Skip to main content
SpringerLink
Log in

Towards Static Assumption Based Cryptosystem in Pairing Setting: Further Applications of DéjàQ and Dual-Form Signature (Extended Abstract)

  • Conference paper
  • First Online:
Provable Security (ProvSec 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11192))

Included in the following conference series:

Abstract

A large number of parameterized complexity assumptions have been introduced in the bilinear pairing setting to design novel cryptosystems and an important question is whether such “q-type” assumptions can be replaced by some static one. Recently Ghadafi and Groth captured several such parameterized assumptions in the pairing setting in a family called bilinear target assumption (BTA). We apply the DéjàQ techniques for all q-type assumptions in the BTA family. In this process, first we formalize the notion of extended adaptive parameter-hiding property and use it in the Chase-Meiklejohn’s DéjàQ framework to reduce those q-type assumptions from subgroup hiding assumption in the asymmetric composite-order pairing. In addition, we extend the BTA family further into BTA1 and BTA2 and study the relation between different BTA variants. We also discuss the inapplicability of DéjàQ techniques on the q-type assumptions that belong to BTA1 or BTA2 family. We then provide one further application of Gerbush et al.’s dual-form signature techniques to remove the dependence on a q-type assumption for which existing DéjàQ techniques are not applicable. This results in a variant of Abe et al.’s structure-preserving signature with security based on a static assumption in composite order setting.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Notes

  1. 1.

    For BTA in \(\mathbb G_T\), the degree of the challenge term polynomials are bounded by 2d, as given the d degree polynomials in both source groups, one can use the pairing to compute the product of these polynomials in \(\mathbb G_T\).

  2. 2.

    We say that the BTA assumption defined in the asymmetric pairing setting is one-sided, if the secret vector \(\mathbf {x}\) associated with the polynomial representation occurs in exactly one of the source groups. Otherwise we say that the assumption is two-sided.

  3. 3.

    Even if \(N=p_1\ldots p_n\), we decompose G using two of its subgroups \(G_1\) and \(G_2\) such that \(G_1\) (resp. \(G_2\)) is a subgroup of order \(p_1\ldots p_{n-1}\) (resp. \(p_n\)).

  4. 4.

    As similar to BTA assumption, hardness of Assumption 4 ensures that the instance and challenge terms should satisfy certain linearly independent condition that corresponds to Eq. 1. However we directly prove the hardness of Assumption 4 in Corollary 1. This guarantees that the above condition automatically satisfies and hence we do not need to explicitly state such condition here.

  5. 5.

    First we check A (resp. D) belongs to G (resp. H) by verifying \(A^N=1_{G}\) (resp. \(D^N=1_{H}\)). Then the pairing equation \(e(A, D)=e(g_1, h_1)\) ensures that D indeed belongs to subgroup \(H_1\).

References

  1. Abe, M., Fuchsbauer, G., Groth, J., Haralambiev, K., Ohkubo, M.: Structure-preserving signatures and commitments to group elements. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 209–236. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_12

    Chapter  Google Scholar 

  2. Abe, M., Groth, J., Haralambiev, K., Ohkubo, M.: Optimal structure-preserving signatures in asymmetric bilinear groups. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 649–666. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_37

    Chapter  MATH  Google Scholar 

  3. Belenkiy, M., Camenisch, J., Chase, M., Kohlweiss, M., Lysyanskaya, A., Shacham, H.: Randomizable proofs and delegatable anonymous credentials. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 108–125. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03356-8_7

    Chapter  Google Scholar 

  4. Belenkiy, M., Chase, M., Kohlweiss, M., Lysyanskaya, A.: P-signatures and noninteractive anonymous credentials. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 356–374. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78524-8_20

    Chapter  Google Scholar 

  5. Boneh, D., Boyen, X.: Short signatures without random oracles. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 56–73. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24676-3_4

    Chapter  Google Scholar 

  6. Boneh, D., Boyen, X., Goh, E.-J.: Hierarchical identity based encryption with constant size ciphertext. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 440–456. Springer, Heidelberg (2005). https://doi.org/10.1007/11426639_26

    Chapter  Google Scholar 

  7. Boneh, D., Gentry, C., Waters, B.: Collusion resistant broadcast encryption with short ciphertexts and private keys. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 258–275. Springer, Heidelberg (2005). https://doi.org/10.1007/11535218_16

    Chapter  Google Scholar 

  8. Boyen, X.: Mesh signatures. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 210–227. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-72540-4_12

    Chapter  Google Scholar 

  9. Boyen, X.: The uber-assumption family. In: Galbraith, S.D., Paterson, K.G. (eds.) Pairing 2008. LNCS, vol. 5209, pp. 39–56. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85538-5_3

    Chapter  Google Scholar 

  10. Boyen, X., Waters, B.: Full-domain subgroup hiding and constant-size group signatures. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 1–15. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71677-8_1

    Chapter  Google Scholar 

  11. Chase, M., Maller, M., Meiklejohn, S.: Déjà Q all over again: tighter and broader reductions of q-type assumptions. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10032, pp. 655–681. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53890-6_22

    Chapter  Google Scholar 

  12. Chase, M., Meiklejohn, S.: Déjà Q: using dual systems to revisit \(q\)-type assumptions. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 622–639. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_34

    Chapter  Google Scholar 

  13. Chatterjee, S., Kabaleeshwaran, R.: Towards static assumption based cryptosystem in pairing setting: further applications of DéjàQ and dual-form signature. IACR Cryptology ePrint Archive 2018/738 (2018)

    Google Scholar 

  14. Chatterjee, S., Menezes, A.: On cryptographic protocols employing asymmetric pairings - the role of \(\Psi \) revisited. Discrete Appl. Math. 159(13), 1311–1322 (2011)

    Article  MathSciNet  Google Scholar 

  15. Cheon, J.H.: Security analysis of the strong Diffie-Hellman problem. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 1–11. Springer, Heidelberg (2006). https://doi.org/10.1007/11761679_1

    Chapter  Google Scholar 

  16. Fuchsbauer, G.: Automorphic signatures in bilinear groups and an application to round-optimal blind signatures. IACR Cryptology ePrint Archive 2009/320 (2009)

    Google Scholar 

  17. Fuchsbauer, G., Hanser, C., Slamanig, D.: Structure-preserving signatures on equivalence classes and constant-size anonymous credentials. IACR Cryptology ePrint Archive 2014/944 (2014)

    Google Scholar 

  18. Fuchsbauer, G., Pointcheval, D., Vergnaud, D.: Transferable constant-size fair e-cash. In: Garay, J.A., Miyaji, A., Otsuka, A. (eds.) CANS 2009. LNCS, vol. 5888, pp. 226–247. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10433-6_15

    Chapter  Google Scholar 

  19. Gerbush, M., Lewko, A., O’Neill, A., Waters, B.: Dual form signatures: an approach for proving security from static assumptions. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 25–42. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-34961-4_4

    Chapter  Google Scholar 

  20. Ghadafi, E.: Efficient distributed tag-based encryption and its application to group signatures with efficient distributed traceability. In: Aranha, D.F., Menezes, A. (eds.) LATINCRYPT 2014. LNCS, vol. 8895, pp. 327–347. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16295-9_18

    Chapter  Google Scholar 

  21. Ghadafi, E.: Stronger security notions for decentralized traceable attribute-based signatures and more efficient constructions. In: Nyberg, K. (ed.) CT-RSA 2015. LNCS, vol. 9048, pp. 391–409. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-16715-2_21

    Chapter  Google Scholar 

  22. Ghadafi, E., Groth, J.: Towards a classification of non-interactive computational assumptions in cyclic groups. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017. LNCS, vol. 10625, pp. 66–96. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70697-9_3

    Chapter  MATH  Google Scholar 

  23. Groth, J., Lu, S.: A non-interactive shuffle with pairing based verifiability. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 51–67. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-76900-2_4

    Chapter  Google Scholar 

  24. Jao, D., Yoshida, K.: Boneh-Boyen signatures and the strong Diffie-Hellman problem. In: Shacham, H., Waters, B. (eds.) Pairing 2009. LNCS, vol. 5671, pp. 1–16. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03298-1_1

    Chapter  Google Scholar 

  25. Lewko, A.: Tools for simulating features of composite order bilinear groups in the prime order setting. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 318–335. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_20

    Chapter  MATH  Google Scholar 

  26. Mitsunari, S., Sakai, R., Kasahara, M.: A new traitor tracing. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 85(2), 481–484 (2002)

    Google Scholar 

  27. Okamoto, T.: Efficient blind and partially blind signatures without random oracles. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 80–99. Springer, Heidelberg (2006). https://doi.org/10.1007/11681878_5

    Chapter  Google Scholar 

  28. Okamoto, T.: Efficient blind and partially blind signatures without random oracles. IACR Cryptology ePrint Archive 2006/102 (2006)

    Google Scholar 

  29. Wee, H.: Déjà Q: encore! Un petit IBE. IACR Cryptology ePrint Archive 2015/1064 (2015)

    Google Scholar 

  30. Zhang, F., Safavi-Naini, R., Susilo, W.: An efficient signature scheme from bilinear pairings and its applications. In: Bao, F., Deng, R., Zhou, J. (eds.) PKC 2004. LNCS, vol. 2947, pp. 277–290. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24632-9_20

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

    Sanjit Chatterjee & R. Kabaleeshwaran

Authors
  1. Sanjit Chatterjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Kabaleeshwaran
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Kabaleeshwaran .

Editor information

Editors and Affiliations

  1. University of Wollongong, Wollongong, NSW, Australia

    Joonsang Baek

  2. University of Wollongong, Wollongong, NSW, Australia

    Willy Susilo

  3. University of Wollongong, Wollongong, NSW, Australia

    Jongkil Kim

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chatterjee, S., Kabaleeshwaran, R. (2018). Towards Static Assumption Based Cryptosystem in Pairing Setting: Further Applications of DéjàQ and Dual-Form Signature (Extended Abstract). In: Baek, J., Susilo, W., Kim, J. (eds) Provable Security. ProvSec 2018. Lecture Notes in Computer Science(), vol 11192. Springer, Cham. https://doi.org/10.1007/978-3-030-01446-9_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-01446-9_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-01445-2

  • Online ISBN: 978-3-030-01446-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation