1. Cryptography Resilient to Continual Memory Leakage Zvika Brakerski Weizmann Institute Yael Tauman Kalai Microsoft Jonathan Katz University of Maryland Vinod Vaikuntanathan IBM

2. Crypto with Leakage secret key is assumed to be truly random and secret RSA (and other schemes) are insecure when a small fraction of the secret key is leaked [Rivest-Shamir85, Coppersmith1996, Heninger-Shacham2009]

3. Computation Leaks Timing [Kocher 96] Power Consumption [Kocher et al. 98] EM Radiation [Quisquater 01]

4. Memory Leaks Cold-boot attack [Halderman-Schoen-Heninger-Clarkson-Paul-Calandrino- Feldman-Appelbaum-Felten 08]

5. Outline Motivation (for studying crypto with leakage) Modeling Leakage Our Results Previous work Our Techniques

6. Modeling Leakage Continual computation leakage (only computation leaks ) [Akavia-Goldwasser-Vaikuntanathan2009] [Dodis-K-Lovett2009] [Naor-Segev2009] [Katz-Vaikuntanathan2009] [Alwen-Dodis-Wichs2009] [Alwen-Dodis-Naor-Segev-Walfish-Wichs2009] [Dodis-Goldwasser-K-Peikert-Vaikuntanathan2010] [Goldwasser-K-Peikert-Vaikuntanathan2010] [Micali-Reyzin2004] [Dziembowski-Pietrzak2008] [Pietrzak2009] [Faust-Kiltz-Pietrzak-Rothblum2009] [Juma-Vahlis2010] [Goldwasswer-Rothblum2010] Bounded memory leakage Drawback! Other models: [Rivest97], [Ishai-Sahai-Wagner2003], [Ishai-Prabhakaran-Sahai-Wagner2006], [Faust-Rabin-Rezin-Tromer-Vaikuntanathan10]

7. Our Model: Continual Memory Leakage L 1 (sk) 1011 00 L 2 (sk) L 3 (sk) 11 0 10 0 110 Is it possible to secure against continual leakage? Note: Must update the secret key

8. Our Model: Continual Memory Leakage L 1 (sk 1 ) 1011 00 L 2 (sk 2 ) L 3 (sk 3 ) 000111 11 0 10 0 110 Challenge: This should be done without changing the public key! Note: Leakage is a function of the entire secret state. Leakage may occur during the update procedure or during the signing process.

9. Example: encryption scheme semantic security with continual memory leakage challenge

10. The updates are oblivious to other users. –Public-key stays the same. –Efficiency does not degrade with the number of updates. No bound on the total leakage over the lifetime of the system. –Amount of leakage is bounded only within each time period. Our Model: Continual Memory Leakage

11. Our Results Cryptographic schemes resilient to continual memory leakage (under the linear assumption over bilinear groups). Public-key encryption scheme Identity based encryption scheme Signature scheme * Thanks to Yevgeniy, Daniel and Gil for pointing us to improved analysis of algebraic lemma ** Thanks to Daniel for pointing us to this assumption

12. Main contributions 1. Efficient signature schemes (and more) in the continual memory leakage model under linear assumption over bilinear groups. Concurrent Work [Dodis-Haralambiev-LopezAlt-Wichs10] 1.Removing the “only computation leaks information” assumption 2. Public-key and identity-based encryption schemes (unknown even assuming “only computation leaks information”)

13. Prior Work: Bounded Memory Leakage [Akavia-Goldwasser-Vaikuntanathan2009]: Regev’s public-key encryption (and IBE) scheme is secure against leakage. [Naor-Segev2009]: several public-key encryption schemes secure against leakage. [Alwen-Dodis-Wichs2009]: Signature schemes secure against leakage in ROM [Katz-Vaikuntanathan2009]: Signature schemes secure against leakage under standard assumptions. [Dodis-K-Lovett2009]: Symmetric-key encryption scheme secure w.r.t. auxiliary input leakage. [Alwen-Dodis-Naor-Segev-Walfish-Wichs2009]: encryption in BRM [Dodis-Goldwasser-K-Peikert-Vaikuntanathan2010]: Several public- key encryption schemes secure w.r.t. auxiliary input leakage.

14. Prior Work: Continual Computation Leakage [MR04] [Dziembowski-Pietrzak08, Pietrzak09] Stream ciphers [Faust-Kiltz-Pietrzak-Rothblum09] Signature schemes [Juma-Vahlis10] [Goldwasser-Rothblum10] Encryption scheme??? Assumption: Only computation leaks information [Micali- Reyzin4] Programs resilient to side-channel attacks (using simple hardware)

15. Today Cryptographic schemes resilient to continual memory leakage (under the linear assumption over bilinear groups). Public-key encryption scheme Identity based encryption scheme Signature scheme

16. Algebraic Lemma: Random Subspaces are Resilient to Continual Leakage Many thanks to Yevgeniy, Daniel and Gil for pointing to an improved analysis [Dodis-Smith2005, Boldyreva-Fehr-O’Neill2008]

17. Algebraic Lemma: Random Subspaces are Resilient to Continual Leakage

18. Algebraic Lemma: Pictorially

19. Candidate Encryption Scheme

20. Update: ?

21. Our 1 st Encryption Scheme d (Thanks Daniel!) Assumption: DDH holds in each group

22. A Step in the Proof random subspace DDH

23. Our 2 nd Encryption Scheme d Cannot distinguish between rank 2 and rank 3 matrices in the exponent

24. Algebraic Lemma: Random Subspaces are Resilient to Continual Leakage

25. Algebraic Lemma: Pictorially

26. Security

27. Pictorial Proof random subspace Linear assumption

28. Security

29. General Proof Template

30. Additional Results 1. Tolerating leakage from the updates. 2. Converting our encryption scheme into an identity based encryption scheme [Brakerski-K10]. 3.General transformation for converting any encryption scheme resilient to continual memory leakage into a signature scheme resilient to continual memory leakage [Katz-Vaikuntanathan09].* * More complicated if we want to tolerate leakage during signing process.

31. We construct –Public key encryption scheme –Identity-based encryption scheme –Signature schemes In continual memory leakage model, under linear assumption over bilinear groups. Summary was not known even if we assume “only computation leaks information”.

32. Thanks !