1. RECENT PROGRESS IN LEAKAGE-RESILIENT CRYPTOGRAPHY Daniel Wichs (NYU) (China Theory Week 2010)

2. Leakage Attacks  Cryptography relies on secrets.  Cryptographic devices:  In reality, many “side-channels”!  Timing, power, radiation, heat, acoustics… Secrets can leak!  Natural response: Not our problem.  Blame the “engineers” – they should fix this!  Theory/Crypto can help! input output Secret keys

3. Cryptography With Leakage  Can we do cryptography with incomplete secrecy?  Need a way to model leakage first!  In this talk: Adv can learn arbitrary information about the secret key as long as its amount is bounded. [AGV09]  Adv specifies any poly-time function Leak : {0,1} * ! {0,1} L.  Learns the output Leak(sk). sk Leak() L = leakage bound Leak(sk)

4. Leakage Resilient Cryptography  Password Login and One-Way Functions.  Identification Schemes and Signatures.  Public-Key Encryption.

5. Password Login Scheme (pk Bob, sk Bob ) pk Bob Prover BobVerifier Alice accept (pk Bob, sk Bob ) pk Bob Impersonation Stage reject! sk Bob sk’ Leakage Stage sk Bob Leak() Leak(sk)

6. Using One-Way Functions (pk Bob = f(x), sk Bob = x ) pk Bob = y Prover BobVerifier Alice Accept iff y = f(x) x  Standard OWF: get y = f(x), hard to find any x’ 2 f -1 (y).  Suffices for regular “password login” security  L-LR OWF: get y = f(x) & Leak(x), hard to find x’ 2 f -1 (y).  Not satisfied by general OWFs (easy counter-examples).  … but can be constructed from general OWFs.

7. OWF ) LR-OWF  OWF: get y = f(x), hard to find any x’ 2 f -1 (y). y=f(x) Domain Range

8. OWF ) LR-OWF  OWF: get y = f(x), hard to find any x’ 2 f -1 (y).  L-LR OWF: also get L bits of leakage about x. y=f(x) x Domain Range

9. OWF ) LR-OWF  OWF: get y = f(x), hard to find any x’ 2 f -1 (y).  L-LR OWF: also get L bits of leakage about x.  SPRF: get x, hard to find any x’ ≠ x s.t. f(x’)=f(x)  Non-triviality: input length n > output length k  Can build from any OWF for any n = poly(k) [Rom90] y=f(x) x x’ Domain Range

10. OWF ) SPRF ) LR-OWF  OWF: get y = f(x), hard to find any x’ 2 f -1 (y).  L-LR OWF: also get L bits of leakage about x.  SPRF: get x, hard to find any x’ ≠ x s.t. f(x’)=f(x)  Non-triviality: input length n > output length k  Can build from any OWF for any n = poly(k) [Rom90] Theorem [ADW09,KV09] : Any SPRF f : {0,1} n → {0,1} k is an L-LR OWF for L ¼ n - k.

11. Proof: Any SPRF is LR-OWF Theorem [ADW09,KV09] : Any SPRF f : {0,1} n → {0,1} k is an L-LR-OWF for L ¼ n – k. y=f(x) x Assume: Can break L-LR-OWF. There is an efficient A s.t. A( f(x), Leak(x) ) = x’ s.t. f(x’) = f(x) Conclude: Can break SPR. Let B(x) = A( f(x), Leak(x) ) B succeeds if (1) A succeeds (2) A does not return x’  x. A has too little info about x. |f(x)| + |Leak(x)| = k + L Pr[A guesses x] < 2 k+L - n

12. Proof: Any SPRF is LR-OWF Theorem [ADW09,KV09] : Any SPRF f : {0,1} n → {0,1} k is an L-LR-OWF for L ¼ n – k. Corollary: If OWF exist then L-LR-OWF exist with L = (1-o(1))n. Open Question: Can we get LR-OWF that are Permutations?

13. Leakage Resilient Cryptography  Password Login and One-Way Functions.  Identification Schemes and Signatures.  Public-Key Encryption.

14. Identification Schemes (pk Bob, sk Bob ) pk Bob Prover BobVerifier Alice accept Learning Stage (pk Bob, sk Bob ) pk Bob Impersonation Stage reject!

15. Leakage-Resilient Identification [ADW09] Learning Stage (pk Bob, sk Bob ) pk Bob Impersonation Stage reject!  Bob’s key can leak !!! (during learning stage, not afterward) sk Bob

16. Tool: Zero-Knowledge Proof of Knowledge Verifier Prover Accept/Reject –Witness Indistinguishable (WI): Even if V dishonest, cannot tell which x is being used by the prover. –Proof of Knowledge (PoK): Even if P dishonest, can extract some valid witness x’ for y from P. Instance y witness x NP relation R

17. ID Schemes from ZK-PoK  Assume: f : {0,1} n → {0,1} k is SPR and  is ZK-PoK for y = f(x). Thm [ADW09]:  is a secure L-LR ID scheme for L ¼ n-k. Pf: Assume Adv breaks ID security.

18. ID Schemes from ZK-PoK  Assume: f : {0,1} n → {0,1} k is SPR and  is ZK-PoK for y = f(x). Thm [ADW09]:  is a secure L-LR ID scheme for L ¼ n-k. Learning Stage (y, x ) yy Impersonation Stage x Pf: Assume Adv breaks ID security.

19. ID Schemes from ZK-PoK  Assume: f : {0,1} n → {0,1} k is SPR and  is ZK-PoK for y = f(x). Thm [ADW09]:  is a secure L-LR ID scheme for L ¼ n-k. Sees: y = f(x) Leakage, interaction with P(x) only k + L < n bits of info on x. Learning Stage y Impersonation Stage K bits L bits 0 bits Pf: Assume Adv breaks ID security. Witness Ind.

20. ID Schemes from ZK-PoK  Assume: f : {0,1} n → {0,1} k is SPR and  is ZK-PoK for y = f(x). Thm [ADW09]:  is a secure L-LR ID scheme for L ¼ n-k. Sees: y = f(x) Leakage, interaction with P(x) only k + L < n bits of info on x. Learning StageImpersonation Stage Extract x’ 2 f -1 (y) Pf: Assume Adv breaks ID security. x’  x Witness Ind.Proof-of-Knowledge

21. ID Schemes from ZK-PoK  Assume: f : {0,1} n → {0,1} k is SPR and  is ZK-PoK for y = f(x). Thm [ADW09]:  is a secure L-LR ID scheme for L ¼ n-k. Pf: Assume Adv breaks ID security. To break SPR: Simulate “Learning Stage” to Adv with x. Extract x’  x.

22. LR Signatures [ADW09,KV09,DHLW09,BSW10]  Similar to ID schemes with two big differences:  Cannot have interaction.  Need to bind each execution to a message.  Solution: use Non-Interactive ZK-PoK for x.  Various techniques to bind proofs to messages (tricky):  Rand Oracles [ADW09]  “Simulation-Sound” Proofs [KV09]  CCA Encryption [DHLW10]

23. Leakage Resilient Cryptography  Password Login and One-Way Functions.  Identification Schemes and Signatures.  Public-Key Encryption.

24. LR Public-Key Encryption [AGV09, NS09] Leakage on the decryption key prior to seeing the ciphertext.

25. Hash Proof Enc Scheme [AGV09, NS09]  Enc scheme with sk = x, pk = f(x) for some SPRF f. PK Public Key Space Secret Key space

26. Hash Proof Enc Scheme [AGV09, NS09]  Enc scheme with sk = x, pk = f(x) for some SPRF f. M DEC C SK M ENC PK

27. Hash Proof Enc Scheme [AGV09, NS09]  Enc scheme with sk = x, pk = f(x) for some SPRF f. DEC M C ENC PK

28. Hash Proof Enc Scheme [AGV09, NS09]  Enc scheme with sk = x, pk = f(x) for some SPRF f.  Correctness  All x 2 f -1 (pk) decrypt C to the correct M. M DEC M C ENC PK M M

29. Hash Proof Enc Scheme [AGV09, NS09]  Enc scheme with sk = x, pk = f(x) for some SPRF f.  Correctness  All x 2 f -1 (pk) decrypt C to the correct M.  Fake Encryption: C= Fake(pk). Decryption depends on x.  Can’t distinguish C from C (even given x). PK C Fake ENC M C Real ENC M1M1 M3M3 M2M2 ≈ DEC PK

30. Proof: Hash Proof Enc is LR [AGV09, NS09] L(SK) M1M1 M3M3 M2M2 C Fake ENC “Fake World”“Real World” M MC Real ENC PK DEC ? PK = y ≈

31. Back to Bigger Picture…

32. Criticism/Extensions  Q: What if leakage depends on complexity?  Bad: more resilience ) more complexity ) more leakage.  Fix: Bounded Retrieval Model [Dzi06,…,ADW09, ADNSWW10] [Complexity does not grow with resilience!]  Q: Why is leakage bounded overall? Should “leak-per-use”!  Continuous Leakage with “Key Updates” [DHLW10, BKKV10]  Q: Why measure leakage in output “bits”?  Noisy Leakage: use “entropy loss” [NS09, DHLW10]  Auxiliary Input: use “hardness of inverting” [DKL09,DGK+10]

33. Conclusions Riv97, Boy99, CDH+00, DSS01, KZ03, ISW03, MR04, DP08, GKR08, Pie09, AGV09, ADW09, DKL09, ADN+10, DGK+10, GKPV10, FKPR10, DHLW10a, FRRTV10, JRV10, GR10, DHLW10b, BKKV10, WL10, BSW10,… Many more models/results (esp. in last 2 years)... Many open questions, much still left to do!