Universal Re-encryption for Mix-Nets and Other Applications

1. Universal Re-encryption:For Mix-Nets and Other Applications(to appear CT-RSA ’04) Markus Jakobsson Ari Juels RSA Labs Paul Syverson NRL Philippe Golle Stanford (now at PARC) UMBC Protocol Meeting 10/01/03

2. Extra Acknowledgements • Most Mix illustrations cribbed from a talk by Ari Juels • Some RFID slides cribbed from Markus Jakobsson’s ACM WiSe’03 talk • See their Web pages for originals

3. Talk Outline • Background and Motivation for mixes and universal re-encryption • Definitions, Security Properties • Implementation using ElGamal • Applications: Bulletin Board, RFID Privacy • Security Arguments

4. message 1 message 2 message 3 message 4 What does a mix network do? Mix network Randomly permutes and decrypts inputs

5. ? What does a mix network do? message 2 Key property: Adversary can’t tell which ciphertext corresponds to a given message

6. From Bob Example application: Anonymizing bulletin board or e-mail From Alice From Charlie

7. “Nobody loves Bob” Is it Bob, Charlie, self-love, or other? “I love Charlie” “I love Alice” Example application: Anonymizing bulletin board or e-mail From Alice From Charlie From Bob

8. A look under the hood

9. Basic Mix (Chaum ‘81) PK3 PK1 PK2 Server 3 Server 2 Server 1

10. Encryption of Message PK3 PK1 PK2 message Ciphertext = EPK1[EPK2[EPK3[message]]]

11. m1 decrypt and permute m2 decrypt and permute m2 m2 decrypt and permute m3 m2 m3 m1 m1 m1 m3 m3 Basic Chaumian Mix Server 1 Server 2 Server 3

12. m1 decrypt and permute m2 decrypt and permute m2 m2 decrypt and permute m3 m2 m3 m1 m1 m1 m3 m3 Basic Chaumian Mix Observe: As long as one server is honest, privacy is preserved

13. ? Basic Chaumian Mix Server 1 Server 3 Server 2 m3

14. SK2 What if one server fails? Server 1 Server 2 Server 3 • Previous solution ideas: • Robustness: Share key among other mixes • Twinning Splitting • Reliability: Track and use reputable mixes

15. SK2 What if one server fails? Server 1 Server 2 Server 3 X • New Idea: Mixing without keys • No need to depend on any server (recovery mechanism) • No need to trust servers to protect keys • No need for PKI

16. A look further under the hood

17. Basic Re-encryption Mixnet • Inputs are ciphertexts • Outputs are a re-encryption of the inputs. • ElGamal public key encryption: • Anyone can encrypt with the public key e • Those who know the secret key d can also decrypt • Malleable: can produce E2(m) from E1(m) without knowing d • Verifiable • Multiplicative homomorphism: given E(m) and E(m’) I can produce E(mm’)

18. Universal Re-encryption Mixnet • Inputs are ciphertexts • Outputs are a re-encryption of the inputs. • ElGamal public key encryption: • Anyone can encrypt without the public key e • Those who know the secret key d can also decrypt • Messages encrypted with different keys are indistinguishable

19. Talk Outline • Background and Motivation for mixes and universal re-encryption • Definitions, Security Properties • Implementation using ElGamal • Applications: Bulletin Board, RFID Privacy • Security Arguments

20. Randomized Public-Key Cryptosystem • (PK, SK)  KG :generate key pairs • C  E(m, r, PK) :encryption of m • m  D(SK, C) :decryption of C

21. Semantic Security • Adversary chooses (m0 , m1 ) • Messages are encrypted • (C0 , C1 ) = (E(m0, r0, PK), E(m1, r1, PK)) and randomly permuted • If adversary determines order correctly no better than by guessing (within ) system is semantically secure

22. Re-encryption • Given: • Randomized public-key cryptosystem • Ciphertexts of form C = E(m, r, PK) • C’  Re(C, r’, PK) :Re-encryption of m

23. Semantic Security underRe-encryption • Adversary chooses (C0 , C1 ) • Messages are re-encrypted • (C0’, C1’) = (Re(C0, r0’, PK), Re(C1, r1’, PK)) and randomly permuted • If adversary gets order correct no better than by guessing (within ) system is semantically secure under re-encryption

24. Key-Privacy (Anonymity) • Adversary chooses message m • m is encrypted under PK0 and PK1 E(m, PK0 ) = C0 E(m, PK1 ) = C1 • If adversary guesses correspondence of ciphertext with public key with negligible advantage, scheme satisfieskey-privacy • Bellare et al. (ASIACRYPT’01) showed ElGamal provides anonymity under chosen-ciphertext assuming DDH.

25. Universal Re-encryption • Given: • Randomized public-key cryptosystem • Ciphertexts of form C = E(m, r, PK) • C’  URe(C, r’, PK) : Universal re-encryption of m X

26. Universal Semantic Security under Re-encryption • Combines semantic security and key-privacy • Given PK0 and PK1 adversary chooses (m0, m1, r0, r1 ) to produce (C0, C1 ) • Messages are universal re-encrypted (C0’, C1’) = (URe(C0, r0’), URe(C1, r1’)) • Ciphertexts are randomly ordered (Cb’, C1-b’) • If adversary gets order correct no better than by guessing (within ) system is universal semantically secure under re-encryption

27. Talk Outline • Background and Motivation for mixes and universal re-encryption • Definitions, Security Properties • ElGamal based Universal Re-encryption • Applications: Bulletin Board, RFID Privacy • Security Arguments

28. ElGamal Encryption • P,Q are prime, P = 2Q + 1 • GQ subgroup of ZP* of order Q • g generator of GQ • x GQ is private key • y = gx mod P is public key • E(m) = (gr, myr) where r GQ ,r random • D(G,M) = M/Gx = myr /gxr = m

29. ElGamal with Re-encryption • Ciphertext (G,M) • Re-encryption (G’,M’) = (Ggr’, Myr’) • Needs public key y but not private key • D(G’,M’) = M’/G’x = myrr’ /gxrr’ = m • Introduced for voting • Much work on efficient provable shuffles

30. Universal Re-encryption • (a,b) = (E[m]; E[1]) E is ElGamal enc • (a’,b’) = (R[b,k]a; R[b,k ’]) • R[*,k] is re-encryption with random k • (E[m]’, E[1]’)= ([(myr ykr’, gr gkr’), (yr’k’, gr’k’)] • D(E[m]’) = M’/G’x = mykrr’ /gxkrr’ = m

31. Symmetric-hybrid Encryption • U[k1],U[1],e[k1,m] • U[1] is universal blank can be converted to U[mi] can be reused • e[k1,m] is symmetric encryption of m • Final message U[k1],U[k2],…U[kn],e[kn,e[kn-1,…e[k1,m]…] • Can also do an asymmetric hybrid

32. Talk Outline • Background and Motivation for mixes and universal re-encryption • Definitions, Security Properties • Implementation using ElGamal • Applications: Low Volume Bulletin Board, RFID Privacy • Security Arguments

33. Universal Mixnet (Bulletin Board) • Senders post messages universally encrypted for recipients • Proof of Knowledge if nonmalleability desired • Any server can download, mix, and repost any or all messages • Servers can be dynamic • Shuffle proof if desired • No PKI and less trust of each server • No robustness/reliability issues with server failure • No overhead or threats from replay (universal semantic security)

34. Low Volume Bulletin Board • Suppose a bulletin board as above • Can mix with previous messages on board • Advantage: less delay retrieving new posts • Advantage: no need to detect replay (sort of) • Disadvantage: Must try decrypting all messages to find ones for you • Mitigate growth with message removal after PoK?

35. RFID Tags • EZ Pass automated toll payment • Supermarket shipment tracking, stock monitoring, theft prevention • Consumer stock monitoring, ordering • Consumer theft-protection of belongings • Implants in family pets • Monitoring cash flows (500 Euro notes)

36. Privacy Problems

37. Privacy Solutions?

38. Privacy Solutions?

39. Privacy Solutions? Method 2: “Put to sleep” RFID tags Problems: • No continuous use • Complexity, key management, trust

40. Privacy Solution: Blocker Tags (Juels, Rivest & Szydlo)

41. Universal Re-encryption for RFID Tag Privacy (Example) • Alice at supermarket checkout. • Uses PKAlice from fidelity card. • Cashier creates universal ciphertexts on Alice’s purchase IDs. • As Alice walks home passes readers that re-encrypt her tags or does it herself. • Alice enters home, tags decrypted for home use.

42. Security of ElGamal based BB • Correctness: Can do shuffle proofs of correct mixing • Communication privacy: If the universal cryptosystem is universal semantic-secure, then bulletin board construct provides communication privacy. • For ElGamal implementation, communication privacy reduced to DDH.

43. Conclusions • Universal Re-encryption: New primitive • Proven Security: • ElGamal BB is correct (wrt mixing) • ElGamal BB reducible to DDH. • Applications: • Reduced trust in mixes • Less complex mixnets (no PKI) • Privacy preserving RFID tags • Future • Reduce receiver overhead in bulletin board • Meteor Mixing (with George Danezis)