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Efficient Ring Signatures: Waters' Scheme Enhanced with GOS NIZKs

Discussing an approach combining Waters' signatures and GOS NIZK techniques for efficient ring signatures without random oracles, focusing on United Chemical Corporation scenarios. This work introduces a modified Waters' signature scheme that offers enhanced security and anonymity. The proposed method allows encryption of public keys and signatures, ensuring verification for encrypted keys and messages. The discussion touches on setup assumptions and ongoing research areas. This paper introduces the first efficient ring signatures without random oracles and highlights the potential for future improvements in eliminating setup assumptions.

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Efficient Ring Signatures: Waters' Scheme Enhanced with GOS NIZKs

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  1. Efficient Ring Signatures Without Random Oracles Hovav Shacham and Brent Waters

  2. Alice’s Dilemma United Chemical Corporation

  3. Option 1: Come Forward United Chemical Corporation

  4. Option 1: Come Forward United Chemical Corporation Alice gets fired!

  5. Option 2: Anonymous Letter United Chemical Corporation Lack of Credibility

  6. Ring Signatures [RST’01] • Alice chooses a set of S public keys (that includes her own) • Signs a message M, on behalf of the “ring” of users • Integrity: Signed by some user in the set • Anonymity: Can’t tell which user signed

  7. Ring Signature Solution United Chemical Corporation

  8. Prior Work • Random Oracle Constructions • RST (Introduced) • DKNS (Constant Size • Generic [BKM’05] • Formalized definitions • Open – Efficient Construction w/o Random Oracles

  9. This work Waters’ Signatures GOS ’06 Style NIZK Techniques + Efficient Group Signatures w/o ROs =

  10. Our Approach • GOS encrypt one of a set of public keys 2) Sign and GOS encrypt message 3) Prove encrypted signature under encrypted key

  11. Bilinear groups of order N=pq [BGN’05] • G: group of order N=pq. (p,q) – secret. bilinear map: e: G  G  GT

  12. BGN encryption, GOS NIZK [GOS’06] • Subgroup assumption: G p Gp • E(m) : r  ZN , C  gm (gp)r  G • GOS NIZK: Statement: C  G Claim: “ C = E(0) or C = E(1) ’’ Proof:   G idea: IF: C = g  (gp)r or C = (gp)r THEN: e(C , Cg-1) = e(gp,gp)r  (GT)q

  13. Upshot of GOS proofs • Prove well-formed in one subgroup • “Hidden” by the other subgroup

  14. Waters’ Signature Scheme (Modified) • Global Setup: g, u’,u1,…,ulg(n), 2 G, A=ga2 G • Key-gen: Choose gb = PK, gab = PrivKey • Sign (M): (s1,s2) = gab(u’ ki=1 uMi)r, g-r • Verify: e(s1,g) e( s2, u’ ki=1 uMi) = e(A,gb)

  15. gb3 gab(u’ ki=1 uMi)r, g-r Our Approach • Alice encrypts her Waters PK • Alice encrypt signature • Prove signature verifies for encrypted key gb1 gb2 gb3

  16. A note on setup assumptions • Common reference string from N=pq for GOS proofs • Common Random String • Linear Assumption -- GOS Crypto ’06 • Upcoming work by Boyen ‘07 • Open: Efficient Ring Signatures w/o setup assumptions

  17. Conclusion • First efficient Ring Signatures w/o random oracles • Combined Waters’ signatures and GOS NIZKs • Encrypted one of several PK’s • Open: Removing setup assumptions

  18. THE END

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