# Privacy-Preserving Monotonicity of Differential Privacy Mechanisms
Thus, privacy-preserving monotonicity shows that the increasing of privacy-preserving level leads to the decreasing of data utility as privacy budget decreases and the decreasing of privacy-preserving level leads to the increasing of data utility as privacy budget increases in differential privacy. Privacy-preserving monotonicity plays an important role in the research and applications of differential privacy. Since computational indistinguishability is the central theory of cryptography, two probability distributions are computationally indistinguishable if no efficient algorithm can distinguish them [32]. Therefore, the closer two probability distributions are, the better indistinguishability is. Conversely, the less close two probability distributions are, the weaker indistinguishability is. In other words, the stronger indistinguishability is, the bigger distinguishing error or uncertainty of a polynomial time algorithm is. Conversely, the weaker indistinguishability is, the smaller distinguishing error or entropy is. Thus, computational indistinguishability can become the foundation of differential privacy. In Figure 1, we presented the research framework of this paper. First, we proposed the definition of privacy-preserving monotonicity of differential privacy based on computational indistinguishability to measure trade-off under a non-interactive model or interactive model. Second, our theoretical and experimental results showed the privacy-preserving monotonicity of several differential privacy mechanisms based on the privacy metrics of the expected estimation error and entropy, such as Laplace mechanism, Gaussian mechanism, exponential mechanism, and randomized response mechanism. In addition, we theoretically and experimentally analyze that the utility monotonicity of these differential privacy mechanisms based on utility metrics of modulus of characteristic function and variant \(1-NE\) of normalized entropy. Finally, based on privacy-preserving monotonicity of differential privacy, we discussed the method of seeking trade-off under a semi-honest model and analyzed unilateral trade-off under rational model. Our main contributions are as follows. * We proposed the definition of privacy-preserving monotonicity of differential privacy based on computational indistinguishability to measure the trade-off between privacy and utility. * We theoretically and experimentally analyzed privacy-preserving monotonicity of existing differential privacy mechanisms, including the Laplace mechanism, Gaussian mechanism,exponential mechanism, and randomized response mechanism. In addition, our theoretical and experimental results showed the utility monotonicity of these differential privacy mechanisms. * We presented the method of seeking trade-off under a semi-honest model and analyzed the unilateral trade-off under rational model for differential privacy. This paper is organized as follows. Section 2 introduces the related works. Section 3 describes the problem statement. Section 4 introduces the preliminary to differential privacy. Section 5 presents the definition of privacy-preserving monotonicity of differential privacy. In Section 6, we theoretically analyze privacy-preserving monotonicity and utility monotonicity of differential privacy mechanisms. Section 7 covers numerical results of privacy-preserving monotonicity and utility monotonicity of differential privacy mechanisms. Section 8 presents the procedure of seeking trade-off under semi-honest model and analyzes unilateral trade-off under rational model. Section 9 concludes this paper.