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Tight Bounds for Unconditional Authentication Protocols in Manual Channel & Shared Key Model

This paper discusses secure wireless device pairing through manual channel authentication, addressing active adversaries and avoiding the need for PKI. It presents a model for manually authenticated communication to minimize string length supported by previous works and optimal forgery probability.

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Tight Bounds for Unconditional Authentication Protocols in Manual Channel & Shared Key Model

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  1. Tight BoundsforUnconditional Authentication Protocolsin the Manual Channel and Shared Key Model s Gil Segev Moni Naor Adam Smith Weizmann Institute of ScienceIsrael

  2. Pairing of Wireless Devices gx Scenario: • Buy a new wireless camera • Want to establish a secure channel for the first time • E.g., Diffie-Hellman key agreement gy

  3. Pairing of Devices Wireless Cable pairing • Simple • Cheap • Authenticated channel “I thought this is a wireless camera…”

  4. Pairing of Wireless Devices Wireless pairing Problem: Active adversaries (“man-in-the-middle”)

  5. Pairing of Wireless Devices Wireless pairing gy gx ga gb Problem: Active adversaries (“man-in-the-middle”)

  6. ^ m Message Authentication • Assure the receiver of a message that it has not been changed by an active adversary m Alice Eve Bob

  7. ^ m = gb || gy Pairing of Wireless Devices gy gx ga gb m = gx || ga

  8. ^ m Message Authentication • Assure the receiver of a message that it has not been changed by an active adversary m Alice Eve Bob • Without additional setup: Impossible !! • Public Key: Signatures • Problem: No trusted PKI This Paper: Manual Channel

  9. The Manual Channel gy gx 141 ga gb 141 User can compare two short strings

  10. Manual Channel Model m Alice Bob s . . . s • Insecure communication channel • Low-bandwidth auxiliary channel: • Enables Alice to “manually” authenticate one short string s s • Adversarial power: • Choose the input message m • Insecure channel: Full control • Manual channel: Read, delay • Delivery timing

  11. Manual Channel Model m Alice Bob s . . . s • Insecure communication channel • Low-bandwidth auxiliary channel: • Enables Alice to “manually” authenticate one short string s s Goal:Minimize the length of the manually authenticated string

  12. Manual Channel Model m Alice Bob s . . . s s • No trusted infrastructure, such as: • Public key infrastructure • Shared secret key • Common reference string • ....... Suitable for ad hoc networks: • Pairing of wireless devices • Wireless USB, Bluetooth • Secure phones • AT&T, PGP, Zfone • Many more...

  13. The Manual Channel 141 141 Constants do matter! So how many bits can we manually authenticate? 20 ?40 ?160 ?????

  14. Previous Work • [Rivest & Shamir `84]: The “Interlock” protocol • Mutual authentication of public keys • No trusted infrastructure • AT&T, PGP,…, Zfone • [Vaudenay `05]: • Formal model • Computationally secure protocol for arbitrary long messages • log(1/)manually authenticated bits • [LAN `05, DDN `00]: Can be based on any one-way function (non-malleable commitments) • Efficient implementations: Forgery probability Optimal ! • Rely on a random oracle or • Assume a common reference string [DIO `98, DKOS `01]

  15. Previous Work • [Rivest & Shamir `84]: The “Interlock” protocol • Mutual authentication of public keys • No trusted infrastructure • AT&T, PGP,…, Zfone Computational Assumptions !! • [Vaudenay `05]: • Formal model • Computationally secure protocol for arbitrary long messages • log(1/)manually authenticated bits • [LAN `05, DDN `00]: Can be based on any one-way function (non-malleable commitments) • Efficient implementations: Forgery probability Optimal ! Are those really necessary? • Rely on a random oracle or • Assume a common reference string [DIO `98, DKOS `01]

  16. Our Results - Tight Bounds m n-bit . . . s ℓ-bit  forgery probability No setup or computational assumptions Only twice as many as [V05] • Upper bound:Constructed log*n-round protocol in which ℓ = 2log(1/) + O(1) • Matching lower bound: n  2log(1/)  ℓ  2log(1/) - 2 • One-way functions are necessary (and sufficient) for breaking the lower bound in the computational setting

  17. Unconditional Security Some advantages over computational security: • Security against unbounded adversaries • Exact evaluation of error probabilities • Protocols are often • easier to compose • more efficient Key agreement protocols

  18. Our Results - Tight Bounds ℓ ℓ = 2log(1/) ℓ = log(1/) One-way functions Unconditional security Computational security Impossible log(1/)

  19. Preliminaries: For m = m1 ... mk GF[Q]k and x GF[Q], let m(x) = mixi k  i = 1 Our Protocol (simplified) • Based on the [GN93] hashing technique • In each round, the parties: • Cooperatively choose a hash function • Reduce to authenticating a shorter message • A short message is manually authenticated ^ Then, for any m ≠ m and for any c, c  GF[Q], ^ ^ ^ Prob x RGF[Q] [ m(x) + c = m(x) + c ]  k/Q

  20. Preliminaries: For m = m1 ... mk GF[Q]k and x GF[Q], let m(x) = mixi k  i = 1 ^ Then, for any m ≠ m and for any c, c  GF[Q], ^ ^ ^ Prob x RGF[Q] [ m(x) + c = m(x) + c ]  k/Q Our Protocol (simplified) x || m(x) + c We hash m to One party chooses x Other party chooses c

  21. Our Protocol (simplified) Alice Bob m a1 a1R GF[Q1] b1R GF[Q1] b2 b1 a2R GF[Q2] b2R GF[Q2] m2 Accept iff m2 is consistent m1 = b1 || m(b1) + a1 Both parties set: Q1 n/ , Q2 log(n)/ m2 = a2 || m1(a2) + b2 2log(1/) + 2loglog(n) + O(1)manually authenticated bits Two GF[Q2]elements • k rounds 2loglog(n) is reduced to 2log(k-1)(n)

  22. Lower Bound - Intuition Alice Bob m, x1 x2 s • mR {0,1}n M, X1, X2, S are well defined random variables

  23. Lower Bound - Intuition Alice Bob M, X1 X2 S • Goal: H(S)  2log(1/) Evolving intuition: • The parties must use at least log(1/) random bits • Each party must use at least log(1/) random bits • Each party must independently reduce H(S) by log(1/) bits Alice’s randomness H(S) = H(S) - H(S | M, X1) + H(S | M, X1) - H(S | M, X1, X2) Bob’s randomness + H(S | M, X1, X2)

  24. Lower Bound - Intuition Alice Bob M, X1 X2 S • Goal: H(S)  2log(1/) H(S) - H(S | M, X1) + H(S | M, X1, X2)  log(1/) H(S | M, X1) - H(S | M, X1, X2)  log(1/) Alice’s randomness H(S) = H(S) - H(S | M, X1) + H(S | M, X1) - H(S | M, X1, X2) Bob’s randomness + H(S | M, X1, X2)

  25. ℓ = 2log(1/) ℓ ℓ = log(1/) One-way functions Unconditional security Computational security Impossible log(1/) Summary • Manual Channel • Computational assumptions are not necessary • Protocol • Matching lower bound • Sharp threshold between unconditional and computational

  26. Thank you ! • Research supported by • Adi Shamir’s Turing Award fund • Israel Science Foundation • Trip to CRYPTO supported by

  27. Backup

  28. Shared Secret Key • Known upper bound: [GN93]Interactive protocol withℓ = 2log(1/) + O(1) • Known lower bound (only non-interactive): ℓ  2log(1/)[GMS74, S84, S85, S88, M00] Our results: • Lower bound (interactive!): ℓ  2log(1/) • Even when authenticating one bit • Again, one-way functions are necessary for breaking the lower bound in the computational setting

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