Secure Implementation of the Cauchy Mechanism

[michaelpatrickpurcell]

The Cauchy mechanism can be used in conjunction with local sensitivity measures to create differentially private release mechanisms. See for example "Smooth Sensitivity and Sampling in Private Data Analysis" (https://cs-people.bu.edu/ads22/pubs/NRS07/NRS07-full-draft-v1.pdf).

Any implementation of the Cauchy mechanism requires secure generation of samples from the Cauchy distribution. It may be possible to generate such samples by exploiting the fact that if X,Y ~ N(0,1) are independent, then X/Y ~ Cauchy(0,1). Better yet, it looks like the Marsaglia polar method can be adapted to show that if X,Y ~ Uniform(0,1) are independent, then conditional on the event X^2 + Y^2 < 1 we have X/Y ~ Cauchy(0,1).