Advertisement

Homomorphic Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based

  • Craig Gentry
  • Amit Sahai
  • Brent Waters
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8042)

Abstract

We describe a comparatively simple fully homomorphic encryption (FHE) scheme based on the learning with errors (LWE) problem. In previous LWE-based FHE schemes, multiplication is a complicated and expensive step involving “relinearization”. In this work, we propose a new technique for building FHE schemes that we call the approximate eigenvector method. In our scheme, for the most part, homomorphic addition and multiplication are just matrix addition and multiplication. This makes our scheme both asymptotically faster and (we believe) easier to understand.

In previous schemes, the homomorphic evaluator needs to obtain the user’s “evaluation key”, which consists of a chain of encrypted secret keys. Our scheme has no evaluation key. The evaluator can do homomorphic operations without knowing the user’s public key at all, except for some basic parameters. This fact helps us construct the first identity-based FHE scheme. Using similar techniques, we show how to compile a recent attribute-based encryption scheme for circuits by Gorbunov et al. into an attribute-based FHE scheme that permits data encrypted under the same index to be processed homomorphically.

References

  1. [ABB10a]
    Agrawal, S., Boneh, D., Boyen, X.: Efficient lattice (h)ibe in the standard model. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 553–572. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. [ABB10b]
    Agrawal, S., Boneh, D., Boyen, X.: Lattice basis delegation in fixed dimension and shorter-ciphertext hierarchical IBE. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 98–115. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. [BB12a]
    Barak, B., Brakerski, Z.: Building the swiss army knife. Windows on Theory Blog (2012), http://windowsontheory.org/2012/05/02/building-the-swiss-army-knife
  4. [BB12b]
    Barak, B., Brakerski, Z.: The swiss army knife of cryptography. Windows on Theory Blog (2012), http://windowsontheory.org/2012/05/01/the-swiss-army-knife-of-cryptography
  5. [BF03]
    Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. SIAM J. of Computing 32(3), 586–615 (2003); Extended abstract in Kilian, J. (ed.): CRYPTO 2001. LNCS, vol. 2139, pp. 586–615. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  6. [BGV12]
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: Fully homomorphic encryption without bootstrapping. In: Innovations in Theoretical Computer Science, ITCS 2012 (2012), http://eprint.iacr.org/2011/277
  7. [BLP+13]
    Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: STOC, pp. 575–584 (2013)Google Scholar
  8. [Bra12]
    Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical gapSVP. In: Safavi-Naini, R. (ed.) CRYPTO 2012. LNCS, vol. 7417, pp. 868–886. Springer, Heidelberg (2012)Google Scholar
  9. [BV11a]
    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. [BV11b]
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS, pp. 97–106 (2011), http://eprint.iacr.org/2011/344
  11. [CHKP10]
    Cash, D., Hofheinz, D., Kiltz, E., Peikert, C.: Bonsai trees, or how to delegate a lattice basis. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 523–552. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. [CHT13]
    Clear, M., Hughes, A., Tewari, H.: Homomorphic encryption with access policies: Characterization and new constructions. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds.) AFRICACRYPT 2013. LNCS, vol. 7918, pp. 61–87. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. [CLT13]
    Coron, J.-S., Lepoint, T., Tibouchi, M.: Practical multilinear maps over the integers. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 476–493. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. [CMNT11]
    Coron, J.-S., Mandal, A., Naccache, D., Tibouchi, M.: Fully homomorphic encryption over the integers with shorter public keys. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 487–504. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. [CNT12]
    Coron, J.-S., Naccache, D., Tibouchi, M.: Public key compression and modulus switching for fully homomorphic encryption over the integers. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 446–464. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  16. [Gen09]
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178 (2009)Google Scholar
  17. [Gen10]
    Gentry, C.: Toward basing fully homomorphic encryption on worst-case hardness. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 116–137. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. [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
  19. [GGH+13b]
    Garg, S., Gentry, C., Halevi, S., Sahai, A., Waters, B.: Attribute-based encryption for circuits from multilinear maps. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 479–499. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  20. [GH11a]
    Gentry, C., Halevi, S.: Fully homomorphic encryption without squashing using depth-3 arithmetic circuits. In: FOCS, pp. 107–109 (2011)Google Scholar
  21. [GH11b]
    Gentry, C., Halevi, S.: Implementing gentry’s fully-homomorphic encryption scheme. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 129–148. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  22. [GHS12a]
    Gentry, C., Halevi, S., Smart, N.P.: Fully homomorphic encryption with polylog overhead. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 465–482. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. [GHS12b]
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R. (ed.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012)Google Scholar
  24. [GHV10]
    Gentry, C., Halevi, S., Vaikuntanathan, V.: A simple BGN-type cryptosystem from LWE. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 506–522. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  25. [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
  26. [GPV08]
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp. 197–206. ACM (2008)Google Scholar
  27. [GVW13]
    Gorbunov, S., Vaikuntanathan, V., Wee, H.: Attribute-based encryption for circuits. In: STOC, pp. 545–554 (2013)Google Scholar
  28. [LATV12]
    López-Alt, A., Tromer, E., Vaikuntanathan, V.: On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: STOC, pp. 1219–1234 (2012)Google Scholar
  29. [LLL82]
    Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Mathematische Annalen 261, 515–534 (1982), doi:10.1007/BF01457454MathSciNetCrossRefzbMATHGoogle Scholar
  30. [LPR10]
    Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  31. [MM11]
    Micciancio, D., Mol, P.: Pseudorandom knapsacks and the sample complexity of LWE search-to-decision reductions. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 465–484. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  32. [MP12]
    Micciancio, D., Peikert, C.: Trapdoors for lattices: Simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  33. [Nac10]
    Naccache, D.: Is theoretical cryptography any good in practice? Invited talk at Crypto/CHES 2010 (2010), http://www.iacr.org/workshops/ches/ches2010
  34. [Pei09]
    Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem: extended abstract. In: STOC, pp. 333–342 (2009)Google Scholar
  35. [RAD78]
    Rivest, R., Adleman, L., Dertouzos, M.L.: On data banks and privacy homomorphisms. In: Foundations of Secure Computation, pp. 169–180 (1978)Google Scholar
  36. [Reg05]
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC, pp. 84–93 (2005)Google Scholar
  37. [Sha84]
    Shamir, A.: Identity-based cryptosystems and signature schemes. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 47–53. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  38. [Str69]
    Strassen, V.: Gaussian elimination is not optimal. Numer. Math. 13, 354–356 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  39. [SV10]
    Smart, N.P., Vercauteren, F.: Fully homomorphic encryption with relatively small key and ciphertext sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  40. [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
  41. [vDGHV10]
    van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. EUROCRYPT, pp. 24–43. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  42. [Wil12]
    Williams, V.V.: Multiplying matrices faster than coppersmith-winograd. In: STOC, pp. 887–898 (2012)Google Scholar

Copyright information

© International Association for Cryptologic Research 2013

Authors and Affiliations

  • Craig Gentry
    • 1
  • Amit Sahai
    • 2
  • Brent Waters
    • 3
  1. 1.IBM ResearchUSA
  2. 2.UCLAUSA
  3. 3.UT AustinUSA

Personalised recommendations