Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Noninteractive Zero Knowledge for NP from (Plain) Learning with Errors

  • Conference paper
  • First Online:
Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11692))

Included in the following conference series:

  • 3519 Accesses

Abstract

We finally close the long-standing problem of constructing a noninteractive zero-knowledge (NIZK) proof system for any NP language with security based on the plain Learning With Errors (LWE) problem, and thereby on worst-case lattice problems. Our proof system instantiates the framework recently developed by Canetti et al.  [EUROCRYPT’18], Holmgren and Lombardi [FOCS’18], and Canetti et al.  [STOC’19] for soundly applying the Fiat–Shamir transform using a hash function family that is correlation intractable for a suitable class of relations. Previously, such hash families were based either on “exotic” assumptions (e.g., indistinguishability obfuscation or optimal hardness of certain LWE variants) or, more recently, on the existence of circularly secure fully homomorphic encryption (FHE). However, none of these assumptions are known to be implied by plain LWE or worst-case hardness.

Our main technical contribution is a hash family that is correlation intractable for arbitrary size-S circuits, for any polynomially bounded S, based on plain LWE (with small polynomial approximation factors). The construction combines two novel ingredients: a correlation-intractable hash family for log-depth circuits based on LWE (or even the potentially harder Short Integer Solution problem), and a “bootstrapping” transform that uses (leveled) FHE to promote correlation intractability for the FHE decryption circuit to arbitrary (bounded) circuits. Our construction can be instantiated in two possible “modes,” yielding a NIZK that is either computationally sound and statistically zero knowledge in the common random string model, or vice-versa in the common reference string model.

This material is based upon work supported by the National Science Foundation under CAREER Award CCF-1054495 and CNS-1606362. The views expressed are those of the authors and do not necessarily reflect the official policy or position of the National Science Foundation or the Sloan Foundation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    For simplicity, here we assume that \(\mathsf {FHE} \) supports unbounded, not just leveled, homomorphic evaluation. Adapting the construction to leveled FHE is straightforward because \(\mathsf {Eval} \) is used only on circuits of bounded depth.

  2. 2.

    The reader might notice that the specific function \(\mathsf {Dec} _{sk}\) is not fixed in advance, but is instead chosen at random by the reduction. This is addressed in the non-uniform setting by “fixing coins” for \(\mathsf {FHE} {.}\mathsf {Gen} \) that maximize the attacker’s success probability, or in the uniform setting by adopting a security definition that lets the adversary declare a (valid) target function before receiving the hash key.

  3. 3.

    Those familiar with the literature will recognize this as the linear transform induced by the “gadget” matrix \(\mathbf {G}\).

  4. 4.

    With this change, the SIS-based proof still goes through, thanks to the technique for ensuring that \(\mathbf {r}\ne \mathbf {0}\).

  5. 5.

    This change also turns the fully homomorphic commitment scheme into the GSW FHE scheme [25, 28], but we do not need its decryption capability.

References

  1. Ajtai, M.: Generating hard instances of lattice problems. Quaderni di Matematica 13, 1–32 (2004). Preliminary version in STOC 1996

    MathSciNet  MATH  Google Scholar 

  2. Alamati, N., Peikert, C., Stephens-Davidowitz, N.: New (and old) proof systems for lattice problems. In: Abdalla, M., Dahab, R. (eds.) PKC 2018. LNCS, vol. 10770, pp. 619–643. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76581-5_21

    Chapter  Google Scholar 

  3. Alperin-Sheriff, J., Peikert, C.: Faster bootstrapping with polynomial error. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 297–314. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_17

    Chapter  Google Scholar 

  4. Applebaum, B., Cash, D., Peikert, C., Sahai, A.: Fast cryptographic primitives and circular-secure encryption based on hard learning problems. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 595–618. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_35

    Chapter  Google Scholar 

  5. Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC\(^1\). J. Comput. Syst. Sci. 38(1), 150–164 (1989). Preliminary version in STOC 1986

    Article  MathSciNet  Google Scholar 

  6. Bellare, M., Goldwasser, S.: New paradigms for digital signatures and message authentication based on non-interactive zero knowledge proofs. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 194–211. Springer, New York (1990). https://doi.org/10.1007/0-387-34805-0_19

    Chapter  Google Scholar 

  7. Ben-Sasson, E., et al.: Zerocash: decentralized anonymous payments from Bitcoin. In: 2014 IEEE Symposium on Security and Privacy, SP 2014, Berkeley, 18–21 May 2014, pp. 459–474 (2014)

    Google Scholar 

  8. Blum, M., De Santis, A., Micali, S., Persiano, G.: Noninteractive zero-knowledge. SIAM J. Comput. 20(6), 1084–1118 (1991). Preliminary version in STOC 1988

    Article  MathSciNet  Google Scholar 

  9. Blum, M., Feldman, P., Micali, S.: Non-interactive zero-knowledge and its applications (extended abstract). In: STOC, pp. 103–112 (1988)

    Google Scholar 

  10. Boneh, D., et al.: Fully key-homomorphic encryption, arithmetic circuit ABE and compact garbled circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 533–556. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_30

    Chapter  Google Scholar 

  11. Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: ITCS, pp. 309–325 (2012)

    Google Scholar 

  12. Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: STOC, pp. 575–584 (2013)

    Google Scholar 

  13. Brakerski, Z., Vaikuntanathan, V.: Lattice-based FHE as secure as PKE. In: ITCS, pp. 1–12 (2014)

    Google Scholar 

  14. Camenisch, J., Chandran, N., Shoup, V.: A public key encryption scheme secure against key dependent chosen plaintext and adaptive chosen ciphertext attacks. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 351–368. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01001-9_20

    Chapter  Google Scholar 

  15. Canetti, R., et al.: Fiat-Shamir: from practice to theory. In: STOC, pp. 1082–1090 (2019)

    Google Scholar 

  16. Canetti, R., Chen, Y., Reyzin, L., Rothblum, R.D.: Fiat-Shamir and correlation intractability from strong KDM-secure encryption. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 91–122. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_4

    Chapter  Google Scholar 

  17. Canetti, R., Goldreich, O., Halevi, S.: The random oracle methodology, revisited. J. ACM 51(4), 557–594 (2004). Preliminary version in STOC 1998

    Article  MathSciNet  Google Scholar 

  18. Chung, K., Dadush, D., Liu, F., Peikert, C.: On the lattice smoothing parameter problem. In: IEEE Conference on Computational Complexity, pp. 230–241 (2013)

    Google Scholar 

  19. Dwork, C., Naor, M.: Zaps and their applications. SIAM J. Comput. 36(6), 1513–1543 (2007)

    Article  MathSciNet  Google Scholar 

  20. Feige, U., Lapidot, D., Shamir, A.: Multiple noninteractive zero knowledge proofs under general assumptions. SIAM J. Comput. 29(1), 1–28 (1999). Preliminary version in FOCS 1990

    Article  MathSciNet  Google Scholar 

  21. Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_12

    Chapter  Google Scholar 

  22. Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: FOCS, pp. 40–49 (2013)

    Google Scholar 

  23. Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009). http://crypto.stanford.edu/craig

  24. Gentry, C., Groth, J., Ishai, Y., Peikert, C., Sahai, A., Smith, A.D.: Using fully homomorphic hybrid encryption to minimize non-interative zero-knowledge proofs. J. Cryptol. 28(4), 820–843 (2015)

    Article  MathSciNet  Google Scholar 

  25. Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_5

    Chapter  Google Scholar 

  26. Goldreich, O., Oren, Y.: Definitions and properties of zero-knowledge proof systems. J. Cryptol. 7(1), 1–32 (1994)

    Article  MathSciNet  Google Scholar 

  27. Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989). Preliminary version in STOC 1985

    Article  MathSciNet  Google Scholar 

  28. Gorbunov, S., Vaikuntanathan, V., Wichs, D.: Leveled fully homomorphic signatures from standard lattices. In: STOC, pp. 469–477 (2015)

    Google Scholar 

  29. Groth, J., Ostrovsky, R., Sahai, A.: New techniques for noninteractive zero-knowledge. J. ACM 59(3), 11:1–11:35 (2012). Preliminary version in EUROCRYPT 2006

    Article  MathSciNet  Google Scholar 

  30. Holmgren, J., Lombardi, A.: Cryptographic hashing from strong one-way functions (or: One-way product functions and their applications). In: FOCS, pp. 850–858 (2018)

    Google Scholar 

  31. Kalai, Y.T., Rothblum, G.N., Rothblum, R.D.: From obfuscation to the security of Fiat-Shamir for proofs. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10402, pp. 224–251. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63715-0_8

    Chapter  Google Scholar 

  32. Kim, S., Wu, D.J.: Multi-theorem preprocessing NIZKs from lattices. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 733–765. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_25

    Chapter  Google Scholar 

  33. 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). https://doi.org/10.1007/978-3-642-29011-4_41

    Chapter  Google Scholar 

  34. Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. SIAM J. Comput. 37(1), 267–302 (2007). Preliminary version in FOCS 2004

    Article  MathSciNet  Google Scholar 

  35. Micciancio, D., Vadhan, S.P.: Statistical zero-knowledge proofs with efficient provers: lattice problems and more. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 282–298. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_17

    Chapter  Google Scholar 

  36. Naor, M., Yung, M.: Public-key cryptosystems provably secure against chosen ciphertext attacks. In: STOC, pp. 427–437 (1990)

    Google Scholar 

  37. Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem. In: STOC, pp. 333–342 (2009)

    Google Scholar 

  38. Peikert, C., Regev, O., Stephens-Davidowitz, N.: Pseudorandomness of Ring-LWE for any ring and modulus. In: STOC, pp. 461–473 (2017)

    Google Scholar 

  39. Peikert, C., Vaikuntanathan, V.: Noninteractive statistical zero-knowledge proofs for lattice problems. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 536–553. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85174-5_30

    Chapter  Google Scholar 

  40. 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). https://doi.org/10.1007/978-3-540-85174-5_31

    Chapter  Google Scholar 

  41. Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. SIAM J. Comput. 40(6), 1803–1844 (2011). Preliminary version in STOC 2008

    Article  MathSciNet  Google Scholar 

  42. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM 56(6), 1–40 (2009). Preliminary version in STOC 2005

    Article  MathSciNet  Google Scholar 

  43. Sahai, A., Waters, B.: How to use indistinguishability obfuscation: deniable encryption, and more. In: STOC, pp. 475–484 (2014)

    Google Scholar 

Download references

Acknowledgments

We thank Alex Lombardi and Daniel Wichs for useful comments.

Author information

Authors and Affiliations

  1. Computer Science and Engineering, University of Michigan, Ann Arbor, USA

    Chris Peikert & Sina Shiehian

Authors
  1. Chris Peikert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sina Shiehian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Chris Peikert or Sina Shiehian .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

Rights and permissions

Reprints and permissions

Copyright information

© 2019 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Peikert, C., Shiehian, S. (2019). Noninteractive Zero Knowledge for NP from (Plain) Learning with Errors. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11692. Springer, Cham. https://doi.org/10.1007/978-3-030-26948-7_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26948-7_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26947-0

  • Online ISBN: 978-3-030-26948-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation