Advertisement

How to Run Turing Machines on Encrypted Data

  • Shafi Goldwasser
  • Yael Tauman Kalai
  • Raluca Ada Popa
  • Vinod Vaikuntanathan
  • Nickolai Zeldovich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8043)

Abstract

Cryptographic schemes for computing on encrypted data promise to be a fundamental building block of cryptography. The way one models such algorithms has a crucial effect on the efficiency and usefulness of the resulting cryptographic schemes. As of today, almost all known schemes for fully homomorphic encryption, functional encryption, and garbling schemes work by modeling algorithms as circuits rather than as Turing machines.

As a consequence of thismodeling, evaluating an algorithmover encrypted data is as slow as the worst-case running time of that algorithm, a dire fact for many tasks. In addition, in settings where an evaluator needs a description of the algorithm itself in some “encoded” form, the cost of computing and communicating such encoding is as large as the worst-case running time of this algorithm.

In this work, we construct cryptographic schemes for computing Turing machines on encrypted data that avoid the worst-case problem. Specifically, we show:

  • An attribute-based encryption scheme for any polynomial-time Turing machine and Random Access Machine (RAM).

  • A (single-key and succinct) functional encryption scheme for any polynomialtime Turing machine.

  • A reusable garbling scheme for any polynomial-time Turing machine. These three schemes have the property that the size of a key or of a garbling for a Turing machine is very short: it depends only on the description of the Turing machine and not on its running time. Previously, the only existing constructions of such schemes were for depthd circuits, where all the parameters grow with d. Our constructions remove this depth d restriction, have short keys, and moreover, avoid the worst-case running time.

  • A variant of fully homomorphic encryption scheme for Turing machines, where one can evaluate a Turing machine M on an encrypted input x in time that is dependent on the running time of M on input x as opposed to the worstcase runtime of M. Previously, such a result was known only for a restricted class of Turing machines and it required an expensive preprocessing phase (with worst-case runtime); our constructions remove both restrictions.

Our results are obtained via a reduction from SNARKs (Bitanski et al) and an “extractable” variant of witness encryption, a scheme introduced by Garg et al.. We prove that the new assumption is secure in the generic group model. We also point out the connection between (the variant of) witness encryption and the obfuscation of point filter functions as defined by Goldwasser and Kalai in 2005.

Keywords

Computing on encrypted data Functional encryption Fully homomorphic encryption Turing machines Input-specific running time 

References

  1. [AGVW13]
    Agrawal, S., Gorbunov, S., Vaikuntanathan, V., Wee, H.: Functional encryption: New perspectives and lower bounds. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 500–518. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. [BCCT13]
    Bitansky, N., Canetti, R., Chiesa, A., Tromer, E.: Recursive composition and bootstrapping for SNARKs and proof-carrying data. In: STOC (2013)Google Scholar
  3. [BGV12]
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: ITCS, pp. 309–325 (2012)Google Scholar
  4. [BGW88]
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: STOC, pp. 1–10 (1988)Google Scholar
  5. [BHR12]
    Bellare, M., Hoang, V.T., Rogaway, P.: Foundations of garbled circuits. In: ACM CCS, pp. 784–796 (2012)Google Scholar
  6. [Bra12]
    Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical gapSVP. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 868–886. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  7. [BSW11]
    Boneh, D., Sahai, A., Waters, B.: Functional encryption: Definitions and challenges. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 253–273. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. [BV11a]
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS, pp. 97–106 (2011)Google Scholar
  9. [BV11b]
    Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 505–524. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. [CCD88]
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: STOC, pp. 11–19 (1988)Google Scholar
  11. [Gen09]
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178 (2009)Google Scholar
  12. [GGH13a]
    Garg, S., Gentry, C., Halevi, S.: Candidate multilinear maps from ideal lattices. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 1–17. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. GGH+13b.
    Garg, S., Gentry, C., Halevi, S., Sahai, A., Waters, B.: Attribute-based encryption for circuits from multilinear maps (2013)Google Scholar
  14. [GGSW13]
    Garg, S., Gentry, C., Sahai, A., Waters, B.: Witness encryption and its applications. In: STOC (2013)Google Scholar
  15. [GK05]
    Goldwasser, S., Kalai, Y.T.: On the impossibility of obfuscation with auxiliary input. In: FOCS, pp. 553–562 (2005)Google Scholar
  16. GKP+13a.
    Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: How to run Turing machines on encrypted data. Cryptology ePrint Archive, Report 2013/229 (2013), http://eprint.iacr.org/
  17. GKP+13a.
    Goldwasser, S., Kalai, Y.T., Popa, R.A., Vaikuntanathan, V., Zeldovich, N.: Reusable garbled circuits and succinct functional encryption. In: STOC (2013)Google Scholar
  18. [GMR88]
    Goldwasser, S., Micali, S., Rivest, R.L.: A digital signature scheme secure against adaptive chosen-message attacks. SIAM J. Comput., 281–308 (1988)Google Scholar
  19. [GMW87]
    Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game. In: STOC, pp. 218–229 (1987)Google Scholar
  20. [GPSW06]
    Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: ACM CCS, pp. 89–98 (2006)Google Scholar
  21. [GVW12]
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Functional encryption with bounded collusions via multi-party computation. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 162–179. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  22. [GVW13]
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Attribute-based encryption for circuits. In: STOC (2013)Google Scholar
  23. [KSW08]
    Katz, J., Sahai, A., Waters, B.: Predicate encryption supporting disjunctions, polynomial equations, and inner products. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 146–162. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  24. [LO12]
    Lu, S., Ostrovsky, R.: How to garble RAM programs? In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 719–734. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  25. LOS+10.
    Lewko, A., Okamoto, T., Sahai, A., Takashima, K., Waters, B.: Fully secure functional encryption: Attribute-based encryption and (Hierarchical) inner product encryption. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 62–91. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  26. [LW12]
    Lewko, A., Waters, B.: New proof methods for attribute-based encryption: Achieving full security through selective techniques. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 180–198. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  27. [PF79]
    Pippenger, N., Fischer, M.J.: Relations among complexity measures. J. ACM 26(2), 361–381 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [SS10]
    Sahai, A., Seyalioglu, H.: Worry-free encryption: functional encryption with public keys. In: ACM CCS, pp. 463–472 (2010)Google Scholar
  29. [SSW09]
    Shen, E., Shi, E., Waters, B.: Predicate privacy in encryption systems. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 457–473. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  30. [SW05]
    Sahai, A., Waters, B.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  31. [Yao86]
    Yao, A.C.: How to generate and exchange secrets (extended abstract). In: FOCS, pp. 162–167 (1986)Google Scholar

Copyright information

© International Association for Cryptologic Research 2013

Authors and Affiliations

  • Shafi Goldwasser
    • 1
  • Yael Tauman Kalai
    • 2
  • Raluca Ada Popa
    • 1
  • Vinod Vaikuntanathan
    • 3
  • Nickolai Zeldovich
    • 1
  1. 1.MIT CSAILUSA
  2. 2.Microsoft ResearchUSA
  3. 3.University of TorontoCanada

Personalised recommendations