Shannon meets Gray: Noise-robust, Low-sensitivity Codes with Applications in Differential Privacy
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Integer data is typically made differentially private by adding noise from a Discrete Laplace (or Discrete Gaussian) distribution. We study the setting where differential privacy of a counting query is achieved using bit-wise randomized response, i.e., independent, random bit flips on the encoding of the query answer. Binary error-correcting codes transmitted through noisy channels with independent bit flips are well-studied in information theory. However, such codes are unsuitable for differential privacy since they have (by design) high sensitivity, i.e., neighboring integers have encodings with a large Hamming distance. Gray codes show that it is possible to create an efficient sensitivity 1 encoding, but are also not suitable for differential privacy due to lack of noise-robustness. Our main result is that it is possible, with a constant rate code, to simultaneously achieve the sensitivity of Gray codes and the noise-robustness of error-correcting codes (down to the noise level required for differential privacy). An application of this new encoding of the integers is a faster, space-optimal differentially private data structure for histograms.
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