Nearly Optimal Private Convolution

  • Nadia Fawaz
  • S. Muthukrishnan
  • Aleksandar Nikolov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)


We study algorithms for computing the convolution of a private input x with a public input h, while satisfying the guarantees of (ε, δ)-differential privacy. Convolution is a fundamental operation, intimately related to Fourier Transforms. In our setting, the private input may represent a time series of sensitive events or a histogram of a database of confidential personal information. Convolution then captures important primitives including linear filtering, which is an essential tool in time series analysis, and aggregation queries on projections of the data. We give an algorithm for computing convolutions which satisfies (ε, δ)-differentially privacy and is nearly optimal for every public h, i.e. is instance optimal with respect to the public input. We prove optimality via spectral lower bounds on the hereditary discrepancy of convolution matrices. Our algorithm is very efficient – it is essentially no more computationally expensive than a Fast Fourier Transform.


Mean Square Error Discrete Fourier Transform Public Input Circular Convolution Matrix Mechanism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barak, B., Chaudhuri, K., Dwork, C., Kale, S., McSherry, F., Talwar, K.: Privacy, accuracy, and consistency too: a holistic solution to contingency table release. In: Proceedings of the Twenty-Sixth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pp. 273–282. ACM (2007)Google Scholar
  2. 2.
    Bhaskara, A., Dadush, D., Krishnaswamy, R., Talwar, K.: Unconditional differentially private mechanisms for linear queries. In: Proceedings of the 44th Symposium on Theory of Computing, STOC 2012, pp. 1269–1284. ACM, New York (2012)CrossRefGoogle Scholar
  3. 3.
    Blum, A., Ligett, K., Roth, A.: A learning theory approach to non-interactive database privacy. In: STOC 2008: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 609–618. ACM, New York (2008)Google Scholar
  4. 4.
    Bolot, J., Fawaz, N., Muthukrishnan, S., Nikolov, A., Taft, N.: Private decayed sum estimation under continual observation. Arxiv preprint arXiv:1108.6123 (2011)Google Scholar
  5. 5.
    Hubert Chan, T.-H., Shi, E., Song, D.: Private and continual release of statistics. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 405–417. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Chandrasekaran, K., Thaler, J., Ullman, J., Wan, A.: Faster private release of marginals on small databases. arXiv preprint arXiv:1304.3754 (2013)Google Scholar
  7. 7.
    Cheraghchi, M., Klivans, A., Kothari, P., Lee, H.: Submodular functions are noise stable. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1586–1592. SIAM (2012)Google Scholar
  8. 8.
    Cormode, G., Procopiuc, C.M., Srivastava, D., Yaroslavtsev, G.: Accurate and efficient private release of datacubes and contingency tablesGoogle Scholar
  9. 9.
    Dinur, I., Nissim, K.: Revealing information while preserving privacy. In: Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pp. 202–210. ACM (2003)Google Scholar
  10. 10.
    Dwork, C., McSherry, F., Nissim, K., Smith, A.: Calibrating noise to sensitivity in private data analysis. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 265–284. Springer, Heidelberg (2006)Google Scholar
  11. 11.
    Dwork, C., Naor, M., Reingold, O., Rothblum, G.N., Vadhan, S.: On the complexity of differentially private data release: efficient algorithms and hardness results. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 381–390. ACM (2009)Google Scholar
  12. 12.
    Dwork, C., Pitassi, T., Naor, M., Rothblum, G.: Differential privacy under continual observation. In: STOC (2010)Google Scholar
  13. 13.
    Gençay, R., Selçuk, F., Whitcher, B.: An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. Elsevier Academic Press (2002)Google Scholar
  14. 14.
    Gray, R.M.: Toeplitz and circulant matrices: a review. Foundations and Trends in Communications and Information Theory 2(3), 155–239 (2006)CrossRefGoogle Scholar
  15. 15.
    Gupta, A., Hardt, M., Roth, A., Ullman, J.: Privately releasing conjunctions and the statistical query barrier. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, pp. 803–812. ACM (2011)Google Scholar
  16. 16.
    Hardt, M., Rothblum, G.: A multiplicative weights mechanism for privacy-preserving data analysis. In: Proc. 51st Foundations of Computer Science (FOCS). IEEE (2010)Google Scholar
  17. 17.
    Hardt, M., Rothblum, G., Servedio, R.: Private data release via learning thresholds. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 168–187. SIAM (2012)Google Scholar
  18. 18.
    Hardt, M., Talwar, K.: On the geometry of differential privacy. In: Proceedings of the 42nd ACM Symposium on Theory of Computing (2010)Google Scholar
  19. 19.
    Kasiviswanathan, S., Rudelson, M., Smith, A., Ullman, J.: The price of privately releasing contingency tables and the spectra of random matrices with correlated rows. In: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 775–784. ACM (2010)Google Scholar
  20. 20.
    Li, C., Hay, M., Rastogi, V., Miklau, G., McGregor, A.: Optimizing linear counting queries under differential privacy. In: Proceedings of the Twenty-Ninth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2010, pp. 123–134. ACM, New York (2010)CrossRefGoogle Scholar
  21. 21.
    Li, C., Miklau, G.: An adaptive mechanism for accurate query answering under differential privacy. PVLDB 5(6), 514–525 (2012)Google Scholar
  22. 22.
    Li, C., Miklau, G.: Measuring the achievable error of query sets under differential privacy. CoRR abs/1202.3399 (2012)Google Scholar
  23. 23.
    Lovász, L., Spencer, J., Vesztergombi, K.: Discrepancy of set-systems and matrices. European Journal of Combinatorics 7(2), 151–160 (1986)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Muthukrishnan, S., Nikolov, A.: Optimal private halfspace counting via discrepancy. In: Proceedings of the 44th ACM Symposium on Theory of Computing (2012)Google Scholar
  25. 25.
    Narayanan, A., Shi, E., Rubinstein, B.: Link prediction by de-anonymization: How we won the kaggle social network challenge. In: The 2011 International Joint Conference on Neural Networks (IJCNN), pp. 1825–1834. IEEE (2011)Google Scholar
  26. 26.
    Narayanan, A., Shmatikov, V.: Robust de-anonymization of large sparse datasets. In: IEEE Symposium on Security and Privacy, SP 2008, pp. 111–125. IEEE (2008)Google Scholar
  27. 27.
    Narayanan, A., Shmatikov, V.: De-anonymizing social networks. In: 2009 30th IEEE Symposium on Security and Privacy, pp. 173–187. IEEE (2009)Google Scholar
  28. 28.
    Nikolov, A., Talwar, K., Zhang, L.: The geometry of differential privacy: the sparse and approximate casesGoogle Scholar
  29. 29.
    Thaler, J., Ullman, J., Vadhan, S.: Faster algorithms for privately releasing marginals. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 810–821. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  30. 30.
    Xiao, X., Wang, G., Gehrke, J.: Differential privacy via wavelet transformsGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Nadia Fawaz
    • 1
  • S. Muthukrishnan
    • 2
  • Aleksandar Nikolov
    • 2
  1. 1.TechnicolorPalo AltoUSA
  2. 2.Rutgers UniversityUSA

Personalised recommendations