Simultaneous Hardcore Bits and Cryptography against Memory Attacks

  • Adi Akavia
  • Shafi Goldwasser
  • Vinod Vaikuntanathan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5444)

Abstract

This paper considers two questions in cryptography.

Cryptography Secure Against Memory Attacks. A particularly devastating side-channel attack against cryptosystems, termed the “memory attack”, was proposed recently. In this attack, a significant fraction of the bits of a secret key of a cryptographic algorithm can be measured by an adversary if the secret key is ever stored in a part of memory which can be accessed even after power has been turned off for a short amount of time. Such an attack has been shown to completely compromise the security of various cryptosystems in use, including the RSA cryptosystem and AES.

We show that the public-key encryption scheme of Regev (STOC 2005), and the identity-based encryption scheme of Gentry, Peikert and Vaikuntanathan (STOC 2008) are remarkably robust against memory attacks where the adversary can measure a large fraction of the bits of the secret-key, or more generally, can compute an arbitrary function of the secret-key of bounded output length. This is done without increasing the size of the secret-key, and without introducing any complication of the natural encryption and decryption routines.

Simultaneous Hardcore Bits. We say that a block of bits of x are simultaneously hard-core for a one-way function f(x), if given f(x) they cannot be distinguished from a random string of the same length. Although any candidate one-way function can be shown to hide one hardcore bit and even a logarithmic number of simultaneously hardcore bits, there are few examples of one-way or trapdoor functions for which a linear number of the input bits have been proved simultaneously hardcore; the ones that are known relate the simultaneous security to the difficulty of factoring integers.

We show that for a lattice-based (injective) trapdoor function which is a variant of function proposed earlier by Gentry, Peikert and Vaikuntanathan, an N − o(N) number of input bits are simultaneously hardcore, where N is the total length of the input.

These two results rely on similar proof techniques.

References

  1. 1.
    Agrawal, D., Archambeault, B., Rao, J.R., Rohatgi, P.: The EM side-channel(s). In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 29–45. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Agrawal, D., Rao, J.R., Rohatgi, P.: Multi-channel attacks. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 2–16. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Ajtai, M.: Generating hard instances of the short basis problem. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 1–9. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Alexi, W., Chor, B., Goldreich, O., Schnorr, C.-P.: Rsa and rabin functions: Certain parts are as hard as the whole. SIAM J. Comput. 17(2), 194–209 (1988)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Alwen, J., Peikert, C.: Generating shorter bases for hard random lattices (manuscript, 2008)Google Scholar
  6. 6.
    Bellare, M., Fischlin, M., O’Neill, A., Ristenpart, T.: Deterministic encryption: Definitional equivalences and constructions without random oracles. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 360–378. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Blum, A., Furst, M., Kearns, M., Lipton, R.J.: Cryptographic primitives based on hard learning problems. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 278–291. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  8. 8.
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM J. Comput. 13(4), 850–864 (1984)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Boldyreva, A., Fehr, S., O’Neill, A.: On notions of security for deterministic encryption, and efficient constructions without random oracles. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 335–359. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Canetti, R., Eiger, D., Goldwasser, S., Lim, D.-Y.: How to protect yourself without perfect shredding. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 511–523. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Catalano, D., Gennaro, R., Howgrave-Graham, N.: Paillier’s trapdoor function hides up to O(n) bits. J. Cryptology 15(4), 251–269 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chari, S., Rao, J.R., Rohatgi, P.: Template attacks. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 13–28. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Coppersmith, D.: Small solutions to polynomial equations, and low exponent rsa vulnerabilities. J. Cryptology 10(4), 233–260 (1997)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dodis, Y., Reyzin, L., Smith, A.: Fuzzy extractors: How to generate strong keys from biometrics and other noisy data. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 523–540. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Dziembowski, S., Pietrzak, K.: Leakage-resilient stream ciphers. In: IEEE Foundations of Computer Science (to appear, 2008)Google Scholar
  16. 16.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197–206 (2008)Google Scholar
  17. 17.
    Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: STOC, pp. 25–32 (1989)Google Scholar
  18. 18.
    Goldreich, O., Rosen, V.: On the security of modular exponentiation with application to the construction of pseudorandom generators. Journal of Cryptology 16, 2003 (2000)Google Scholar
  19. 19.
    Goldwasser, S., Kalai, Y., Peikert, C., Vaikuntanathan, V (manuscript in preparation, 2008)Google Scholar
  20. 20.
    Goldwasser, S., Kalai, Y.T., Rothblum, G.N.: One-time programs. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 39–56. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Halderman, A., Schoen, S., Heninger, N., Clarkson, W., Paul, W., Calandrino, J., Feldman, A., Appelbaum, J., Felten, E.: Lest we remember: Cold boot attacks on encryption keys. In: Usenix Security Symposium (2008)Google Scholar
  23. 23.
    Håstad, J., Näslund, M.: The security of individual rsa bits. In: FOCS, pp. 510–521 (1998)Google Scholar
  24. 24.
    Håstad, J., Schrift, A.W., Shamir, A.: The discrete logarithm modulo a composite hides o(n) bits. J. Comput. Syst. Sci. 47(3), 376–404 (1993)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ishai, Y., Prabhakaran, M., Sahai, A., Wagner, D.: Private circuits II: Keeping secrets in tamperable circuits. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 308–327. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    Ishai, Y., Sahai, A., Wagner, D.: Private circuits: Securing hardware against probing attacks. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 463–481. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  27. 27.
    Kaliski Jr., B.S.: A pseudo-random bit generator based on elliptic logarithms. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 84–103. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  28. 28.
    Kocher, P.C.: Timing attacks on implementations of diffie-hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996)Google Scholar
  29. 29.
    Kocher, P.C., Jaffe, J., Jun, B.: Differential power analysis. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 388–397. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  30. 30.
    Liu, Y.-K., Lyubashevsky, V., Micciancio, D.: On bounded distance decoding for general lattices. In: APPROX-RANDOM, pp. 450–461 (2006)Google Scholar
  31. 31.
    Long, D.L., Wigderson, A.: The discrete logarithm hides o(log n) bits. SIAM J. Comput. 17(2), 363–372 (1988)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Side-Channel Cryptanalysis Lounge (2008), http://www.crypto.rub.de/en_sclounge.html.
  33. 33.
    Micali, S., Reyzin, L.: Physically observable cryptography. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 278–296. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  34. 34.
    Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem. Cryptology ePrint Archive, Report 2008/481 (2008), http://eprint.iacr.org/
  35. 35.
    Peikert, C., Vaikuntanathan, V., Waters, B.: A framework for efficient and composable oblivious transfer. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 554–571. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  36. 36.
    Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. In: STOC, pp. 187–196 (2008)Google Scholar
  37. 37.
    Petit, C., Standaert, F.-X., Pereira, O., Malkin, T., Yung, M.: A block cipher based pseudo random number generator secure against side-channel key recovery. In: ASIACCS, pp. 56–65 (2008)Google Scholar
  38. 38.
    Pietrzak, K., Vaikuntanathan, V.: Personal Communication (2009)Google Scholar
  39. 39.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC, pp. 84–93 (2005)Google Scholar
  40. 40.
    Rosen, A., Segev, G.: Chosen-ciphertext security via correlated products. Cryptology ePrint Archive, Report 2008/116 (2008)Google Scholar
  41. 41.
    Vazirani, U.V., Vazirani, V.V.: Efficient and secure pseudo-random number generation. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 193–202. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  42. 42.
    Yao, A.C.: Theory and application of trapdoor functions. In: Symposium on Foundations of Computer Science, pp. 80–91 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Adi Akavia
    • 1
  • Shafi Goldwasser
    • 2
  • Vinod Vaikuntanathan
    • 3
  1. 1.IAS and DIMACSUSA
  2. 2.MIT and Weizmann InsituteUSA
  3. 3.MIT and IBM ResearchUSA

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