Mehdi Azadi Motlagh m.azadim@gmail Tarbiat Moallem university Tehran, Iran May 2011

2. FRAME PROOF CODES Mehdi AzadiMotlagh m.azadim@gmail.com TarbiatMoallem university Tehran, Iran May 2011

3. intoduction

4. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION In order to protect a product (such as digital data, computer software, etc.), a distrib­utor marks each copy with some code word and then ships each user his data marked with that code word. codes may be embedded in the content or codes may be associated with the keys used to recover the content. This marking (digital fingerprint ) can offer protection by providing some form of traceability (TA) for pirated data.

5. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Strong versions of TA allow at least one member of a coalition that constructs a “pirate decoder” to be traced. Weaker versions of this concept ensure that no coalition can “frame” a disjoint user or group of users. However, a coalition of users may detect some of the marks, namely the ones where their copies differ. They can then change these marks arbitrarily.

6. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION The original logarithm table

7. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Digital fingerprinting was introduced for the first time by Wagner in 1983. N. R. Wagner. Fingerprinting. In Proceedings of the 1983 Symposium on Security and Privacy, 1983.

8. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION In digital fingerprinting the vendor embeds a secret unique mark in each copy of the digital object. This mark, the fingerprint, makes it possible to trace the guilty buyers, which we call the pirates.

9. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION The marking assumption a distributor must embed the fingerprint into the digital copy. This embedding must ensure that: 1 - A non-colluding pirate cannot detect the marks. 2 - The pirates cannot change the state of an undetected mark without rendering the object useless. The marking assumption defines what a pirate collusion is allowed to do.

10. FRAME PROOF CODES

11. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Frame proof codes prevent a coalition from framing a user not in the coalition. we might try to construct a code such that, given an illegal copy, at least one user in the coalition that created it can be found. Unfortunately, such a code cannot exist. (Boneh and Shaw-1995). we consider a slightly weaker property , it is impossible that an illegal copy could be cre­ated by two disjoint coalitions.

12. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Defnition 1:A set on an alphabet Q with is called a and each is called a code word. A binary v-tuple is called an unregistered word. Given a code , the incidence matrix will be the matrix in which the rows are the code words in

13. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Defnition 2Let be a -code. Suppose . For , we say that bit position is undetectable for C if Let be the set of undetectable bit positions for . Then is called the feasible set of .

14. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION if there is a code word C, then user could be “framed” if the coalition C produces the . Definition 3: - feasible set of , denoted , as follows: The set consists of the -tuples that could be produced by some coalition of size at most .

15. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 4 : A is called a frame proof code if, for every such that , we have We will say that is a . in a frame proof code, the only code words in the feasible set of a coalition of at most users are the code words of members of the coalition. Thus, no coalition of at most users can frame a user who is not a member of the coalition.

16. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Example 1 : to be the containing all bit binary words with exactly one 1. Claim: is a .

17. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 5 : Suppose that is a . For any , define . Suppose that is an unregistered word, if and there exists a code word such that for all , then we would at least be able to identify user as being guilty. Unfortunately, this is hoping for too much

18. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Theorem 1 : Suppose is a with .Suppose where . Then there exists an unregistered wordsuch that for any with

19. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Proof : Let For , define for any with . It remains to show that is an unregistere word.

20. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION We present a

21. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Suppose not; then for some . Let with . Then , which contradicts the fact that isframeproof. The above theorem says that we cannot be guaranteed of identifying a guilty user in a . For, if for some where , then

22. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 6: Suppose thatis a code. is said to be a -secure frame proof code if for any such that and we have that . We will say that is a for short. Lemma 1: A is a .

23. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Example 2 :Let denote the following incidence matrix for a code: We will show that is a by computing for all such that

24. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION

25. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION A does not permit traceability, but it does afford some security, as follows: 1 - It is impossible for a coalition of size at most to implicate a disjoint coalition of size at most by constructing an unregistered word . 2 - If is an unregistered word that has been constructed by a coalition of size at most c, then any contains at least one guilty user.

26. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 7 : Suppose that is a code , is an integer, Let be all the subsets of such that ,then is a (identifiable parent property) code provided that for all , it holds that . A code has the if no coalition of size at most can produce an -tuple that cannot be traced back to at least one member of the coalition. Lemma 2 : A is a.

27. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION cannot exist for . Lemma 3: Suppose is any code with and .Then is not a code. Proof: Let be chosen such that

28. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Then let . Now, it is easy to see that for any . is not .

29. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 8 : For , define . Suppose that is a code , is an integer, Let be all the subsets of such that , Definition 9 : is a (traceability) code provided that, for all and for all , there is at least one code word such that Lemma 4 : A is a

30. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Example 3 : Let is . Then is , since the symbols in the first position of all the code words are different. let ,Then . However, and . Thus the code is not a.

31. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION we cannot expect to identify all the traitors, except for certain “trivial” codes. Lemma 5 : Suppose is any code code with .Then there exist three code words such that Proof: There exist three code words and a coordinate such that .

32. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION x Then define as follows: Clearly .

33. Combinatorial descriptions

34. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Theorem 2 : For any integer ; there is a . Proof. We define the incidence matrix . The rows of are indexed by the elements in the set . The columns are indexed by the subsets such that . Denote these subsets as , where . Now, the entry in row and column of is defined to be

35. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION suppose , and . Since b =2c, it follows that . Without loss of generality, suppose that . Now, there is a unique bit position such that . If follows that . Hence, .

36. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Example4 : We present a .

37. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Defnition 10: A set system is a pair where is a set of elements called points, and is a set of subsets of , the members of which are called blocks. Let be a set system where and ,The incidence matrix of is the matrix , where Conversely, given an incidence matrix, we can define an associated set system in an obvious way.

38. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION If is a code, then the matrix is a incidence matrix of a set system . For any , denote the associated block in the corresponding set system Example 5 :

39. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Lemma 6 :Let and let . Then if and only if

40. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Theorem 3 : There exists a if and only if there exists a set system such that and for any subset of blocks ; there does not exist a block such that

41. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 11 : A set system is an sandwich-free family provided that, for any two disjoint subsets the following property holds: An sandwich-free family, , will be denoted as an if

42. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Theorem 4: A ) exists if and only if there exists a . Proof :Suppose that is a set system. is not a if and only if there is a set such that

43. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION are sets of code words in the associated code ,

44. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 12 : A set system is an separating systemprovide that, for any , there exists a such that . An separating system,, will be denoted as an if

45. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Theorem 5 : There exists an if and only if there exists an . Corollary 1: A exists if and only if there exists a , and a exists if and only if there exists a .

46. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 13 : An perfect hash family is a set of functions , such that and for any such that , there exists at least one such that is one-to-one. When an -perfect hash family will be denoted by

47. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION A PHF(v, n, m , c) can be depicted as an matrix with entries from , such that for any w columns there exists at least one row such that the w entries in the given c columns are distinct.

48. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Given a “small” , we can recursively construct a “large” by using perfect hash families. We present our construction using of sandwich-free families. Theorem 6 : If there exists an and a ; then there exists an .

49. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION Definition 14 : An separating hash family is a set of functions , , and for any such that ,and , there exists at least one suchthat . The notation will be used to denote an separating hash family with .

50. INTRODUCTION FRAME PROOOF CODES COMBINATORIAL DESCRIPTION