Fully Collusion Resistant Traitor Tracing with Short Ciphertexts and Private Keys

1. Fully Collusion Resistant Traitor Tracing with Short Ciphertexts and Private Keys Dan Boneh, Amit Sahai, and Brent Waters

2. Broadcast Systems Distribute content to a large set of users • Commercial Content Distribution • File systems • Military Grade GPS • Multicast IP

3. Tracing Pirate Devices[CFN’94] • Attacker creates “pirated device” • Want to trace origin of device

4. FAQ-1 “The Content can be Copied?” • DRM- Impossibility Argument • Protecting the service • Goal: Stop attacker from creating devices that access the original broadcast

5. FAQ 2-Why black-box tracing? [BF’99] • D: may contain unrecognized keys, is obfuscated, or tamper resistant. • All we know: Pr[ M  G, C  Encrypt (PK, M) : D(C)=M] > 1- K1 D: K3 K$*JWNFD&RIJ$ K2 R R

6. S  {1, …, n } PK, TK, { Kj| j  S} RunSetup(n) Pirate Decoder D TraceD( TK ) i  {1,…,n} Formally: Secure TT systems • (1) Semantically secure, and (2) Traceable: Challenger Attacker Adversary wins if: (1) Pr[D(C)=M] > 1-, and (2) i  S

7. Brute Force System • Setup (n): Generate n PKE pairs (PKi, Ki) Output private keys K1 , …, KnPK (PK1, …, PKn) , TK PK . • Encrypt (PK, M): C  ( EPK1(M), …, EPKn(M) ) • Tracing: next slide. • This is the best known TT system secure under arbitrary collusion. … until now

8. n n i=1 i=1 TraceD(PK): [BF99, NNL00, KY02] R • For i = 1, …, n+1 define for M  G : pi := Pr[D( EPK1(), …, EPKi-1(), EPKi(M), …, EPKn(M) ) = M] • Then: p1 > 1-  ; pn+1  0 • 1- = |pn+1 – p1 | = | pi+1 – pi|   |pi+1 – pi|  Exists i{1,…,n} s.t. | pi+1 – pi |  (1- )/n User i must be one of the pirates.

9. Security Theorem  • Tracing algorithm estimates: | pi - pi | < (1-)/4n • Need O(n2) samples per pi. (D – stateless) • Cubic time tracing. • Can be improved to quadratic in |S| . • Thm: underlying PKE system is semantically secure  No eff. adv wins tracing game with non-neg adv.

10. Linear Broadcast Encryption Private B.E. Abstracting the Idea [BSW’06] Properties needed: • For i = 1 ,… , n+1 need to encrypt M so: • Without Ki adversary cannot distinguish: Enc(i, PK, M) from Enc(i+1, PK, M) n 1 i-1 i users cannot decrypt users can decrypt

11. Private Linear Broadcast Enc (PLBE) • Setup(n): outputs private keys K1 , …, Kn and public-key PK. • Encrypt( u, PK, M): Encrypt M for users {u, u+1, …, n} Output ciphertext CT. • Decrypt(CT, j, Kj, PK): If j  u, output M • Broadcast-Encrypt(PK,M) := Encrypt( 1, PK, M) • Note: slightly more complicated defs in [BSW’06]

12. PK, { Kj| j  u} m C*  Enc( u+b, PK, m) b’  {0,1} Security definition • Message hiding: given all private keys: Encrypt( n+1 , M, PK) PEncrypt( n+1 , , PK) • Index hiding: for u = 1, … , n : Challenger Attacker RunSetup(n) b{0,1}

13. Results • Thm: Secure PLBE  Secure TT Same size CT and priv-keys (black-box and publicly traceable) • New PLBE system: CT-size = O(n) ; priv-key size = O(1) enc-time = O(n) ; dec-time = O(1)

14. n PLBE Construction: hints • Arrange users in matrix • Key for user (x,y): Kx,y  Rx  Cy • CT: one tuple per row, one tuple per col. size = O(n) • CT to user (i,j): User (x,y) can dec. if (x > i) OR [ (x=i) AND (y  j) ] n=36 users Encrypt to user (4,3)

15. Bilinear groups of order N=pq [BGN’05] • G: group of order N=pq. (p,q) – secret. bilinear map: e: G  G  GT • G = Gp  Gq . gp = gq  Gp ; gq = gp  Gq • Facts: h  G  h = (gq)a  (gp)b e( gp , gq ) = e(gp , gq) = e(g,g)N = 1 e( gp , h ) = e( gp , gp)b !!

16. A n size PLBE • Ciphertext: ( C1, …, Cn, R1, …, Rn) • User (x,y) must pair Rx and Cy to decrypt Well-formed Malformed/Random Zero

17. Summary and Open Problems FCR • New results:[BGW’05, BSW’06, BW’06] • Full collusion resistance: • B.E: O(1) CT, O(1) priv-keys … but O(n) PK • T.T: O(n) CT, O(1) priv-keys. • T.R.:O(n) CT, O(n) priv-keys. • Open questions: • Private linear B.E. with O(log n) CT. • Private B.E. with short ciphertexts.

18. THE END

19. BGN encryption • Subgroup assumption: G p Gp • E(m) : r  ZN , C  gm (gp)r  G • Additive hom: E(m1+m2) = C1  C2  (gp)r • One mult hom: E(m1m2) = e(C1,C2)  e(gp,gp)r

20. Results • Thm: Secure PLBE  Secure TT Same size CT and priv-keys (black-box and publicly traceable) • New PLBE system: CT-size = O(n) ; priv-key size = O(1) enc-time = O(n) ; dec-time = O(1) • Applications: • Tracing Traitors : O(n) CTs and O(1) keys. • Adaptive BE. (need Augmented PLBE) • Comparison searches on encrypted data.

21. T.T: a popular problem 32 papers from 49 authors

22. i M A Simple System • n users in system, each gets separate key • User i gets Ki • Encrypt message to separately to user –lump it • (Use “hybrid encryption” and encrypt an AES key) … E(Ki , M) … E(K1 , M) E(K2 , M) E(Kn , M)

23. Device works 100 User j is an attacker 35 Everything Random Tracing • Let E’(i, M) => Encrypt R to 1,…,i-1 and M to i,…n … … E(K1 , R) E(K2 , R) E(Ki-1 , R) E(Ki , M) E(Kn , M) • Pi = prob. pirate device decrypts E’(i,M) • Can learn Pi’s from probing the device