Document Open Access Logo

Exact zCDP Characterizations for Fundamental Differentially Private Mechanisms

Authors Charlie Harrison , Pasin Manurangsi

PDF

File

Thumbnail PDF
PDF
LIPIcs.FORC.2026.3.pdf
  • Filesize: 1.85 MB
  • 18 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Security and privacy
Keywords
  • Zero-Concentrated Differentially Privacy
  • Laplace Mechanism
  • Randomized Response

Metrics

Abstract

Zero-concentrated differential privacy (zCDP) is a variant of differential privacy (DP) that is widely used partly due to its nice composition property. While a tight conversion from ε-DP to zCDP exists for the worst-case mechanism, many common algorithms satisfy stronger guarantees. In this work, we derive tight zCDP characterizations for several fundamental mechanisms. We prove that the tight zCDP bound for the ε-DP Laplace mechanism is exactly ε + e^{-ε} - 1, confirming a recent conjecture by Wang [Yu-Xiang Wang, 2022]. We further provide tight bounds for the discrete Laplace mechanism, k-Randomized Response (for k ≤ 6), and RAPPOR. Lastly, we also provide a tight zCDP bound for the worst case bounded range mechanism.

Cite As Get BibTex

Charlie Harrison and Pasin Manurangsi. Exact zCDP Characterizations for Fundamental Differentially Private Mechanisms. In 7th Symposium on Foundations of Responsible Computing (FORC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 368, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FORC.2026.3

Author Details

Charlie Harrison
  • Google, Austin, TX, USA
Pasin Manurangsi
  • Google Research, Bangkok, Thailand

Acknowledgements

We would like to thank Thomas Steinke for insightful discussions, and for encouraging us to find a tight bound for bounded range mechanisms.

References

  1. Mark Bun and Thomas Steinke. Concentrated differential privacy: Simplifications, extensions, and lower bounds. In TCC, pages 635-658, 2016. URL: https://doi.org/10.1007/978-3-662-53641-4_24.
  2. Mark Cesar and Ryan Rogers. Bounding, concentrating, and truncating: Unifying privacy loss composition for data analytics. In ALT, pages 421-457, 2021. URL: https://proceedings.mlr.press/v132/cesar21a.html.
  3. Google Differential Privacy Team. Privacy loss distributions, 2026. URL: https://github.com/google/differential-privacy/blob/main/common_docs/Privacy_Loss_Distributions.pdf.
  4. Google Differential Privacy Team. Privacy loss distributions, 2026. URL: https://github.com/google/differential-privacy/blob/9b03b79908ba4c37437b09f40ff3e1fc9682c7ee/python/dp_accounting/dp_accounting/rdp/rdp_privacy_accountant.py#L973.
  5. Zeyu Ding, Daniel Kifer, Thomas Steinke, Yuxin Wang, Yingtai Xiao, Danfeng Zhang, et al. The permute-and-flip mechanism is identical to report-noisy-max with exponential noise. arXiv preprint, 2021. URL: https://arxiv.org/abs/2105.07260.
  6. Jinshuo Dong, David Durfee, and Ryan Rogers. Optimal differential privacy composition for exponential mechanisms. In ICML, pages 2597-2606, 2020. URL: https://proceedings.mlr.press/v119/dong20a.html.
  7. David Durfee and Ryan Rogers. Practical differentially private top-k selection with pay-what-you-get composition. In NeurIPS, pages 3527-3537, 2019. URL: https://proceedings.neurips.cc/paper/2019/hash/b139e104214a08ae3f2ebcce149cdf6e-Abstract.html.
  8. Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In TCC, pages 265-284, 2006. URL: https://doi.org/10.1007/11681878_14.
  9. Cynthia Dwork, Moni Naor, Omer Reingold, Guy N. Rothblum, and Salil P. Vadhan. On the complexity of differentially private data release: efficient algorithms and hardness results. In STOC, pages 381-390, 2009. URL: https://doi.org/10.1145/1536414.1536467.
  10. Cynthia Dwork and Aaron Roth. The algorithmic foundations of differential privacy. Found. Trends Theor. Comput. Sci., 9(3–4):211-407, August 2014. URL: https://doi.org/10.1561/0400000042.
  11. Cynthia Dwork and Guy N. Rothblum. Concentrated differential privacy. CoRR, abs/1603.01887, 2016. URL: https://arxiv.org/abs/1603.01887.
  12. Hamid Ebadi, David Sands, and Gerardo Schneider. Differential privacy: Now it’s getting personal. In Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '15, pages 69-81, New York, NY, USA, 2015. Association for Computing Machinery. URL: https://doi.org/10.1145/2676726.2677005.
  13. Úlfar Erlingsson, Vasyl Pihur, and Aleksandra Korolova. RAPPOR: randomized aggregatable privacy-preserving ordinal response. In CCS, pages 1054-1067, 2014. URL: https://doi.org/10.1145/2660267.2660348.
  14. Vitaly Feldman and Tijana Zrnic. Individual privacy accounting via a rényi filter. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 28080-28091. Curran Associates, Inc., 2021. URL: https://proceedings.neurips.cc/paper_files/paper/2021/file/ec7f346604f518906d35ef0492709f78-Paper.pdf.
  15. Quan Geng and Pramod Viswanath. The optimal mechanism in differential privacy. In ISIT, pages 2371-2375, 2014. URL: https://doi.org/10.1109/ISIT.2014.6875258.
  16. Arpita Ghosh, Tim Roughgarden, and Mukund Sundararajan. Universally utility-maximizing privacy mechanisms. SIAM J. Comput., 41(6):1673-1693, 2012. URL: https://doi.org/10.1137/09076828X.
  17. Charlie Harrison. Tight rdp and zcdp bounds for rappor. https://charlieharrison.xyz/2024/08/20/rdp-rappor.html, 2025. Accessed: 2025-10-17.
  18. Peter Kairouz, Kallista A. Bonawitz, and Daniel Ramage. Discrete distribution estimation under local privacy. In ICML, volume 48, pages 2436-2444, 2016. URL: http://proceedings.mlr.press/v48/kairouz16.html.
  19. Shiva Prasad Kasiviswanathan, Homin K. Lee, Kobbi Nissim, Sofya Raskhodnikova, and Adam D. Smith. What can we learn privately? SIAM J. Comput., 40(3):793-826, 2011. URL: https://doi.org/10.1137/090756090.
  20. Mathias Lécuyer. Practical privacy filters and odometers with rényi differential privacy and applications to differentially private deep learning, 2021. URL: https://arxiv.org/abs/2103.01379.
  21. Ryan McKenna and Daniel R Sheldon. Permute-and-flip: A new mechanism for differentially private selection. Advances in Neural Information Processing Systems, 33:193-203, 2020. Google Scholar
  22. Frank McSherry and Kunal Talwar. Mechanism design via differential privacy. In FOCS, pages 94-103, 2007. URL: https://doi.org/10.1109/FOCS.2007.66.
  23. Ilya Mironov. Rényi differential privacy. In CSF, pages 263-275, 2017. URL: https://doi.org/10.1109/CSF.2017.11.
  24. Alfréd Rényi. On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, 1961. Google Scholar
  25. Ryan Rogers and Thomas Steinke. A better privacy analysis of the exponential mechanism. DifferentialPrivacy.org, July 2021. URL: https://differentialprivacy.org/exponential-mechanism-bounded-range/.
  26. Ryan M Rogers, Aaron Roth, Jonathan Ullman, and Salil Vadhan. Privacy odometers and filters: Pay-as-you-go composition. In D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016. URL: https://proceedings.neurips.cc/paper_files/paper/2016/file/58c54802a9fb9526cd0923353a34a7ae-Paper.pdf.
  27. Thomas Steinke. Tight rdp & zcdp bounds from pure dp. DifferentialPrivacy.org, May 2024. URL: https://differentialprivacy.org/pdp-to-zcdp/.
  28. Tianhao Wang, Jeremiah Blocki, Ninghui Li, and Somesh Jha. Locally differentially private protocols for frequency estimation. In USENIX Security, pages 729-745, 2017. URL: https://www.usenix.org/conference/usenixsecurity17/technical-sessions/presentation/wang-tianhao.
  29. Yu-Xiang Wang. For laplace mechanism, ε + exp(-ε) -1 is the formula. and for randresp, it is the difference btw. cross-entropy and entropy. what’s in common? they are the kl-div and the derivative of renyidp at α → 0. @shortstein is this trivial observation published somewhere (or well-known)?, September 2022. Tweet. URL: https://x.com/yuxiangw_cs/status/1565765510872018950.
  30. Proof Wiki. Hyperbolic tangent less than x, 2025. URL: https://proofwiki.org/wiki/Hyperbolic_Tangent_Less_than_X.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact