Skip to main content
SpringerLink
Log in

Additive-Homomorphic Functional Commitments and Applications to Homomorphic Signatures

  • Conference paper
  • First Online:
Advances in Cryptology – ASIACRYPT 2022 (ASIACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13794))

Abstract

Functional Commitments (FC) allow one to reveal functions of committed data in a succinct and verifiable way. In this paper we put forward the notion of additive-homomorphic FC and show two efficient, pairing-based, realizations of this primitive supporting multivariate polynomials of constant degree and monotone span programs, respectively. We also show applications of the new primitive in the contexts of homomorphic signatures: we show that additive-homomorphic FCs can be used to realize homomorphic signatures (supporting the same class of functionalities as the underlying FC) in a simple and elegant way. Using our new FCs as underlying building blocks, this leads to the (seemingly) first expressive realizations of multi-input homomorphic signatures not relying on lattices or multilinear maps.

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 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

Notes

  1. 1.

    Their functionality resembles commit-and-prove SNARKs except that FCs are evaluation binding rather than (knowledge) sound.

  2. 2.

    sQAPs is in \(\textsf{P}\) as it can be decided via Gaussian elimination.

  3. 3.

    Even if one starts from an additive-homomorphic FC for linear maps, one can notice that the transformation to FCs for linearizable functions does not preserve the additive-homomorphism.

  4. 4.

    We use the bracket notation for bilinear groups of [9].

  5. 5.

    Strictly speaking the signature does not need to be structure preserving as long as it allows to (homomorphically) sign group elements.

  6. 6.

    This notion is similar in spirit to the basic security of accumulators [3].

  7. 7.

    Since \(\boldsymbol{x}^{(\delta )}\) has several terms repeated multiple times (e.g., after one product, the resulting vector contains both \(x_ix_j\) and \(x_jx_i\)), we assume \(\boldsymbol{\hat{f}}_i\) to always use the first of them, according to lexicographic order, and have 0 coefficients for the others.

  8. 8.

    It is known that a circuit in the class \({\textsf{NC}^{1}}\) can be converted into a polynomial-size boolean formula, and the latter can be turned into a monotone span program of equivalent size, e.g. [20, Appendix G].

References

  1. Boneh, D., Freeman, D.M.: Homomorphic signatures for polynomial functions. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 149–168. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_10

    Chapter  Google Scholar 

  2. 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 

  3. Camenisch, J., Lysyanskaya, A.: Dynamic accumulators and application to efficient revocation of anonymous credentials. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 61–76. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45708-9_5

    Chapter  Google Scholar 

  4. Catalano, D., Fiore, D.: Vector commitments and their applications. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 55–72. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36362-7_5

    Chapter  Google Scholar 

  5. Catalano, D., Fiore, D., Messina, M.: Zero-knowledge sets with short proofs. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 433–450. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_25

    Chapter  Google Scholar 

  6. Catalano, D., Fiore, D., Nizzardo, L.: On the security notions for homomorphic signatures. In: Preneel, B., Vercauteren, F. (eds.) ACNS 2018. LNCS, vol. 10892, pp. 183–201. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-93387-0_10

    Chapter  MATH  Google Scholar 

  7. Catalano, D., Fiore, D., Warinschi, B.: Homomorphic signatures with efficient verification for polynomial functions. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014, Part I. LNCS, vol. 8616, pp. 371–389. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44371-2_21

    Chapter  Google Scholar 

  8. Catalano, D., Marcedone, A., Puglisi, O.: Authenticating computation on groups: new homomorphic primitives and applications. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part II. LNCS, vol. 8874, pp. 193–212. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45608-8_11

    Chapter  Google Scholar 

  9. 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, Part II. LNCS, vol. 8043, pp. 129–147. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_8

    Chapter  Google Scholar 

  10. Gennaro, R., Gentry, C., Parno, B., Raykova, M.: Quadratic span programs and succinct NIZKs without PCPs. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 626–645. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38348-9_37

    Chapter  Google Scholar 

  11. Gennaro, R., Wichs, D.: Fully homomorphic message authenticators. In: Sako, K., Sarkar, P. (eds.) ASIACRYPT 2013, Part II. LNCS, vol. 8270, pp. 301–320. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-42045-0_16

    Chapter  Google Scholar 

  12. Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: Fortnow, L., Vadhan, S.P. (eds.) 43rd ACM STOC, pp. 99–108. ACM Press (2011). https://doi.org/10.1145/1993636.1993651

  13. Gorbunov, S., Vaikuntanathan, V., Wichs, D.: Leveled fully homomorphic signatures from standard lattices. In: Servedio, R.A., Rubinfeld, R. (eds.) 47th ACM STOC, pp. 469–477. ACM Press (2015). https://doi.org/10.1145/2746539.2746576

  14. Groth, J., Kohlweiss, M., Maller, M., Meiklejohn, S., Miers, I.: Updatable and universal common reference strings with applications to zk-SNARKs. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part III. LNCS, vol. 10993, pp. 698–728. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_24

    Chapter  Google Scholar 

  15. Karchmer, M., Wigderson, A.: On span programs. In: Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pp. 102–111 (1993)

    Google Scholar 

  16. Kate, A., Zaverucha, G.M., Goldberg, I.: Constant-size commitments to polynomials and their applications. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 177–194. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17373-8_11

    Chapter  Google Scholar 

  17. Katsumata, S., Nishimaki, R., Yamada, S., Yamakawa, T.: Designated verifier/prover and preprocessing NIZKs from Diffie-Hellman assumptions. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019, Part II. LNCS, vol. 11477, pp. 622–651. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_22

    Chapter  Google Scholar 

  18. Lai, R.W.F., Malavolta, G.: Subvector commitments with application to succinct arguments. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019, Part I. LNCS, vol. 11692, pp. 530–560. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_19

    Chapter  Google Scholar 

  19. Lewko, A., Okamoto, T., Sahai, A., Takashima, K., Waters, B.: Fully secure functional encryption: attribute-based encryption and (hierarchical) inner product encryption. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 62–91. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_4

    Chapter  Google Scholar 

  20. Lewko, A., Waters, B.: Unbounded HIBE and attribute-based encryption. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 547–567. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_30

    Chapter  Google Scholar 

  21. Libert, B., Peters, T., Joye, M., Yung, M.: Linearly homomorphic structure-preserving signatures and their applications. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 289–307. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_17

    Chapter  Google Scholar 

  22. Libert, B., Ramanna, S.C., Yung, M.: Functional commitment schemes: from polynomial commitments to pairing-based accumulators from simple assumptions. In: Chatzigiannakis, I., Mitzenmacher, M., Rabani, Y., Sangiorgi, D. (eds.) ICALP 2016. LIPIcs, vol. 55, pp. 30:1–30:14. Schloss Dagstuhl (2016). https://doi.org/10.4230/LIPIcs.ICALP.2016.30

  23. Libert, B., Yung, M.: Concise mercurial vector commitments and independent zero-knowledge sets with short proofs. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 499–517. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11799-2_30

    Chapter  Google Scholar 

  24. Lipmaa, H., Pavlyk, K.: Succinct functional commitment for a large class of arithmetic circuits. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020, Part III. LNCS, vol. 12493, pp. 686–716. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64840-4_23

    Chapter  Google Scholar 

  25. Peikert, C., Pepin, Z., Sharp, C.: Vector and functional commitments from lattices. In: Nissim, K., Waters, B. (eds.) TCC 2021, Part III. LNCS, vol. 13044, pp. 480–511. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90456-2_16

    Chapter  Google Scholar 

  26. Waters, B.: Ciphertext-policy attribute-based encryption: an expressive, efficient, and provably secure realization. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 53–70. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19379-8_4

    Chapter  Google Scholar 

Download references

Acknowledgements

We would like to thank Pierre Bourse for initial discussions that inspired this work, and Ignacio Cascudo for a useful discussion. This work has received funding in part from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under project PICOCRYPT (grant agreement No. 101001283), by the Spanish Government under projects SCUM (ref. RTI2018-102043-B-I00), and RED2018-102321-T, by the Madrid Regional Government under project BLOQUES (ref. S2018/TCS-4339), by a research grant from Nomadic Labs and the Tezos Foundation, and by the Programma ricerca di ateneo UNICT 35 2020-22 linea 2 and by research gifts from Protocol Labs.

Author information

Authors and Affiliations

  1. University of Catania, Catania, Italy

    Dario Catalano

  2. IMDEA Software Institute, Madrid, Spain

    Dario Fiore & Ida Tucker

Authors
  1. Dario Catalano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dario Fiore
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ida Tucker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dario Fiore .

Editor information

Editors and Affiliations

  1. Indian Institute of Technology Madras, Chennai, India

    Shweta Agrawal

  2. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

Rights and permissions

Reprints and permissions

Copyright information

© 2022 International Association for Cryptologic Research

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Catalano, D., Fiore, D., Tucker, I. (2022). Additive-Homomorphic Functional Commitments and Applications to Homomorphic Signatures. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13794. Springer, Cham. https://doi.org/10.1007/978-3-031-22972-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-22972-5_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22971-8

  • Online ISBN: 978-3-031-22972-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation