Traitor Tracing

1. Traitor Tracing Papers Benny Chor, Amos Fiat and Moni Naor, Tracing Traitors (1994) Moni Naor and Benny Pinkas, Threshold Traitor Tracing (1998) Presented By: Anukool Lakhina, Keren Pinkas and Scott Savarese

2. How this Presentation is Organized • First, we motivate and introduce the General Traitor Tracing problem that we want to solve. • Next, we introduce two methods to solve this problem. • We then analyze the efficiency of each method. • We conclude with a concrete example.

3. Motivation We want to trace the source of leaks when sensitive or proprietary data is made available to a large set of parties.

4. Typical Scenario • We are Cablevision. We only want to broadcast to legal subscribers (all of which have a special decrypting key). • Suppose Professor Itkis is a subscriber who with other subscribers designs a device which will allow people to view our broadcasts without paying. • The Goal: After confiscating this device, how do we figure out who supplied the keys which decrypt our broadcasts. • This is the basic idea of Traitor Tracing.

5. Basic Definitions • Data Provider: Cablevision (Us). • Traitor (Pirate): Professor Itkis and his friends.  • Content: Our encrypted broadcasts. • Pirate Decoder: Device used by the pirates to decrypt our encrypted broadcasts.

6. Basic Assumptions • Two types of pirate decoders: • 1) Created by obtaining keys from legitimate users. • 2) Created by breaking the underlying encryption. • We assume that our encryption scheme is difficult to break. So, we only care about Type 1. • We only want to find the traitor who contributed the largest number of keys.

7. Addressing the Problem • Two methods: • 1) k-Resilient Traitor Tracing (Fully Resilient Traitor Tracing) • 2) Threshold Traitor Tracing • k-Resilient Traitor TracingScheme catches anyone who can illegally decrypt our encrypted broadcast. • Threshold Traitor Tracing Scheme catches anyone who can illegally decrypt more than a specified fraction of our encrypted broadcast.

8. Efficiency Parameters We measure the efficiency of these solutions in terms of the following parameters: • (a) Memory and Computation requirements for the user. • (b) Memory and Computation requirements for the Data Provider • (c) Data Redundancy Overhead – How much more data do we need to broadcast in order to be trace traitors.

9. k-Resilient Traitor Tracing(Fully Resilient Traitor Tracing)

10. k-Resilient Tracing • A scheme is k-resilient if it can correctly identify a traitor and not an innocent user even if k traitors combine and collude. • We are only able to catch the traitor who submits the most keys to the pirate decoder.

11. How Data is Broadcasted • Broadcast is broken up into pieces • Each piece contains two parts: the enabling block and the cipher block. Message = <Enabling Block, Cipher Block> • Cipher Block is created using a secret key or one time pad obtained by decrypting the information in the enabling block.

12. One Level Open SchemeThe simplest • Maps n users into a set of 2k2encryption keys • Users Keys, P(u) = O(k2log n) • Enabling Block = O(k4 log n )

13. Initialization • We create l first-level hash functions <h1,h2,…hl>. • Each hi maps a particular user, u into one of 2k2 sets. • Thus the personal key for a user contains l keys <h1(u), h2(u), … hl(u)>

14. Distribution of Secret • The cipher block is encrypted with either a one time pad or secret key s. • Key s is broken into l pieces such that s = s1 XOR s2 XOR … si … XOR sl • Each siis encrypted with each of the 2k2keys.

15. Decryption of Cipher Block • Each user has a key for each row i in the enabling block. • They are able to decrypt si and thus are able to obtain s • With s they obtain the information in the cipher block

16. Creation of a Pirate Decoder • At most k people get together. • For each i from 1 to l, the create a set of keys F. • Without keys for each of the l rows they are unable to decrypt the cipher block. • With all l keys they are able to decrypt every secret they receive.

17. Detection of Traitors • Using black box techniques the set of keys F is determined. • For each row i we perform h-1(fi). This gives us a set of users that map to that key. We mark each user. • After obtaining the list of users for all l keys, the user seen the most is the traitor.

18. Proof • Each traitor in coalition gives at most l/k keys. • For each row i the coalition has at most k keys. The probability that a particular user’s key is one of the k keys is 1/2k. • Must create l such that the number of an innocent user’s keys that are exposed is less than l/k.

19. Results • We determine l to be 4k2log n • Thus, the number of keys a user has is 4k2log n • The enabling block consists of 8k4 log n

20. Secret One-Level Scheme • Keeps the hash mapping secret • Lower costs then the one-level open scheme by a factor of k. • Simpler construction • Introduces a probability p which is the probability that pirates will create a device that is untraceable.

21. Secret scheme (contd.) • Same as one-level open scheme exact that instead of 2k2 groups there are only4k. • The number of keys that a user has is (4/3)k log (n/p) • The number of keys in the enabling block is (16/3)k2 log (n/p)

22. Threshold Traitor Tracing

23. Threshold Traitor Tracing • Suppose Cablevision divides a program into 1 minute segments. An illegal decoder which can decrypt 90% of these segments will fail to decode one minute out of ten minutes. Will you pay for such a decoder? • So, for many applications, a decoder which can decrypt with a low success probability is useless. • So the real threat are decoders which can decrypt, say, 99% of all the segments. Threshold Traitor Tracing only concerns with these decoders. • We want to be able to catch a true traitor with probability 1-p. (So ideally, we want p to be very very small.)

24. How do we distribute the Content • We generate a meta-key which contains a base set A of random keys and we assign l keys to each user. • These l keys form the user’s Personal Key. (Two users cannot have exactly the same set of keys.) • A program is always broadcasted in segments. Each segment consists of two parts: an enabling block and a cipher block. Message = <enabling block, cipher block> • Cipher Block is the encrypted program segment, using some secret key s. • Enabling Block allows authorized users to obtain the secret key, s.

25. A One-Level q-Threshold Scheme • Specify our threshold by q. (That is, we want to catch all decoders that can decode q of the broadcast segments.) • Let n be the number of legal subscribers. • Let k be the number of traitors.

26. We address the following about One-Level Threshold Traitor Tracing • Initialization • Distribution of Secret • Decryption Procedure • Parameters Involved • Tracing Procedure • Analysis

27. 1) Initialization: • We have a set of l hash functions {h1, h2, … ,hl} which are chosen at random. • Each hash functionmaps a particular user, u into one of a 4k random keys. • So, user u receives l keys: {h1(u), h2(u), … , hl(u)}. • All this can be represented very nicely in a l x 4k matrix A.

28. 2) Distribution of Secret • Let s be the secret key to be distributed. We (The Data Provider) divide the secret key, into t shares, where t is random, and 0 < t <= l. • We ensure that s = s0xor s1 xor … xor st • Eachsi is encrypted using each of the 4k keys of the corresponding row in matrix A. (continued…)

29. Distribution of Secret (contd.) • Let w be a fraction such that q <= w < 1. • The scheme divides the secret into t shares and ensures that a decoder which contain keys from a fraction of at least w of the l rows would be able to decrypt the secret with probability greater than q.

30. 3) Decryption • Each authorized user has one key from every row and is therefore always able to decrypt every siand compute s.

31. 4) Parameters • Memory Required per user is m=l keys. • Amount of work that each user performs to reveal a key is O(t). • Data Redundancy Overhead is r=4kt.

32. 5) Tracing • We are only concerned with decoders that have keys from wl rows. (Since only these decoders can decrypt with probability q). • Suppose we have the set of keys F that a pirate decoder uses to crack our encrypted broadcast. Suppose F contains at least one key from each of the wl rows of Matrix A. Denote these rows by r1, r2,…, rwlanddenote the key common to F and row rias fri. Since we know the hash function, hri we can compute its inverse and determine the users of that key . • The user with the largest number of marks is our traitor.

33. 6) Analysis of One-Level Threshold • There are k traitors. • On average, each traitor contributes wl/k keys to F. • How do we know that an innocent user say, Alice, is not identified as a traitor? • The probability that friequals the key mapped to Alice is 1/4k. So, the probability that at least wl/k of the keys of Alice are in F is at most 2^-3wl/4k. We choose an l such that the probability of this happening is very very small.

34. Results! • Recall q is our threshold value. k is the number of traitors. n is the number of users. 1-p is the probability of catching a true traitor. We have the following: • Personal Key, l, consists of (4k/3w) * log(n/p) keys. • Data Redundancy Overhead, 4kt, is: 4k* log(1/q) / log (1/w) keys. • Number of decryptions, that each user must performis log(1/q) / log (1/w) decryptions. (So if w=q, number of decryptions needed is 1.)

35. Two Level k-Resilient Traitor Tracing(Fully Resilient TraitorTracing)

36. Two Level Open Scheme • Much more complicated than a one-level scheme. • More efficient by a factor of k. • User has 2k2log2k log n keys. • 4k3log4k log n keys in the enabling block.

37. Two Level Threshold Traitor Tracing

38. Two Level Threshold Scheme • Two-Level Threshold Schemes are constructed from One-Level Threshold Schemes by using many One-Level Schemes and applying a hash function to map users to schemes • Advantages: Shorter key length than one-level • Disadvantages: Higher Data Redundancy than one-level. • In one-level, q is predefined. Two-level threshold schemes allow us to have q as a function of other parameters.

39. Results

40. Some Numbers: • Suppose: • number of users, n = 106 • number of traitors, k = 1000 • Our threshold, • q = 0.75 • q = 0.95 • Probability of finding the true traitor is 1-p (where p=10-3) • We have the following results 

41. Results

42. Conclusions: • For many applications, there is no need to have a fully resilient tracing scheme. • Threshold Tracing Schemes are more efficient.