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

Traitor Tracing with \(N^{1/3}\)-Size Ciphertexts and O(1)-Size Keys from k-Lin

  • Conference paper
  • First Online:
Advances in Cryptology – EUROCRYPT 2023 (EUROCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14006))

Abstract

We present a pairing-based traitor tracing scheme for N users with

$$\begin{aligned} | \textsf{pk}| = | \textsf{ct}| = O(N^{1/3}), \quad | \textsf{sk}| = O(1). \end{aligned}$$

This is the first pairing-based scheme to achieve \(| \textsf{pk}| \cdot | \textsf{sk}| \cdot | \textsf{ct}| = o(N)\). Our construction relies on the (bilateral) k-Lin assumption, and achieves private tracing and full collusion resistance. Our result simultaneously improves upon the sizes of \( \textsf{pk}, \textsf{ct}\) in Boneh–Sahai–Waters [Eurocrypt ’06] and the size of \( \textsf{sk}\) in Zhandry [Crypto ’20], while further eliminating the reliance on the generic group model in the latter work.

J. Gong—Partially supported by National Natural Science Foundation of China (62002120), Innovation Program of Shanghai Municipal Education Commission (2021-01-07-00-08-E00101) and the “Digital Silk Road” Shanghai International Joint Lab of Trustworthy Intelligent Software (22510750100). Part of this work was done while visiting NTT Research.

J. Luo—Partially supported by NSF grants CNS-1936825 (CAREER), CNS-2026774, a JP Morgan AI Research Award, a Cisco Research Award, and a Simons Collaboration on the Theory of Algorithmic Fairness. Part of this work was done during an internship at NTT Research.

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.

    In a bit more detail, the construction starts with a variant of \((O(1),N_2)\)-rPLBE with parameters

    $$\begin{aligned} | \textsf{pk}| = O(N_2), \quad | \textsf{ct}| = O(N_2\kappa ), \quad | \textsf{sk}| = O(1), \end{aligned}$$

    which yields a “\(1/N_1\)-risky” traitor tracing scheme for \(N_1N_2\) users following [18]. That is, tracing succeeds with probability \(1/N_1\). This is then amplified to a standard traitor tracing scheme with a blow-up in \( \textsf{sk}\).

  2. 2.

    In MBME, ciphertexts are associated with \((z_1,\ldots ,z_\ell ) \in \{0,1\}^\ell \) and keys with \((y_1,\ldots ,y_\ell ) \in \{0,1\}^\ell \) and decryption is possible iff

    $$\textstyle \bigwedge _{i=1}^\ell z_i \vee y_i = 1.$$

    Security requires both attribute and function hiding. MBME for \(\ell \)-bit vectors can be instantiated from attribute-hiding function-hiding inner product predicate encryption for \(O(\ell )\)-dimensional vectors, since

    $$\textstyle \bigwedge _{i=1}^\ell z_i \vee y_i = 1 \Longleftrightarrow \sum _{i=1}^\ell (1-z_i)(1-y_i) {\mathop {=}\limits ^{?}} 0. $$

    .

  3. 3.

    As \(\kappa =\omega (\log \lambda )\) and \(N={\text {poly}}(\lambda )\), any statistical error \(2^{-\Omega (\kappa )}\) is absorbed by \(\lambda ^{-\omega (1)}\) when combined with a computational argument, and thus omitted in such case.

  4. 4.

    Claim 3 does not care about whether \({\varepsilon _0\ge \varepsilon (\lambda )}\).

  5. 5.

    \(N_1\) is a random variable due to the random coins of \(\mathcal {A}\), so it is impossible to write \(N_1\) outside probability or expectation. For non-uniform security we may assume \(N_1\) is fixed for every \(\lambda \), yet it is better to present the more general proof.

  6. 6.

    It is important that we do not assume \(\rho \) is the identity map by enlarging \(\textbf{M}\), so that we capture key size dependency in m, the locality. The scheme will be instantiated for \(\kappa \)-local roMSPs (\({\kappa \ll n}\)), which is crucial for the efficiency of our application.

References

  1. Abdalla, M., Catalano, D., Gay, R., Ursu, B.: Inner-product functional encryption with fine-grained access control. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12493, pp. 467–497. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64840-4_16

    Chapter  Google Scholar 

  2. Agrawal, S., Libert, B., Stehlé, D.: Fully secure functional encryption for inner products, from standard assumptions. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 333–362. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_12

    Chapter  Google Scholar 

  3. Baltico, C.E.Z., Catalano, D., Fiore, D., Gay, R.: Practical functional encryption for quadratic functions with applications to predicate encryption. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 67–98. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_3

    Chapter  Google Scholar 

  4. Billet, O., Phan, D.H.: Efficient traitor tracing from collusion secure codes. In: Safavi-Naini, R. (ed.) ICITS 2008. LNCS, vol. 5155, pp. 171–182. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85093-9_17

    Chapter  Google Scholar 

  5. Boneh, D., Naor, M.: Traitor tracing with constant size ciphertext. In: Ning, P., Syverson, P.F., Jha, S. (eds.) ACM CCS 2008, pp. 501–510. ACM Press, October 2008. https://doi.org/10.1145/1455770.1455834

  6. Boneh, D., Sahai, A., Waters, B.: Fully collusion resistant traitor tracing with short ciphertexts and private keys. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 573–592. Springer, Heidelberg (2006). https://doi.org/10.1007/11761679_34

    Chapter  Google Scholar 

  7. Boneh, D., Shoup, V.: A Graduate Course in Applied Cryptography. Draft (2015). version 0.2. https://toc.cryptobook.us/

  8. Boneh, D., Waters, B.: A fully collusion resistant broadcast, trace, and revoke system. In: Juels, A., Wright, R.N., De Capitani di Vimercati, S. (eds.) ACM CCS 2006. pp. 211–220. ACM Press, October/November 2006. https://doi.org/10.1145/1180405.1180432

  9. 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 

  10. Chen, J., Gay, R., Wee, H.: Improved dual system ABE in prime-order groups via predicate encodings. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 595–624. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46803-6_20

    Chapter  Google Scholar 

  11. Chen, J., Gong, J., Kowalczyk, L., Wee, H.: Unbounded ABE via bilinear entropy expansion, revisited. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 503–534. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_19

    Chapter  Google Scholar 

  12. Chen, Y., Vaikuntanathan, V., Waters, B., Wee, H., Wichs, D.: Traitor-tracing from LWE made simple and attribute-based. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018. LNCS, vol. 11240, pp. 341–369. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03810-6_13

    Chapter  Google Scholar 

  13. Chor, B., Fiat, A., Naor, M.: Tracing traitors. In: Desmedt, Y.G. (ed.) Advances in Cryptology — CRYPTO 1994. LNCS, vol. 839, pp. 257–270. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-48658-5_25

    Chapter  Google Scholar 

  14. 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 

  15. Etesami, O., Mahloujifar, S., Mahmoody, M.: Computational concentration of measure: Optimal bounds, reductions, and more. In: Chawla, S. (ed.) 31st SODA, pp. 345–363. ACM-SIAM, January 2020. https://doi.org/10.1137/1.9781611975994.21

  16. Gong, J., Luo, J., Wee, H.: Traitor tracing with \(N^{1/3}\)-size ciphertexts and \(O(1)\)-size keys from \(k\)-Lin. Cryptology ePrint Archive, Report 2023/256 (2023). https://eprint.iacr.org/2023/256

  17. Gong, J., Wee, H.: Adaptively secure ABE for DFA from k-Lin and more. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 278–308. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_10

    Chapter  Google Scholar 

  18. Goyal, R., Koppula, V., Russell, A., Waters, B.: Risky traitor tracing and new differential privacy negative results. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 467–497. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_16

    Chapter  Google Scholar 

  19. Goyal, R., Koppula, V., Waters, B.: Collusion resistant traitor tracing from learning with errors. In: Diakonikolas, I., Kempe, D., Henzinger, M. (eds.) 50th ACM STOC, pp. 660–670. ACM Press, June 2018. https://doi.org/10.1145/3188745.3188844

  20. Goyal, R., Quach, W., Waters, B., Wichs, D.: Broadcast and trace with \(N^{\varepsilon }\) ciphertext size from standard assumptions. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11694, pp. 826–855. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26954-8_27

    Chapter  Google Scholar 

  21. Ishai, Y., Wee, H.: Partial garbling schemes and their applications. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 650–662. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43948-7_54

    Chapter  MATH  Google Scholar 

  22. Karchmer, M., Wigderson, A.: On span programs. In: Proceedings of Structures in Complexity Theory, pp. 102–111 (1993)

    Google Scholar 

  23. Kowalczyk, L., Wee, H.: Compact adaptively secure ABE for NC\(^{1}\) from \(k\)-Lin. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11476, pp. 3–33. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17653-2_1

  24. Lin, H., Luo, J.: Succinct and adaptively secure ABE for ABP from \(k\)-Lin. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020. LNCS, vol. 12493, pp. 437–466. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64840-4_15

    Chapter  Google Scholar 

  25. Mahloujifar, S., Mahmoody, M.: Can adversarially robust learning leverage computational hardness? CoRR abs/1810.01407 (2018). http://arxiv.org/abs/1810.01407

  26. Shamir, A.: How to share a secret. Commun. Assoc. Comput. Mach. 22(11), 612–613 (1979)

    MathSciNet  MATH  Google Scholar 

  27. Waters, B.: Dual system encryption: realizing fully secure IBE and HIBE under simple assumptions. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 619–636. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_36

    Chapter  Google Scholar 

  28. Wee, H.: Attribute-hiding predicate encryption in bilinear groups, revisited. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 206–233. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70500-2_8

    Chapter  Google Scholar 

  29. Wee, H.: Functional encryption for quadratic functions from \(k\)-Lin, revisited. In: Pass, R., Pietrzak, K. (eds.) TCC 2020. LNCS, vol. 12550, pp. 210–228. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64375-1_8

    Chapter  Google Scholar 

  30. Zhandry, M.: New techniques for traitor tracing: size \(N^{1/3}\) and more from pairings. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12170, pp. 652–682. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56784-2_22

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. East China Normal University, Shanghai, China

    Junqing Gong

  2. Shanghai Qi Zhi Institute, Shanghai, China

    Junqing Gong

  3. Paul G. Allen School of Computer Science & Engineering, University of Washington, Seattle, USA

    Ji Luo

  4. NTT Research, Sunnyvale, USA

    Hoeteck Wee

Authors
  1. Junqing Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hoeteck Wee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Junqing Gong , Ji Luo or Hoeteck Wee .

Editor information

Editors and Affiliations

  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay

  2. Simula UiB, Bergen, Norway

    Martijn Stam

Rights and permissions

Reprints and permissions

Copyright information

© 2023 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gong, J., Luo, J., Wee, H. (2023). Traitor Tracing with \(N^{1/3}\)-Size Ciphertexts and O(1)-Size Keys from k-Lin. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14006. Springer, Cham. https://doi.org/10.1007/978-3-031-30620-4_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-30620-4_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-30619-8

  • Online ISBN: 978-3-031-30620-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation