Skip to main content
SpringerLink
Log in

Round-Robin is Optimal: Lower Bounds for Group Action Based Protocols

  • Conference paper
  • First Online:
Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14372))

Included in the following conference series:

  • 214 Accesses

Abstract

An hard homogeneous space (HHS) is a finite group acting on a set with the group action being hard to invert and the set lacking any algebraic structure. As such HHS could potentially replace finite groups where the discrete logarithm is hard for building cryptographic primitives and protocols in a post-quantum world.

Threshold HHS-based primitives typically require parties to compute the group action of a secret-shared input on a public set element. On one hand this could be done through generic MPC techniques, although they incur in prohibitive costs due to the high complexity of circuits evaluating group actions known to date. On the other hand round-robin protocols only require black box usage of the HHS. However these are highly sequential procedures, taking as many rounds as parties involved. The high round complexity appears to be inherent due the lack of homomorphic properties in HHS, yet no lower bounds were known so far.

In this work we formally show that round-robin protocols are optimal. In other words, any at least passively secure distributed computation of a group action making black-box use of an HHS must take a number of rounds greater or equal to the threshold parameter. We furthermore study fair protocols in which all users receive the output in the same round (unlike plain round-robin), and prove communication and computation lower bounds of \(\varOmega (n \log _2 n)\) for n parties. Our results are proven in Shoup’s Generic Action Model (GAM), and hold regardless of the underlying computational assumptions.

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 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
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.

    Including one round to broadcast the result to other users.

  2. 2.

    Or, in honest majority, to publicly reconstruct this user’s secret share.

  3. 3.

    Due to our first result, this is also the best round complexity in this case.

  4. 4.

    More precisely we later introduce a relation “\(\rightarrow \)” among query indices.

  5. 5.

    Because it implies that computing \(a \mathop {\star }E_0\) can only be done through sequential applications of the group action starting from \(E_0\).

  6. 6.

    Up to choosing paths satisfying a rather technical minimality condition.

  7. 7.

    Even though broadcast could be achieved simply sending the same message to all parties, as we only focus on semi-honest adversaries.

  8. 8.

    These can be defined for \(\mathbb {G}\) if \(\mathbb {Z}_N\) has an exceptional set of size at least n, with N being the order of \(\mathbb {G}\).

  9. 9.

    Note |S| may be smaller than t if there are repetitions among \(i_1, \ldots , i_t\).

  10. 10.

    r and \(r'\) are the first component of respectively the LHS and the RHS.

References

  1. Atapoor, S., Baghery, K., Cozzo, D., Pedersen, R.: CSI-SharK: CSI-FiSh with sharing-friendly keys. In: Simpson, L., Rezazadeh Baee, M.A. (eds.) ACISP 2023. LNCS, vol. 13915, pp. 471–502. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-35486-1_21

    Chapter  Google Scholar 

  2. Atapoor, S., Baghery, K., Cozzo, D., Pedersen, R.: Practical robust DKG protocols for CSIDH. In: Tibouchi, M., Wang, X. (eds.) ACNS 2023. LNCS, pp. 219–247. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-33491-7_9

    Chapter  Google Scholar 

  3. Atapoor, S., Baghery, K., Cozzo, D., Pedersen, R.: VSS from distributed ZK proofs and applications. IACR Cryptology ePrint Archive, p. 992 (2023). https://eprint.iacr.org/2023/992

  4. Baghery, K., Cozzo, D., Pedersen, R.: An isogeny-based ID protocol using structured public keys. In: Paterson, M.B. (ed.) IMACC 2021. LNCS, vol. 13129, pp. 179–197. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92641-0_9

    Chapter  Google Scholar 

  5. Beullens, W., Disson, L., Pedersen, R., Vercauteren, F.: CSI-RAShi: distributed key generation for CSIDH. In: Cheon, J.H., Tillich, J.-P. (eds.) PQCrypto 2021 2021. LNCS, vol. 12841, pp. 257–276. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-81293-5_14

    Chapter  Google Scholar 

  6. Beullens, W., Dobson, S., Katsumata, S., Lai, Y.F., Pintore, F.: Group signatures and more from isogenies and lattices: generic, simple, and efficient. In: Dunkelman, O., Dziembowski, S. (eds.) EUROCRYPT 2022, Part II. LNCS, vol. 13276, pp. 95–126. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-07085-3_4

    Chapter  Google Scholar 

  7. Beullens, W., Katsumata, S., Pintore, F.: Calamari and Falafl: logarithmic (linkable) ring signatures from isogenies and lattices. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020, Part II. LNCS, vol. 12492, pp. 464–492. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_16

    Chapter  Google Scholar 

  8. Beullens, W., Kleinjung, T., Vercauteren, F.: CSI-FiSh: efficient isogeny based signatures through class group computations. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019, Part I. LNCS, vol. 11921, pp. 227–247. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34578-5_9

    Chapter  Google Scholar 

  9. Boneh, D., Guan, J., Zhandry, M.: A lower bound on the length of signatures based on group actions and generic isogenies. In: Hazay, C., Stam, M. (eds.) EUROCRYPT 2023, Part V. LNCS, vol. 14008, pp. 507–531. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-30589-4_18

    Chapter  Google Scholar 

  10. Campos, F., Muth, P.: On actively secure fine-grained access structures from isogeny assumptions. In: Cheon, J.H., Johansson, T. (eds.) PQCrypto 2022. LNCS, vol. 13512, pp. 375–398. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-17234-2_18

    Chapter  Google Scholar 

  11. Castryck, W., Decru, T.: An efficient key recovery attack on SIDH. In: Hazay, C., Stam, M. (eds.) EUROCRYPT 2023, Part V. LNCS, vol. 14008, pp. 423–447. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-30589-4_15

    Chapter  Google Scholar 

  12. Castryck, W., Lange, T., Martindale, C., Panny, L., Renes, J.: CSIDH: an efficient post-quantum commutative group action. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018, Part III. LNCS, vol. 11274, pp. 395–427. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03332-3_15

    Chapter  Google Scholar 

  13. Catalano, D., Fiore, D., Gennaro, R., Giunta, E.: On the impossibility of algebraic vector commitments in pairing-free groups. In: Kiltz, E., Vaikuntanathan, V. (eds.) TCC 2022, Part II. LNCS, vol. 13748, pp. 274–299. Springer, Heidelberg (2022). https://doi.org/10.1007/978-3-031-22365-5_10

    Chapter  MATH  Google Scholar 

  14. Cozzo, D., Smart, N.P.: Sashimi: cutting up CSI-FiSh secret keys to produce an actively secure distributed signing protocol. In: Ding, J., Tillich, J.-P. (eds.) PQCrypto 2020. LNCS, vol. 12100, pp. 169–186. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-44223-1_10

    Chapter  Google Scholar 

  15. De Feo, L., Galbraith, S.D.: SeaSign: compact isogeny signatures from class group actions. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019, Part III. LNCS, vol. 11478, pp. 759–789. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17659-4_26

    Chapter  Google Scholar 

  16. De Feo, L., Meyer, M.: Threshold schemes from isogeny assumptions. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020, Part II. LNCS, vol. 12111, pp. 187–212. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45388-6_7

    Chapter  Google Scholar 

  17. Döttling, N., Hartmann, D., Hofheinz, D., Kiltz, E., Schäge, S., Ursu, B.: On the impossibility of purely algebraic signatures. In: Nissim, K., Waters, B. (eds.) TCC 2021, Part III. LNCS, vol. 13044, pp. 317–349. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90456-2_11

    Chapter  MATH  Google Scholar 

  18. Duman, J., Hartmann, D., Kiltz, E., Kunzweiler, S., Lehmann, J., Riepel, D.: Generic models for group actions. In: Boldyreva, A., Kolesnikov, V. (eds.) PKC 2023, Part I. LNCS, vol. 13940, pp. 406–435. Springer, Heidelberg (2023). https://doi.org/10.1007/978-3-031-31368-4_15

    Chapter  MATH  Google Scholar 

  19. El Kaafarani, A., Katsumata, S., Pintore, F.: Lossy CSI-FiSh: efficient signature scheme with tight reduction to decisional CSIDH-512. In: Kiayias, A., Kohlweiss, M., Wallden, P., Zikas, V. (eds.) PKC 2020, Part II. LNCS, vol. 12111, pp. 157–186. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45388-6_6

    Chapter  Google Scholar 

  20. Fouotsa, T.B., Petit, C.: SHealS and HealS: isogeny-based PKEs from a key validation method for SIDH. In: Tibouchi, M., Wang, H. (eds.) ASIACRYPT 2021, Part IV. LNCS, vol. 13093, pp. 279–307. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-92068-5_10

    Chapter  Google Scholar 

  21. Giunta, E.: On the impossibility of algebraic NIZK in pairing-free groups. In: Handschuh, H., Lysyanskaya, A. (eds.) CRYPTO 2023. LNCS, vol. 14084, pp. 702–730. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-38551-3_22

    Chapter  Google Scholar 

  22. Lai, Y.-F., Galbraith, S.D., Delpech de Saint Guilhem, C.: Compact, efficient and UC-secure isogeny-based oblivious transfer. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021, Part I. LNCS, vol. 12696, pp. 213–241. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77870-5_8

    Chapter  Google Scholar 

  23. Maino, L., Martindale, C., Panny, L., Pope, G., Wesolowski, B.: A direct key recovery attack on SIDH. In: Hazay, C., Stam, M. (eds.) EUROCRYPT 2023. LNCS, vol. 14008, pp. 448–471. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-30589-4_16

    Chapter  Google Scholar 

  24. Maurer, U.: Abstract models of computation in cryptography (invited paper). In: Smart, N.P. (ed.) Cryptography and Coding. LNCS, vol. 3796, pp. 1–12. Springer, Heidelberg (2005). https://doi.org/10.1007/11586821_1

    Chapter  MATH  Google Scholar 

  25. Moriya, T., Onuki, H., Takagi, T.: SiGamal: a supersingular isogeny-based PKE and its application to a PRF. In: Moriai, S., Wang, H. (eds.) ASIACRYPT 2020, Part II. LNCS, vol. 12492, pp. 551–580. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64834-3_19

    Chapter  Google Scholar 

  26. Papakonstantinou, P.A., Rackoff, C., Vahlis, Y.: How powerful are the DDH hard groups? Electron. Colloquium Comput. Complex. 167 (2012). https://eccc.weizmann.ac.il/report/2012/167

  27. Robert, D.: Breaking SIDH in polynomial time. In: Hazay, C., Stam, M. (eds.) EUROCRYPT 2023, Part V. LNCS, vol. 14008, pp. 472–503. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-30589-4_17

    Chapter  Google Scholar 

  28. Rotem, L., Segev, G., Shahaf, I.: Generic-group delay functions require hidden-order groups. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020, Part III. LNCS, vol. 12107, pp. 155–180. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_6

    Chapter  Google Scholar 

  29. Schul-Ganz, G., Segev, G.: Accumulators in (and beyond) generic groups: non-trivial batch verification requires interaction. In: Pass, R., Pietrzak, K. (eds.) TCC 2020, Part II. LNCS, vol. 12551, pp. 77–107. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-64378-2_4

    Chapter  Google Scholar 

  30. Shaw, S., Dutta, R.: Identification scheme and forward-secure signature in identity-based setting from isogenies. In: Huang, Q., Yu, Yu. (eds.) ProvSec 2021. LNCS, vol. 13059, pp. 309–326. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90402-9_17

    Chapter  MATH  Google Scholar 

  31. Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pp. 124–134 (1994)

    Google Scholar 

  32. 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 

  33. Stolbunov, A.: Cryptographic schemes based on isogenies (2012)

    Google Scholar 

  34. Tairi, E., Moreno-Sanchez, P., Maffei, M.: Post-quantum adaptor signature for privacy-preserving off-chain payments. In: Borisov, N., Diaz, C. (eds.) FC 2021, Part II. LNCS, vol. 12675, pp. 131–150. Springer, Heidelberg (2021). https://doi.org/10.1007/978-3-662-64331-0_7

    Chapter  Google Scholar 

  35. Zhandry, M.: To label, or not to label (in generic groups). In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO 2022, Part III. LNCS, vol. 13509, pp. 66–96. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15982-4_3

    Chapter  MATH  Google Scholar 

Download references

Acknowledgment

This work received funding from projects from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under project PICOCRYPT (grant agreement No. 101001283), from SECURING Project (ref. PID2019-110873RJ-I00), from the Spanish Government under projects PRODIGY (TED2021-132464B-I00) and ESPADA (PID2022-142290OB-I00). The last two projects are co-funded by European Union EIE, and NextGenerationEU/PRTR fund.

Author information

Authors and Affiliations

  1. IMDEA Software Institute, Madrid, Spain

    Daniele Cozzo & Emanuele Giunta

  2. Universidad Politecnica de Madrid, Madrid, Spain

    Emanuele Giunta

Authors
  1. Daniele Cozzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Emanuele Giunta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniele Cozzo .

Editor information

Editors and Affiliations

  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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

Cozzo, D., Giunta, E. (2023). Round-Robin is Optimal: Lower Bounds for Group Action Based Protocols. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14372. Springer, Cham. https://doi.org/10.1007/978-3-031-48624-1_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-48624-1_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-48623-4

  • Online ISBN: 978-3-031-48624-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Societies and partnerships

Search

Navigation