Shannon meets Gray: Noise-robust, Low-sensitivity Codes with Applications in Differential Privacy

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., neighbouring 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 an asymptotically faster, space-optimal differentially private data structure for histograms.