Short Traceable Signatures Based on Bilinear Pairings

1. Short Traceable Signatures Based on Bilinear Pairings Seung Geol Choi Columbia University joint work with Kunsoo Park Seoul National University Moti Yung Columbia University

2. Contents • Overview of Traceable Signatures • Motivation • Preliminaries • ZK for SDH Representation • Construction • Security • Conclusion

3. Overview of Traceable Signatures

4. Traceable Signatures • Can be regarded as an extension of group signatures. • Provides all the operations of group signatures- setup, join, sign, verify, open • Provides stronger revocation of anonymity- tracing (reveal, trace) • Provides claiming (claim, claim_verify)

5. Why do we need traceable sig.? • Consider following setting: • Anonymous Users • Typical Abstract Large System: • Many users • Many remote verification points. • Users issue signatures that get aggregated and verified in remote points.

6. Scenario #1: Verification Points Authority Tracing Request: “open signature”

7. Scenario #2 Verification Points Tracing Request: “USER X” needs to be traced Authority Using the opening mechanism from scenario #1: all signatures must be aggregated and the Authority will have to Open all to discover the ones signed by user X

8. Shortcomings of group sig. • Signatures from remote verification points must be aggregated. Load Balancing Concerns • Authority must open all signatures thus severely (and unnecessarily) violating the privacy of many users. Privacy Concerns • Authority is typically a distributed entity so that opening requires the collaboration of many agents. Efficiency Concerns • Outcome: Scenario #1 is insufficient for dealing with the above tracing request.

9. Scenario #3 User wants to claim a signature as his

10. Features of Traceable Sig.(1) • Anonymity • A user (group member) signs on behalf of the group. • Verification is done using the group’s public-key. • Claiming • A user can claim his own signature.

11. Features of Traceable Sig.(1) • Revocation of Anonymity • The group manager can opena problematic signature and find out who signed it. • The tracing agents can trace all the signatures of a suspicious user.

12. Motivation

13. Motivation • Previous constructions were quite long. • [KTY04] : 1206 bytes • [NS04]: 917 bytes • Adapt the short group signature [BBS04] to traceable signature. • Ours: 362 bytes • 1.5 ~ 3 times the length of the RSA sig.

14. Basic Tools

15. Three main basic tools • Bilinear Pairings • One more SDH (Strong Diffie-Hellman) representation problem • Linear Encryption Scheme

16. Basic Tools – Bilinear Pairings • G1, G2, GT : cyclic groups of prime order p • P1 , P2 : generator of G1 , G2 • ψ: G2 G1 (isomorphism mapping) • Def: Bilinear pairing e : G1 x G2 GT is: • Bilinear: e(aP1, bP2) = e(P1, P2)ab for all a, b  Z • Non-degenerate: e(P1, P2) ≠ 1 • Efficiently computable

17. Basic Tools – One More SDH Representation Problem (1) • SDH Representation • Given P1, P2, Q, R where Q  G1 , R = γP2 • SDH Representation: (A, x, t) s.t. A = (xP1 + Q)/ (t+γ) or equivalently e(A, tP2+R) = e(xP1 + Q, P2) • One more SDH representation problem • Given K SDH representations, output another valid SDH representation

18. Basic Tools – One More SDH Representation Problem (2) • Under q-SDH Assumption, One more representation problem is hard. • q-SDH Assumption [BB04] • The following q-SDH problem is hard: P1, P2, γP2, γ2P2, …, γqP2? (A, x) s.t.(γ +x)A = P1 where A  G1 , x  Zp

19. Basic Tools – Linear Encryption [BBS04] (1) • Keys: • Encryption Key: X, Y, Z  G1 • Decryption Key: ξ1, ξ2 s.t. ξ1 X = Z, ξ2 Y=Z • Encryption/Decryption • E(M) = ( r1X, r2Y, M+(r1+r2)Z ) • D(C1, C2, C3) = C3 – ξ1C1 – ξ2C2

20. Basic Tools – Linear Encryption [BBS04] (2) • Semantic Security: • Under DLDH (Decisional Linear Diffie-Hellman) Assumption [BBS04], linear encryption is semantically secure. • DLDH Assumption • The following problem is hard: • Given X, Y, Z, aX, bY, cZc = a + b? or c is randomly chosen?

21. ZK for SDH Representation

22. Basic Idea • Why do we need this? • Come up with zk proof for the rep, and use the proof as a sig (FS transform)  Anonymity • The rep is a witness of a proof a signing key

23. Basic Setting • Proof: • PK{(A,x,t): e(xP1 + Q, P2) = (A, tP2+R)} • Instance: P1, Q, P2, R • Where P1 (gen. of G1), Q (random point) P2 (gen. of G2), R (= γP2) • Prover’s aux input (SDH rep./witness): (A, x, t) s.t. e(xP1 + Q, P2) = (A, tP2+R) • Other Public Parameters • For linear enc. : X, Y, Z (gen. of G1) • Etc. : W (gen. of G2)

24. ZK for SDH Representation (1) • Prover constructs T1, … T5: • T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z (linear enc. of A)T4 = r3W, T5 = e(P1, T4)x (DLP of x ) • Sub-proof • PK{(a1, a2, b1, b2, u, v): T1 = a1X, T2 = a2Y, uT1 = b1X, uT2 = b2Y, T5 = e(P1, T4)v , e (T3, P2)u e(T3, R) = e(Z, P2) (b1+b2) e(Z, R) (a1+a2) e(P1, P2)v e(Q, P2) }

25. ZK for SDH Representation (2) • Exists a Simulator (i.e. it is ZK) • T1, …, T5 : • From semantic security of linear enc:- Pick a random A’- T1 = r1X, T2 = r2Y, T3 = A’ + (r1+r2)Z • From DDH:- pick a random x’ - T4 = r3W, T5 = e(P1, T4)x’ • Indistinguishable from the original transcript • Sub-Proof: • Runs the simulator of Sub-Proof

26. ZK for SDH Representation (3) • Exists an extractor (i.e. it is POK) • Sub-Proof: Simple 3-move honest verifier DLP ZK-POK exists an extractor for the Sub-Proof • Using the extractor of DLP proof, we can also extract an SDH Rep. • Specifically • Let (a1, a2, b1, b2, u, v) be the extracted witness. • b1 + b2 = u(a1 + a2)

27. e (T3, P2)u e(T3, R) = e(Z, P2) (b1+b2) e(Z, R) (a1+a2) e(P1, P2)v e(Q, P2) e(T3, uP2+R) = e(Z, (b1+b2)P2+(a1+a2) R) • e(vP1+Q, P2)e(T3, uP2+R) / e(Z, u(a1+a2)P2+(a1+a2) R) = e(vP1+Q, P2)e(T3, uP2+R) / e((a1+a2)Z, uP2+ R) = e(vP1+Q, P2)e(T3 - (a1+a2)Z, uP2+ R) = e(vP1+Q, P2) If we Let A = T3 – (a1+a2)Z, e(A, uP2+ R) = e(vP1+Q, P2) (A, u, v) is an SDH rep. ZK for SDH Representation (4)

28. Construction

29. Procedures of Traceable sig. • Setup • Join/Iss • Sign/Verify • Open • Reveal/Trace • Claim/Claim_Verify

30. Construction - Setup • Generate public parameters for ZK for SDH Rep. • P1, Q, P2, R, X, Y, Z, W • For SDH rep. : P1, Q, P2, R • For linear enc. : X, Y, Z s.t. ξ1 X = Z, ξ2 Y=Z • Etc. : W • The group manager’s private key:(γ, ξ1, ξ2) • γ : for the generation of SDH rep (join proc.) • ξ1, ξ2 : dec. key for linear enc. (opening)

31. Construction – Join/Iss • Interactive Protocol between a user (Join) and the group manager (Iss) • Ui (user i)  GM : xiP1 • GM  Ui: (Ai, ti) s.t. e(Ai, tiP2+ R) = e(xiP1+Q, P2) • Note that GM can generate (Ai, ti) without knowing the value xi. • Let Ci = xiP1 • A = (Ci + Q)/ (t+γ) • Ui now has an SDH rep: (Ai, xi, ti) • GM stores the joining record: (Ai, Ci, ti)

32. Construction – Sign/Verify (1) • Big Picture of ZK Protocol for SDH Rep: • 3 move honest verifier proof for DLP • Instance: T1, …, T5 • P (Prover)  V (Verifier): B1, …, B6 • V  P : c • P  V : sa1, sa2, sb1, sb2, su, sv • V : checks if sa1, sa2, sb1, sb2, su, sv are consistent.

33. Construction – Sign/Verify (2) • Details • T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z T4 = r3W, T5 = e(P1, T4)x • d1 = r1t , d2 = r2t • B1 = br1X, B2 = br2X, B3 = btT1 – bd1X, B4 = btT2 – bd2YB5 = e(P1, T4) bxB6 = e(T3, P2)bt e(Z, P2)-bd1-bd2 e(Z, R)-br1-br2 e(P1, P2)-bx • sr1 = br1 + cr1,sr2 = br2 + cr2,sd1 = bd1 + cd1, sd2 = bd2 + cd2,sx = bx + cx,st = bt + ct,

34. Construction – Sign/Verify (3) • Apply the variant of Fiat-Shamir to the protocol (Schnorr type sig.) • Sign: • Replace B1, …, B6 of the verifier with hash function: c = H(m, T1, …, T5, B1, …, B6) • The signature will be:(T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx ) • 362 bytes: T5 = 1024 bits, all others 170 bits. • Verification: • construct B’1, …, B’6 from the signature. • check if H(m, T1, …, T5, B’1, …, B’6) =? c.

35. Construction – Open • Given a signature: • (T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx ) • The GM use his decryption key for linear enc. to recover A from T1, T2,T3. • T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z • Dec(T1, T2, T3) = T3 – ξ1T1 – ξ2T2 = A • Look up the user j from the join records {(Ai, Ci, ti)} such that Aj = A

36. Construction – Tracing a user (Reveal/Trace) • Reveal • Given the identity j of a certain user Uj , returns an information to be used for tracing • The GM returns Cj from his join record (Aj, Cj, tj). • Trace • Given Cj (from Reveal) the tracing info of Uj,and a sig. (T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx ), decides whether it’s Uj’s sig. or not. • e(Cj, T4) =? T5 ( Note that T5 = e(P1, T4)x )

37. Construction - Claiming a Sig.(Claim/Claim_Verify) • Claim: • Given a sig. (T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx ) • The signer returns a NIZK proof.PK{ y: T5 = e(P1, T4)y} • Claim_Verify: Verify the proof.

38. Security

39. Security Model [KTY04] • There are three kind of attacks • Misidentification: the adv. forges a valid signature that is opened/traced to no one. • Framing: the adv. forges a valid signature that is opened/traced to an innocent user even if the adv. corrupts the GM. • Anonymity: the adv. distinguishes a sig. of user A from a sig. of user B. • The adv. is allowed to access oracles.

40. Oracles Executes a Iss procedure. (Adv is playing the role of user. Oracle is playing the role of GM.) QY Returns the Public-key Qa-join Returns the GM’s private key Qs Executes a join dialog internally Qb-join Executes a Join procedure. (Adv is playing the role of GM. Oracle is playing the role of user.) Qp-join Given <i, m>, returns a signature on m by the i-th user Qsig Given <i>, returns the tracing info. Ci. Qreveal

41. Misidentification attack Oracles Adv QY, Qp-join, Qa-join, Qsig, Qreveal Represents the system collectively: good users and GM • Forges a sig. satisfying • it opens to none of the controlled group or • it tracesto none of the controlled group. • Secure against Misidentification from the hardness of one-more SDH rep. problem

42. Framing attack T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z T4 = r3W, T5 = e(P1, T4)x Oracles Adv QY, QS, Qb-join, Qsig Represents the system collectively: good users and GM • Forges a sig. satisfying • it opens to an innocent useror • it traces to an innocent user. • Secure against Framing • from the hardness of DLP.

43. Anonymity attack Oracles Adv T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z T4 = r3W, T5 = e(P1, T4)x QY, Qp-join, Qa-join, Qsig, Qreveal i0, i1 Selects two users i0 i1 (by name) Pick b randomly from {0,1} Generate a sig. σ of ib σ Guess b • The adv is not allowed to call Qreveal(i0) or Qreveal(i1) before or after i0 and i1 are chosen. • Secure against Anonymity from semantic security of linear encryption and the DDH

44. Security of Our scheme • Theorem : Under the q-SDH and DLDH assumption, our scheme is secure in the random oracle model.

45. Conclusion

46. Conclusion • Invented a New Technical Tool • One more SDH rep. problem based on q-SDH assumption • Constructed a Short Scheme • Ours: 362 bytes • 1.5 ~ 3 times the length of the RSA sig. • [KTY04] : 1206 bytes, [NS04]: 917 bytes • Proved the security formally