Abstract
In this work we investigate the problem of private statistical analysis of time-series data in the distributed and semi-honest setting. In particular, we study some properties of Private Stream Aggregation (PSA), first introduced by Shi et al. 2011. This is a computationally secure protocol for the collection and aggregation of data in a distributed network and has a very small communication cost. In the non-adaptive query model, a secure PSA scheme can be built upon any key-homomorphic weak pseudo-random function as shown by Valovich 2017, yielding security guarantees in the standard model which is in contrast to Shi et al. We show that every mechanism which preserves \((\epsilon ,\delta )\)-differential privacy in effect preserves computational \((\epsilon ,\delta )\)-differential privacy when it is executed through a secure PSA scheme. Furthermore, we introduce a novel perturbation mechanism based on the symmetric Skellam distribution that is suited for preserving differential privacy in the distributed setting, and find that its performances in terms of privacy and accuracy are comparable to those of previous solutions. On the other hand, we leverage its specific properties to construct a computationally efficient prospective post-quantum protocol for differentially private time-series data analysis in the distributed model. The security of this protocol is based on the hardness of a new variant of the Decisional Learning with Errors (DLWE) problem. In this variant the errors are taken from the symmetric Skellam distribution. We show that this new variant is hard based on the hardness of the standard Learning with Errors (LWE) problem where the errors are taken from the discrete Gaussian distribution. Thus, we provide a variant of the LWE problem that is hard based on conjecturally hard lattice problems and uses a discrete error distribution that is similar to the continuous Gaussian distribution in that it is closed under convolution. A consequent feature of the constructed prospective post-quantum protocol is the use of the same noise for security and for differential privacy.
The research was supported by the DFG Research Training Group GRK 1817/1.
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Notes
- 1.
- 2.
Although the uniform distribution is reproducible as well, the result from [10] does not provide a proper error distribution for our DLWE-based PSA scheme, since a differentially private mechanism with uniform noise provides no accuracy to statistical data analyses.
- 3.
Due to the use of a cryptographic protocol, the plaintexts have to be discrete. This is the reason why we use discrete distributions for generating noise.
- 4.
These mechanisms work in the centralised setting, where a trusted curator sees the full database in the clear and perturbs it properly.
- 5.
In [7] it was shown that the sum of n discrete Gaussians each with parameter \(\sigma ^2\) is statistically close to a discrete Gaussian with parameter \(\nu =n\sigma ^2\) if \(\sigma >\sqrt{n}\eta _\varepsilon (\varLambda )\) for some smoothing parameter \(\eta _\varepsilon (\varLambda )\) of the underlying lattice \(\varLambda \). However, as pointed out in [28], this approach is less suitable for our purpose if the number of users is large, since the aggregated decryption outcome would have a an error with a variance of order \(\nu =\Omega (n^2)\) (in Example 2 the variance is only of order \(O(\lambda ^2\kappa n)\)).
- 6.
See for instance Theorem 3 in [20].
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Valovich, F., Aldà, F. (2018). Computational Differential Privacy from Lattice-Based Cryptography. In: Kaczorowski, J., Pieprzyk, J., Pomykała, J. (eds) Number-Theoretic Methods in Cryptology. NuTMiC 2017. Lecture Notes in Computer Science(), vol 10737. Springer, Cham. https://doi.org/10.1007/978-3-319-76620-1_8
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