Advertisement

Short Discrete Log Proofs for FHE and Ring-LWE Ciphertexts

  • Rafael del PinoEmail author
  • Vadim Lyubashevsky
  • Gregor Seiler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11442)

Abstract

In applications of fully-homomorphic encryption (FHE) that involve computation on encryptions produced by several users, it is important that each user proves that her input is indeed well-formed. This may simply mean that the inputs are valid FHE ciphertexts or, more generally, that the plaintexts m additionally satisfy \(f(m)=1\) for some public function f. The most efficient FHE schemes are based on the hardness of the Ring-LWE problem and so a natural solution would be to use lattice-based zero-knowledge proofs for proving properties about the ciphertext. Such methods, however, require larger-than-necessary parameters and result in rather long proofs, especially when proving general relationships.

In this paper, we show that one can get much shorter proofs (roughly 1.25 KB) by first creating a Pedersen commitment from the vector corresponding to the randomness and plaintext of the FHE ciphertext. To prove validity of the ciphertext, one can then prove that this commitment is indeed to the message and randomness and these values are in the correct range. Our protocol utilizes a connection between polynomial operations in the lattice scheme and inner product proofs for Pedersen commitments of Bünz et al. (S&P 2018). Furthermore, our proof of equality between the ciphertext and the commitment is very amenable to amortization – proving the equivalence of k ciphertext/commitment pairs only requires an additive factor of \(O(\log {k})\) extra space than for one such proof. For proving additional properties of the plaintext(s), one can then directly use the logarithmic-space proofs of Bootle et al. (Eurocrypt 2016) and Bünz et al. (IEEE S&P 2018) for proving arbitrary relations of discrete log commitment.

Our technique is not restricted to FHE ciphertexts and can be applied to proving many other relations that arise in lattice-based cryptography. For example, we can create very efficient verifiable encryption/decryption schemes with short proofs in which confidentiality is based on the hardness of Ring-LWE while the soundness is based on the discrete logarithm problem. While such proofs are not fully post-quantum, they are adequate in scenarios where secrecy needs to be future-proofed, but one only needs to be convinced of the validity of the proof in the pre-quantum era. We furthermore show that our zero-knowledge protocol can be easily modified to have the property that breaking soundness implies solving discrete log in a short amount of time. Since building quantum computers capable of solving discrete logarithm in seconds requires overcoming many more fundamental challenges, such proofs may even remain valid in the post-quantum era.

Notes

Acknowledgements

We thank the reviewers for their careful reading of the proofs and useful suggestions. This work was supported in part by the SNSF ERC Transfer Grant CRETP2-166734 FELICITY and H2020 FutureTPM.

References

  1. [ABB+18]
    Androulaki, E., et al.: Hyperledger fabric: a distributed operating system for permissioned blockchains. CoRR, abs/1801.10228 (2018)Google Scholar
  2. [ADPS16]
    Alkim, E., Ducas, L., Pöppelmann, T., Schwabe, P.: Post-quantum key exchange - a new hope. In: USENIX, pp. 327–343 (2016)Google Scholar
  3. [BBB+17]
    Bünz, B., Bootle, J., Boneh, D., Poelstra, A., Wuille, P., Maxwell, G.: Bulletproofs: short proofs for confidential transactions and more. IACR Cryptology ePrint Archive, 2017:1066 (2017)Google Scholar
  4. [BCC+16]
    Bootle, J., Cerulli, A., Chaidos, P., Groth, J., Petit, C.: Efficient zero-knowledge arguments for arithmetic circuits in the discrete log setting. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 327–357. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_12CrossRefzbMATHGoogle Scholar
  5. [BCK+14]
    Benhamouda, F., Camenisch, J., Krenn, S., Lyubashevsky, V., Neven, G.: Better zero-knowledge proofs for lattice encryption and their application to group signatures. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8873, pp. 551–572. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-45611-8_29CrossRefGoogle Scholar
  6. [BGV12]
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: ITCS, pp. 309–325 (2012)Google Scholar
  7. [CS03]
    Camenisch, J., Shoup, V.: Practical verifiable encryption and decryption of discrete logarithms. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 126–144. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-45146-4_8CrossRefGoogle Scholar
  8. [FGP14]
    Fiore, D., Gennaro, R., Pastro, V.: Efficiently verifiable computation on encrypted data. In: Proceedings of the 2014 ACM SIGSAC Conference on Computer and Communications Security, Scottsdale, AZ, USA, 3–7 November 2014, pp. 844–855 (2014)Google Scholar
  9. [FMMC12]
    Fowler, A.G., Mariantoni, M., Martinis, J.M., Cleland, A.N.: Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012)CrossRefGoogle Scholar
  10. [Gid18]
    Gidney, C.: Why will quantum computers be slow? (2018). http://algassert.com/post/1800. Accessed 6 Mar 2019
  11. [LLNW18]
    Libert, B., Ling, S., Nguyen, K., Wang, H.: Lattice-based zero-knowledge arguments for integer relations. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 700–732. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-96881-0_24CrossRefGoogle Scholar
  12. [LM06]
    Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006, Part II. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006).  https://doi.org/10.1007/11787006_13CrossRefGoogle Scholar
  13. [LN17]
    Lyubashevsky, V., Neven, G.: One-shot verifiable encryption from lattices. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10210, pp. 293–323. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-56620-7_11CrossRefGoogle Scholar
  14. [LPR13]
    Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. J. ACM 60(6), 43 (2013). Preliminary Version Appeared in EUROCRYPT 2010MathSciNetCrossRefGoogle Scholar
  15. [LTV12]
    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
  16. [LWF+17]
    Lekitsch, B., et al.: Blueprint for a microwave trapped ion quantum computer. Sci. Adv. 3(2), e1601540 (2017)CrossRefGoogle Scholar
  17. [MW16]
    Mukherjee, P., Wichs, D.: Two round multiparty computation via multi-key FHE. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 735–763. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_26CrossRefGoogle Scholar
  18. [Pip80]
    Pippenger, N.: On the evaluation of powers and monomials. SIAM J. Comput. 9(2), 230–250 (1980)MathSciNetCrossRefGoogle Scholar
  19. [PR06]
    Peikert, C., Rosen, A.: Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145–166. Springer, Heidelberg (2006).  https://doi.org/10.1007/11681878_8CrossRefGoogle Scholar
  20. [PS16]
    Peikert, C., Shiehian, S.: Multi-key FHE from LWE, revisited. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 217–238. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53644-5_9CrossRefGoogle Scholar
  21. [WTS+18]
    Wahby, R.S., Tzialla, I., Shelat, A., Thaler, J., Walfish, M.: Doubly-efficient zkSNARKs without trusted setup. In: Proceedings of the 2018 IEEE Symposium on Security and Privacy, SP 2018, San Francisco, California, USA, 21–23 May 2018, pp. 926–943 (2018)Google Scholar

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  • Rafael del Pino
    • 1
    Email author
  • Vadim Lyubashevsky
    • 2
  • Gregor Seiler
    • 2
    • 3
  1. 1.ENS ParisParisFrance
  2. 2.IBM Research – ZurichZurichSwitzerland
  3. 3.ETH ZurichZurichSwitzerland

Personalised recommendations