Differential privacy using Gamma distribution

The Laplace mechanism is a commonly employed approach that offers privacy guarantees within the framework of differential privacy. Nevertheless, the Laplace mechanism exhibits two limitations. Firstly, the privacy leakage of data can be exacerbated when the general differential private mechanism is accessed repeatedly with the same input owing to the sequential property of differential privacy. Secondly, the Laplace mechanism may not be suitable for some applications that solely involve positive samples as it can yield unwanted negative samples from the Laplace distribution.We address these issues by utilizing the Gamma distribution to handle database entries that must be consist of positive values ranging from 0 to infinity. In our approach, the epsilon parameter of our mechanism is determined by the value with noise according to the definition of differential privacy. Notably, the range of the noise is unbounded on the right thereby epsilon to approach infinity as the value with noise increases. To mitigate this, we impose constraints on the range of the noise in order to reasonably restrict the epsilon value of the mechanism. However, it should be noted that these constraints may impact the probability of ensuring epsilon-differential privacy and necessitate the imposition of a minimum boundary on the values of dataset. Additionally, we propose a new noise parameter that can be used to adjust the probability of ensuring differential privacy for a fixed epsilon.