,,,, ) secure frameproof code ((, ) SFP), if for any subsets, of of cardinality at most, desc desc implies. 3. (, ) Traceability code: is an (,,,, )-traceability code ((, ) TA) if for any of cardinality at most and for any desc, there exists such that,, for all. 4. (, ) Identifiable parent property code: is an (,,,, ) identifiable parent property code ((, ) IPP) provided that, for all desc,, it holds that :,desc Note that the original definitions of FP, SFP, IPP and TA codes will be obtained if we take,...,, where,1, is the natural projection on the th component. Our first result determines the relations between our generalized traceability codes. Proposition 2.2. The following relationships hold for any (,, ) code. (, ) TA (, ) IPP (, ) SFP (, ) FP. Proof. Assume that is an (,,,, ) TA code. We show that is an (, ) IPP code. Let desc,. Then there exists,, such that desc. Let such that,, for all. Thus,, for any by assumption. We show that for any,, desc implies, which proves that is an (, ) IPP code. By assumption, there is such that,, for any. If, then we will have,,, which is a contradiction. Hence. It is easy to see that any (, ) IPP code is an (, ) SFP code. We just prove that if C is an (, ) SFP code, then is an (, ) FP code. Let be a subset of of size at most and desc. Thus there exists a code word such that. Let. Then desc. By assumption, we have showing that. Hence which completes the proof. Proposition 2.3. Let be an (,, ) code and let min,,,. Assume that 2 is an integer such that 1. Then is an (, ) TA code.

4 208 Majid Mazrooei, Ali Zaghian Proof. Let 1 and 1. Assume that,, be a set of code words and desc 0. For any desc 0, let max, and min desc. For any and, we have, which shows that,. On the other hand, we have,. Thus for any desc 0. Hence. Now let be a code word such that,. Then for any we have,,. This completes the proof. Example 2.4. Let and 100, 411, 511. Define the maps 1,2,3 such that 100 2, and for 1, 2, 3. Then we have 3 4. This shows that is an (3, 3, 11,, 2) TA code while is not a TA code. 3. Generalized Traceability Codes and Hash Families A finite set of functions, where, is called an (,, ) hash family, denoted by (,, ) HF. An (,, ) HF can be presented as an matrix on symbols, where each column of the matrix corresponds to one of the functions in. Sometimes it is easier to consider hash families as such matrices. Definition 3.1. We consider the following kinds of hash families. 1. Perfect hash family: An (,, ) HF is called an (,,, ) perfect hash family ((,,, ) PHF) if for any subset of of size, there is a function H such that is injective on. 2. Separating hash family: An (,, ) HF is called an (,,,, ) separating hash family ((,,,, ) SHF) if for any disjoint subsets, of with and, there is a function such that and are also disjoint. The following theorem, due to Staddon, Stinson and Wei [9], makes a connection between perfect hash families and IPP codes. Theorem 3.2. Let be an (,, ) code whose matrix representation is C. If C is an (,,, 2 /4) PHF, then is an (,,, ) IPP code. Stinson, Van Trung and Wei [10] obtained another connections between (secure) frame-proof codes and separating hash families, which is stated in the following theorem. Theorem A code is an (,,, ) FP code iff is an (,,,, 1) SHF. 2. A code is an (,,, ) SFP code iff is an (,,,, ) SHF.

5 Generalized traceability codes 209 In this section, we will derive similar results of our generalizations of FP, SFP and IPP codes. Before it, we need the following definition. Definition 3.4. Let be an (,, ) code,,..., be maps and,...,. We say that is an injective family if for any,, implies. Proposition 3.5. Let be an (,, ) code and 2 be an integer. If is an (,,,, 1) SHF, then is an (, ) FP code. The converse holds if is an injective family. Proof. Assume that, and desc. Thus there exists such that. We claim that which show that is an (, ) FP code. Let. if then. Hence, by assumption, there is 1 such that, which is a contradiction because desc. Now assume that is an (, ) FP code and α is an injective family. We prove that is an (,,,, 1) SHF. If not, then there exist disjoint subsets, of with 1 and such that for any 1, 1. This shows that there is a code word such that desc. Since is an (, ) FP code,. Thus there exists such that, which, by our assumption, implies. Hence, a contradiction. Thus is an (,,,, 1) SHF. Proposition 3.6. Let be an (,, ) code and 2 be an integer. Then is an (, ) SFP code iff is an (,,,, ) SHF. Proof. Assume that is an (, ) SFP code. We show that is an (,,,, ) SHF. If not, then there exist disjoint subsets, of, both of size, such that 0 1 for all 1. This shows that for any 1, there exists and such that. Hence desc desc, which is a contradiction because is an (, ) SFP code. Hence should be an (,,,, ) SHF. Conversely, assume that is an (,,,, ) SHF. We prove that is an (, ) SFP code. Let, be disjoint subsets of both of size. We shall show that desc desc. By assumption, there exists 1 such that. This shows that desc desc, as desired. Proposition 3.7. Let be an (,,,, ) IPP code where 1. Then the set,..., is an (,,, 1) PHF. Proof. If not, then there is such that 1 and for any 1, there are such that. Let,...,. Then for any, we have desc where.

6 210 Majid Mazrooei, Ali Zaghian But for any, and we have which contradicts the fact that is an (, ) IPP code. Hence the set is an (,,, 1) PHF. Proposition 3.8. A code is an (,,,, 2) IPP code iff the family is both an (,,, 3) PHF and an (,,, 2, 2) SHF. Proof. If is an (,,,, 2) IPP code then is both an (,,, 3) PHF and an (,,, 2, 2) SHF by propositions 3.6 and 3.7. Assume that is both an (,,, 3) PHF and an (,,, 2, 2) SHF. We show that is an (,,,, 2) IPP code. Let desc, 1,...,, where,, 1, are distinct subsets of. If, then by assumption, there exists 1 such that, which is a contradiction because desc desc. Hence. Without loss of generality we may assume that. Note that in this case we have. We claim that for all 3,...,. If not, then there exists 3 such that. Since, we will have. Similarly,. Hence,. By assumption, there exists 1 such that, and are distinct. Now there are 2 cases: Case 1. : In this case we will have desc, which is a contradiction. Case 2. : In this case we will have desc, which is a contradiction. Hence for all 1,...,, as desired. Theorem 3.9. Let be an (,, ) code. If the family,..., is an (,,, 2 /4) PHF, then is an (, ) IPP code. Proof. If not, then there exist desc, such that. : desc C, C Let : desc C, C and define min :,. Without loss of generality, we may assume that,..., is a family of distinct elements of which have no common code words and for any 1, desc. Let,...,. Hence for any 1, there exists, such that. This shows that for any 1, we have,..., 1. which implies 1 2. Since is an integer, the last inequality shows that 22 /4. Now, since is an (,,, ) PHF,

7 Generalized traceability codes 211 there exists 1 such that,..., are distinct. On the other hand, there exists 1 such that. Also there is some such that. Since there is no code word such that, we have desc which is a contradiction. Hence is an (, ) IPP code. In [5], it is proved that if, then there exists an (,,, ) PHF. Hence we have the following corollary. Corollary Let be an (,, ) code and 2 be an integer such that that is an (, ) IPP code. 4. Conclusion. Then there are maps,..., such We generalized and studied codes providing some forms of traceability (FP, SFP, TA and IPP codes) to generate new families of such codes. Our generalized traceability codes enjoy the basic combinatorial properties of original ones. This motivates us to study decoding algorithms of our families in the future. Acknowledgement The first author is partially supported by the University of Kashan under grant number /1. R E F E R E N C E S [1]. A. Barg, G. Cohen, S. Encheva, G. Kabatiansky and G. Zemor, A hypergraph approach to the identifying parent property: the case of multiple parents, SIAM J. Discrete. Math., 14 (2001), [2]. D. Boneh and M. Franklin, An efficient public key traitor tracing schemes, Advances in Cryptography - Crypto' 94 (Lecture Notes in Computer Science), Springer-Verlag 839 (1994), [3]. B. Chor, A. Fiat and M. Naor, Tracing Traitors, Advances in Cryptography - Crypto' 94 (Lecture Notes in Computer Science), Springer-Verlag 839 (1994), [4]. H. D. L. Hollmann, J. H. Van Lint, J. P. LinnartzL. M. G. M. Tulhuizen, On codes with identifiable parent property, J. Comp. Theory A, 82 (1998), [5]. K. Mehlhorn, On the program size of perfect and universal hash functions, Proceedings of the 23rd IEEE Symposium of Foundations of Computer Science (1982), [6]. R. Safavi-Naini and Y. Wang, New results on frameproof codes and traceability schemes, IEEE Trans. Infom. Theory, 47 (2001), [7]. P. Sarkar, D. R. Stinson, Frameproof and IPP codes, Progress in Cryptography (Lecture Notes in Computer Science), 2247 (2001),

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