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

Multiparty Noninteractive Key Exchange from Ring Key-Homomorphic Weak PRFs

  • Conference paper
  • First Online:
Topics in Cryptology – CT-RSA 2023 (CT-RSA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13871))

Included in the following conference series:

  • 546 Accesses

Abstract

A weak pseudorandom function \(F: \mathcal {K} \times \mathcal {X} \rightarrow \mathcal {Y}\) is said to be ring key-homomorphic if, given \(F \left( k_{1}, x \right) \) and \(F \left( k_{2}, x \right) \), there are efficient algorithms to compute \(F \left( k_{1} \oplus k_{2}, x \right) \) and \(F \left( k_{1} \otimes k_{2}, x \right) \) where \(\oplus \) and \(\otimes \) are the addition and multiplication operations in the ring \(\mathcal {K}\), respectively. In this work, we initiate the study of ring key-homomorphic weak PRFs (RKHwPRFs). As our main result, we show that any RKHwPRF implies multiparty noninteractive key exchange (NIKE) for an arbitrary number of parties in the standard model.

Our analysis of RKHwPRFs in a sense takes a major step towards the goal of building cryptographic primitives from Minicrypt primitives with structure, which has been studied in a recent line of works. With our result, most of the well-known asymmetric cryptographic primitives can be built from a weak PRF with either a group or ring homomorphism over either the input space or the key space.

N. Alamati—Most of the work was done while the author was at University of Michigan.

S. Patranabis—Most of the work was done while the author was at ETH Zürich.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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.

    We note that the recent work of [JLS21] constructed iO based on subexponential hardness of certain problems, and building iO from polynomial hardness of standard problems has so far remained out of reach. We refer to [GLSW15] for a discussion on the necessity of superpolynoimal security loss for realizing iO.

  2. 2.

    We use the terminology of [Imp95], which used “Minicrypt” and “Cryptomania” to describe the worlds of symmetric-key and asymmetric-key cryptography, respectively.

  3. 3.

    Recall that a weak PRF is a weakened version of a normal/strong PRF where an adversary gets to see the outputs on randomly chosen inputs.

  4. 4.

    Informally, a module is a generalization of vector space where the “scalars” form a ring (rather than a field).

  5. 5.

    As a simple example, one can consider two-party NIKE from learning with rounding problem [BPR12], for which a uniform matrix is generated during setup.

  6. 6.

    We assume that each public message also includes the index i.

  7. 7.

    The proof of the lemma is attributed to Rackoff, as pointed out by [IZ89, Mic02].

  8. 8.

    We remark that we use \((\boxplus , \boxtimes )\) operations for the ring \(\text{ M}_m(R)\), and these operations are inherited from R. Later, we drop this notation for simplification.

  9. 9.

    Note that such an alternative way of sampling is possible because for any finite ring R and arbitrary vector \(\textbf{v}\in R^n\), any R-linear function defined by \(f_{\textbf{v}}(\textbf{x}) = \sum _{i = 1}^{n} v_i x_i\) is regular over the (left) ideal of R generated by \(\textbf{v}\), i.e., any possible output in the ideal has the same number of preimages. Without regularity, the alternative sampling may yield a skewed distribution that is far from uniform. The regularity naturally extends to functions defined by any matrix of ring elements.

  10. 10.

    This is simply a weaker version of Lemma 4 in which \(\textbf{u}\) is not given publicly.

  11. 11.

    As in Lemma 4, we remark that such an alternative way of sampling is possible because of regularity of R-linear functions for vectors/matrices over any finite ring R. See the footnote on page 13 for more details.

  12. 12.

    A field KHwPRF \(F: K \times X \rightarrow Y\) is a stronger version of RKHwPRF where K, Y are fields and for any input \(x \in X\) we have a field homomorphism from K to Y induced by \(F(\cdot , x)\).

References

  1. Arvind, V., Das, B., Mukhopadhyay, P.: The complexity of black-box ring problems. In: Chen, D.Z., Lee, D.T. (eds.) COCOON 2006. LNCS, vol. 4112, pp. 126–135. Springer, Heidelberg (2006). https://doi.org/10.1007/11809678_15

    Chapter  Google Scholar 

  2. Alamati, N., Montgomery, H., Patranabis, S.: Symmetric primitives with structured secrets. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 650–679. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_23

    Chapter  Google Scholar 

  3. Alamati, N., Montgomery, H., Patranabis, S., Roy, A.: Minicrypt primitives with algebraic structure and applications. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11477, pp. 55–82. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_3

    Chapter  Google Scholar 

  4. Asharov, G., Segev, G.: Limits on the power of indistinguishability obfuscation and functional encryption. In: Guruswami, V., (ed.), 56th FOCS, pp. 191–209. IEEE Computer Society Press (2015)

    Google Scholar 

  5. Barak, B.: The complexity of public-key cryptography. In: Tutorials on the Foundations of Cryptography, pp. 45–77 (2017)

    Google Scholar 

  6. Barak, B., et al.: On the (Im)possibility of Obfuscating Programs. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 1–18. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44647-8_1

    Chapter  Google Scholar 

  7. Barak, B., Garg, S., Kalai, Y.T., Paneth, O., Sahai, A.: Protecting obfuscation against algebraic attacks. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 221–238. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_13

    Chapter  Google Scholar 

  8. Boneh, D., et al.: Multiparty non-interactive key exchange and more from isogenies on elliptic curves. J. Math. Cryptol. 14(1), 5–14 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Boneh, D., Lewi, K., Montgomery, H., Raghunathan, A.: Key homomorphic PRFs and their applications. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 410–428. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_23

    Chapter  Google Scholar 

  10. Banerjee, A., Peikert, C., Rosen, A.: Pseudorandom functions and lattices. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 719–737. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_42

    Chapter  Google Scholar 

  11. Brakerski, Z., Rothblum, G.N.: Virtual black-box obfuscation for all circuits via generic graded encoding. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 1–25. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_1

    Chapter  Google Scholar 

  12. Boneh, D., Silverberg, A.: Applications of multilinear forms to cryptography. Contemp. Math. 324(1), 71–90 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boneh, D., Zhandry, M.: Multiparty Key exchange, efficient traitor tracing, and more from indistinguishability obfuscation. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 480–499. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_27

    Chapter  Google Scholar 

  14. Cheon, J.H., Han, K., Lee, C., Ryu, H., Stehlé, D.: Cryptanalysis of the multilinear map over the integers. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 3–12. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46800-5_1

    Chapter  Google Scholar 

  15. Coron, J.-S., Lee, M.S., Lepoint, T., Tibouchi, M.: Cryptanalysis of GGH15 multilinear maps. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 607–628. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53008-5_21

    Chapter  Google Scholar 

  16. Coron, J.-S., Lepoint, T., Tibouchi, M.: Practical multilinear maps over the integers. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 476–493. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_26

    Chapter  Google Scholar 

  17. Diffie, W., Hellman, M.E.: New directions in cryptography. IEEE Trans. Inf. Theory, IT 22(6), 644–654 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dodis, Y., Kiltz, E., Pietrzak, K., Wichs, D.: Message authentication, revisited. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 355–374. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_22

    Chapter  Google Scholar 

  19. Escala, A., Herold, G., Kiltz, E., Ràfols, C., Villar, J.: An algebraic framework for diffie-hellman assumptions. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 129–147. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_8

    Chapter  Google Scholar 

  20. Freire, E.S.V., Hesse, J., Hofheinz, D.: Universally composable non-interactive key exchange. In: Abdalla, M., De Prisco, R. (eds.) SCN 2014. LNCS, vol. 8642, pp. 1–20. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10879-7_1

    Chapter  Google Scholar 

  21. Freire, E.S.V., Hofheinz, D., Kiltz, E., Paterson, K.G.: Non-interactive key exchange. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 254–271. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36362-7_17

    Chapter  MATH  Google Scholar 

  22. 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). https://doi.org/10.1007/978-3-642-38348-9_1

    Chapter  Google Scholar 

  23. Gentry, C., Gorbunov, S., Halevi, S.: Graph-induced multilinear maps from lattices. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9015, pp. 498–527. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46497-7_20

    Chapter  Google Scholar 

  24. Gentry, C., Lewko, A.B., Sahai, A., Waters, B.: Indistinguishability obfuscation from the multilinear subgroup elimination assumption. In: Guruswami, V., (ed.) 56th FOCS, pp. 151–170. IEEE Computer Society Press (2015)

    Google Scholar 

  25. Garg, S., Pandey, O., Srinivasan, A., Zhandry, M.: Breaking the sub-exponential barrier in obfustopia. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 156–181. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56617-7_6

    Chapter  MATH  Google Scholar 

  26. Hesse, J., Hofheinz, D., Kohl, L.: On tightly secure non-interactive key exchange. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10992, pp. 65–94. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_3

    Chapter  Google Scholar 

  27. Hesse, J., Hofheinz, D., Kohl, L., Langrehr, R.: Towards tight adaptive security of non-interactive key exchange. In: Nissim, K., Waters, B. (eds.) TCC 2021. LNCS, vol. 13044, pp. 286–316. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90456-2_10

    Chapter  MATH  Google Scholar 

  28. Håstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28(4), 1364–1396 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hu, Y., Jia, H.: Cryptanalysis of GGH map. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9665, pp. 537–565. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49890-3_21

    Chapter  Google Scholar 

  30. Impagliazzo, R.: A personal view of average-case complexity. In: Proceedings of Structure in Complexity Theory, 10th Annual IEEE Conference, pp. 134–147 (1995)

    Google Scholar 

  31. Impagliazzo, R., Zuckerman, D.: How to recycle random bits. In: 30th FOCS, pp. 248–253. IEEE Computer Society Press (1989)

    Google Scholar 

  32. Jager, T.: On black-box models of computation in cryptology. Ph.D. thesis, Ruhr University Bochum (2012)

    Google Scholar 

  33. Jain, A., Lin, H., Sahai, A.: Indistinguishability obfuscation from well-founded assumptions (2021)

    Google Scholar 

  34. Joux, A.: A one round protocol for tripartite Diffie-Hellman. J. Cryptol. 17(4), 263–276 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Koppula, V., Waters, B., Zhandry, M.: Adaptive multiparty nike. Cryptology ePrint Archive, Paper 2022/1324 (to appear in TCC 2022) (2022). https://eprint.iacr.org/2022/1324

  36. Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. In: 43rd FOCS, pp. 356–365. IEEE Computer Society Press (2002)

    Google Scholar 

  37. Maurer, U., Raub, D.: Black-Box extension fields and the inexistence of field-homomorphic one-way permutations. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 427–443. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-76900-2_26

    Chapter  Google Scholar 

  38. Miles, E., Sahai, A., Zhandry, M.: Annihilation attacks for multilinear maps: cryptanalysis of indistinguishability obfuscation over GGH13. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 629–658. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53008-5_22

    Chapter  Google Scholar 

  39. Naor, M., Pinkas, B., Reingold, O.: Distributed pseudo-random functions and KDCs. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 327–346. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48910-X_23

    Chapter  Google Scholar 

  40. Naor, M., Reingold, O.: Synthesizers and their application to the parallel construction of pseudo-random functions. In: 36th FOCS, pp. 170–181. IEEE Computer Society Press (1995)

    Google Scholar 

  41. Peikert, C.: Lattice cryptography for the internet. In: Mosca, M. (ed.) PQCrypto 2014. LNCS, vol. 8772, pp. 197–219. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11659-4_12

    Chapter  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Ring key-homomorphic weak PRFs and applications. Preprint/Manuscript (2020)

    Google Scholar 

  44. Shoup, V.: Lower bounds for discrete logarithms and related problems. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 256–266. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-69053-0_18

    Chapter  Google Scholar 

  45. Yamakawa, T., Hanaoka, G., Kunihiro, N.: Generalized hardness assumption for self-bilinear map with auxiliary information. In: Liu, J.K., Steinfeld, R. (eds.) ACISP 2016. LNCS, vol. 9723, pp. 269–284. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40367-0_17

    Chapter  MATH  Google Scholar 

  46. Yamakawa, T., Yamada, S., Hanaoka, G., Kunihiro, N.: Self-bilinear map on unknown order groups from indistinguishability obfuscation and its applications. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 90–107. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_6

    Chapter  Google Scholar 

  47. Yamakawa, T., Yamada, S., Hanaoka, G., Kunihiro, N.: Generic hardness of inversion on ring and its relation to self-bilinear map. Cryptology ePrint Archive, Report 2018/463, 2018. https://eprint.iacr.org/2018/463

  48. Yamakawa, T., Yamada, S., Hanaoka, G., Kunihiro, N.: Generic hardness of inversion on ring and its relation to self-bilinear map. Theor. Comput. Sci. 820, 60–84 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. VISA Research, Palo Alto, USA

    Navid Alamati

  2. Linux Foundation, San Francisco, USA

    Hart Montgomery

  3. IBM Research India, Bangalore, India

    Sikhar Patranabis

Authors
  1. Navid Alamati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hart Montgomery
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sikhar Patranabis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sikhar Patranabis .

Editor information

Editors and Affiliations

  1. Oregon State University, Corvallis, OR, USA

    Mike Rosulek

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Alamati, N., Montgomery, H., Patranabis, S. (2023). Multiparty Noninteractive Key Exchange from Ring Key-Homomorphic Weak PRFs. In: Rosulek, M. (eds) Topics in Cryptology – CT-RSA 2023. CT-RSA 2023. Lecture Notes in Computer Science, vol 13871. Springer, Cham. https://doi.org/10.1007/978-3-031-30872-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-30872-7_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-30871-0

  • Online ISBN: 978-3-031-30872-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation