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The approximation introduced by finite-precision representation of continuousdata can induce arbitrarily large information leaks even when the computationusing exact semantics is secure. Such leakage can thus undermine design effortsaimed at protecting sensitive information. We focus here on differentialprivacy, an approach to privacy that emerged from the area of statisticaldatabases and is now widely applied also in other domains. In this approach,privacy is protected by the addition of noise to a true (private) value. Todate, this approach to privacy has been proved correct only in the ideal casein which computations are made using an idealized, infinite-precisionsemantics. In this paper, we analyze the situation at the implementation level,where the semantics is necessarily finite-precision, i.e. the representation ofreal numbers and the operations on them, are rounded according to some level ofprecision. We show that in general there are violations of the differentialprivacy property, and we study the conditions under which we can stillguarantee a limited (but, arguably, totally acceptable) variant of theproperty, under only a minor degradation of the privacy level. Finally, weillustrate our results on two cases of noise-generating distributions: thestandard Laplacian mechanism commonly used in differential privacy, and abivariate version of the Laplacian recently introduced in the setting ofprivacy-aware geolocation.

Preserving differential privacy under finite-precision semantics