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The Consensus Number of Untraceable Cryptocurrencies

Authors Christian Cachin , David Lehnherr , Juan Villacis , François-Xavier Wicht

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Concurrent algorithms
  • Security and privacy → Pseudonymity, anonymity and untraceability
Keywords
  • Consensus number
  • untraceability
  • privacy-preserving cryptocurrencies
  • wait-freedom
  • masking sets
  • concurrent objects

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Abstract

Privacy-preserving cryptocurrencies hide which account funds a transfer among a set of candidate accounts, called its masking set. To prevent the hidden account from being spent twice, classical schemes retain the candidates and append a nullifier that marks the spent account without revealing it. Their ledgers therefore grow with every transfer. Constant-state schemes instead consume and replace the entire masking set. Here, we study the synchronization cost of this second design.
We introduce the constant untraceable asset transfer (CUAT) object. A CUAT transfer atomically replaces every account in its masking set; hence two transfers with intersecting sets cannot both succeed. We capture this contention by a conflict graph on transfer invocations, whose edges join invocations with intersecting masking sets. In one-round protocols, the relevant sets form a clique at a critical configuration. In general protocols, only sets from opposite valency classes must intersect.
Under weak untraceability, which considers one transaction in isolation, CUAT has unbounded consensus number even for one-round protocols. Under strong untraceability, which considers the complete history, accounts in a common masking set must have equal incidence. This condition makes both the one-round consensus power and CUAT’s consensus number grow quadratically with the masking-set size. Cyclic and projective-plane constructions establish the corresponding one-round lower bounds, while a recursive grid construction shows that CUAT’s consensus number is exactly the square of the masking-set size. CUAT is also not starvation-free.
The full version studies the complementary linear untraceable asset transfer (LUAT) object, which retains its masking set and records a nullifier. Its state grows, but its consensus number is two for every masking-set size and under either untraceability notion, and it is starvation-free. Thus LUAT pays for untraceability in storage, whereas CUAT pays in synchronization and fairness.

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Christian Cachin, David Lehnherr, Juan Villacis, and François-Xavier Wicht. The Consensus Number of Untraceable Cryptocurrencies. In 8th Conference on Advances in Financial Technologies (AFT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 395, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AFT.2026.5

Author Details

Christian Cachin
  • Institute of Computer Science, University of Bern, Switzerland
David Lehnherr
  • Institute of Computer Science, University of Bern, Switzerland
Juan Villacis
  • Institute of Computer Science, University of Bern, Switzerland
François-Xavier Wicht
  • Institute of Computer Science, University of Bern, Switzerland

Funding

This work was supported by the Swiss National Science Foundation (SNSF) under grant agreement Nr. 219403 (Emerging Consensus) and by the Initiative for Cryptocurrencies and Contracts (IC3).

References

  1. Kurt M. Alonso and The Monero Community. Zero to Monero: Second Edition - Second Edition: First Principles and Cryptonote Cryptography. The Monero Project, 2020. Version 2.0.0, April 2020. URL: https://web.getmonero.org/library/Zero-to-Monero-2-0-0.pdf.
  2. Orestis Alpos, Christian Cachin, Giorgia Azzurra Marson, and Luca Zanolini. On the synchronization power of token smart contracts. In 41st IEEE International Conference on Distributed Computing Systems, ICDCS 2021, Washington DC, USA, July 7-10, 2021, pages 640-651. IEEE, 2021. URL: https://doi.org/10.1109/ICDCS51616.2021.00067.
  3. Jayamine Alupotha, Mathieu Gestin, and Christian Cachin. Nopenena untraceable payments: Defeating graph analysis with small decoy sets. IACR Cryptol. ePrint Arch., page 903, 2024. URL: https://eprint.iacr.org/2024/903.
  4. Mariarosaria Barbaraci and Christian Cachin. Byzantine reliable broadcast with causal ordering: (research statement). In Dennis Olivetti and Eric Ruppert, editors, ApPLIED, pages 13:1-13:5. ACM, 2026. URL: https://doi.org/10.1145/3820355.3820392.
  5. Mathieu Baudet, Alberto Sonnino, Mahimna Kelkar, and George Danezis. Zef: Low-latency, scalable, private payments. arXiv:2201.05671, 2022. URL: https://arxiv.org/abs/2201.05671.
  6. Eli Ben-Sasson, Alessandro Chiesa, Christina Garman, Matthew Green, Ian Miers, Eran Tromer, and Madars Virza. Zerocash: Decentralized anonymous payments from bitcoin. In 2014 IEEE Symposium on Security and Privacy, SP 2014, Berkeley, CA, USA, May 18-21, 2014, pages 459-474. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/SP.2014.36.
  7. Kenneth P. Birman, André Schiper, and Pat Stephenson. Lightweight causal and atomic group multicast. ACM Trans. Comput. Syst., 9(3):272-314, 1991. URL: https://doi.org/10.1145/128738.128742.
  8. Sam Blackshear, Andrey Chursin, George Danezis, Anastasios Kichidis, Lefteris Kokoris-Kogias, Xun Li, Mark Logan, Ashok Menon, Todd Nowacki, Alberto Sonnino, Brandon Williams, and Lu Zhang. Sui lutris: A blockchain combining broadcast and consensus. arXiv:2310.18042, 2023. URL: https://doi.org/10.48550/arXiv.2310.18042.
  9. Benedikt Bünz, Shashank Agrawal, Mahdi Zamani, and Dan Boneh. Zether: Towards privacy in a smart contract world. In Joseph Bonneau and Nadia Heninger, editors, Financial Cryptography and Data Security - 24th International Conference, FC 2020, Kota Kinabalu, Malaysia, February 10-14, 2020 Revised Selected Papers, volume 12059 of Lecture Notes in Computer Science, pages 423-443. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-51280-4_23.
  10. Christian Cachin, David Lehnherr, Juan Villacis, and François-Xavier Wicht. The consensus number of untraceable cryptocurrencies. Cryptology ePrint Archive, Paper 2026/1503, 2026. URL: https://eprint.iacr.org/2026/1503.
  11. Christian Cachin, Jovana Micic, Nathalie Steinhauer, and Luca Zanolini. Quick order fairness. In Financial Cryptography, pages 316-333, 2022. URL: https://doi.org/10.1007/978-3-031-18283-9_15.
  12. Miranda Christ and Joseph Bonneau. Limits on revocable proof systems, with implications for stateless blockchains. In Foteini Baldimtsi and Christian Cachin, editors, Financial Cryptography and Data Security - 27th International Conference, FC 2023, Bol, Brač, Croatia, May 1-5, 2023, Revised Selected Papers, Part II, volume 13951 of Lecture Notes in Computer Science, pages 54-71. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-47751-5_4.
  13. Eli Daian, Giuliano Losa, Yehuda Afek, and Eli Gafni. A wealth of sub-consensus deterministic objects. In Ulrich Schmid and Josef Widder, editors, 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15-19, 2018, volume 121 of LIPIcs, pages 17:1-17:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.DISC.2018.17.
  14. Benjamin E. Diamond. Many-out-of-many proofs and applications to anonymous zether. In SP, pages 1800-1817, 2021. URL: https://doi.org/10.1109/SP40001.2021.00026.
  15. Christoph Egger, Russell W. F. Lai, Viktoria Ronge, Ivy K. Y. Woo, and Hoover H. F. Yin. On defeating graph analysis of anonymous transactions. Proc. Priv. Enhancing Technol., 2022(3):538-557, 2022. URL: https://doi.org/10.56553/POPETS-2022-0085.
  16. Prastudy Fauzi, Sarah Meiklejohn, Rebekah Mercer, and Claudio Orlandi. Quisquis: A new design for anonymous cryptocurrencies. In Steven D. Galbraith and Shiho Moriai, editors, Advances in Cryptology - ASIACRYPT 2019 - 25th International Conference on the Theory and Application of Cryptology and Information Security, Kobe, Japan, December 8-12, 2019, Proceedings, Part I, volume 11921 of Lecture Notes in Computer Science, pages 649-678. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-34578-5_23.
  17. Michael J. Fischer, Nancy A. Lynch, and Mike Paterson. Impossibility of distributed consensus with one faulty process. J. ACM, 32(2):374-382, April 1985. URL: https://doi.org/10.1145/3149.214121.
  18. Davide Frey, Mathieu Gestin, and Michel Raynal. The synchronization power (consensus number) of access-control objects: the case of allowlist and denylist. In Rotem Oshman, editor, 37th International Symposium on Distributed Computing, DISC 2023, October 10-12, 2023, L'Aquila, Italy, volume 281 of LIPIcs, pages 21:1-21:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.DISC.2023.21.
  19. Zoltán Füredi. Maximum degree and fractional matchings in uniform hypergraphs. Combinatorica, 1(2):155-162, 1981. URL: https://doi.org/10.1007/BF02579271.
  20. David K. Gifford. Weighted voting for replicated data. In Michael D. Schroeder and Anita K. Jones, editors, Proceedings of the Seventh Symposium on Operating System Principles, SOSP 1979, Asilomar Conference Grounds, Pacific Grove, California, USA, 10-12, December 1979, pages 150-162. ACM, 1979. URL: https://doi.org/10.1145/800215.806583.
  21. Rachid Guerraoui, Petr Kuznetsov, Matteo Monti, Matej Pavlovic, and Dragos-Adrian Seredinschi. The consensus number of a cryptocurrency. Distributed Comput., 35(1):1-15, 2022. URL: https://doi.org/10.1007/S00446-021-00399-2.
  22. Timothy L. Harris, Keir Fraser, and Ian A. Pratt. A practical multi-word compare-and-swap operation. In Dahlia Malkhi, editor, Distributed Computing, 16th International Conference, DISC 2002, Toulouse, France, October 28-30, 2002 Proceedings, volume 2508 of Lecture Notes in Computer Science, pages 265-279. Springer, 2002. URL: https://doi.org/10.1007/3-540-36108-1_18.
  23. Maurice Herlihy. Wait-free synchronization. ACM Trans. Program. Lang. Syst., 13(1):124-149, 1991. URL: https://doi.org/10.1145/114005.102808.
  24. Maurice Herlihy, Victor Luchangco, and Mark Moir. Obstruction-free synchronization: Double-ended queues as an example. In 23rd International Conference on Distributed Computing Systems (ICDCS 2003), pages 522-529. IEEE Computer Society, 2003. URL: https://doi.org/10.1109/ICDCS.2003.1203503.
  25. Maurice Herlihy and Jeannette M. Wing. Axioms for concurrent objects. In Conference Record of the Fourteenth Annual ACM Symposium on Principles of Programming Languages, Munich, Germany, January 21-23, 1987, pages 13-26. ACM Press, 1987. URL: https://doi.org/10.1145/41625.41627.
  26. Ferdinand Ihringer and Andrey Kupavskii. Regular intersecting families. Discrete Applied Mathematics, 270:142-152, 2019. URL: https://doi.org/10.1016/j.dam.2019.07.009.
  27. Mahimna Kelkar, Fan Zhang, Steven Goldfeder, and Ari Juels. Order-fairness for byzantine consensus. In CRYPTO (3), pages 451-480, 2020. URL: https://doi.org/10.1007/978-3-030-56877-1_16.
  28. Amrit Kumar, Clément Fischer, Shruti Tople, and Prateek Saxena. A traceability analysis of monero’s blockchain. In ESORICS (2), pages 153-173, 2017. URL: https://doi.org/10.1007/978-3-319-66399-9_9.
  29. Varun Madathil and Alessandra Scafuro. sfpri tt fhesfte: Achieving full-privacy in account-based cryptocurrencies is possible. In Goichiro Hanaoka and Bo-Yin Yang, editors, Advances in Cryptology - ASIACRYPT 2025 - 31st International Conference on the Theory and Application of Cryptology and Information Security, Melbourne, VIC, Australia, December 8-12, 2025, Proceedings, Part VII, volume 16251 of Lecture Notes in Computer Science, pages 199-229. Springer, 2025. URL: https://doi.org/10.1007/978-981-95-5122-4_7.
  30. Silvio Micali, Michael O. Rabin, and Salil P. Vadhan. Verifiable random functions. In FOCS, pages 120-130, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814584.
  31. Ian Miers, Christina Garman, Matthew Green, and Aviel D. Rubin. Zerocoin: Anonymous distributed e-cash from bitcoin. In 2013 IEEE Symposium on Security and Privacy, SP 2013, Berkeley, CA, USA, May 19-22, 2013, pages 397-411. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/SP.2013.34.
  32. Malte Möser, Kyle Soska, Ethan Heilman, Kevin Lee, Henry Heffan, Shashvat Srivastava, Kyle Hogan, Jason Hennessey, Andrew Miller, Arvind Narayanan, and Nicolas Christin. An empirical analysis of traceability in the monero blockchain. Proc. Priv. Enhancing Technol., 2018(3):143-163, 2018. URL: https://doi.org/10.1515/POPETS-2018-0025.
  33. Robert H. Thomas. A majority consensus approach to concurrency control for multiple copy databases. ACM Trans. Database Syst., 4(2):180-209, 1979. URL: https://doi.org/10.1145/320071.320076.
  34. Douglas B West. Introduction to graph theory. Prentice Hall, Upper Saddle River, NJ, 2 edition, August 2000. Google Scholar
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