The paper presents a method for calibrating noise in private data analysis to ensure privacy while minimizing the amount of noise required. It introduces the concept of ε-indistinguishability, where a privacy mechanism ensures that the probability distribution of transcripts remains close under small changes to the database. The key idea is to calibrate the standard deviation of the noise based on the sensitivity of the function being analyzed. Sensitivity is defined as the maximum change in the output of a function when a single database entry is modified. The paper shows that by adding noise according to a Laplace distribution with standard deviation S(f)/ε, where S(f) is the sensitivity of the function, ε-indistinguishability can be achieved. This approach allows for the analysis of general functions, not just sums, and reduces the amount of noise needed compared to previous methods. The paper also discusses the limitations of non-interactive mechanisms and shows that interactive mechanisms can handle a broader range of functions with less noise. It provides examples of functions with low sensitivity, such as histograms and distance functions, and demonstrates how they can be analyzed privately with minimal noise. The paper concludes with a separation result showing that non-interactive mechanisms require very large databases to answer certain types of queries, while interactive mechanisms can handle these queries more efficiently.The paper presents a method for calibrating noise in private data analysis to ensure privacy while minimizing the amount of noise required. It introduces the concept of ε-indistinguishability, where a privacy mechanism ensures that the probability distribution of transcripts remains close under small changes to the database. The key idea is to calibrate the standard deviation of the noise based on the sensitivity of the function being analyzed. Sensitivity is defined as the maximum change in the output of a function when a single database entry is modified. The paper shows that by adding noise according to a Laplace distribution with standard deviation S(f)/ε, where S(f) is the sensitivity of the function, ε-indistinguishability can be achieved. This approach allows for the analysis of general functions, not just sums, and reduces the amount of noise needed compared to previous methods. The paper also discusses the limitations of non-interactive mechanisms and shows that interactive mechanisms can handle a broader range of functions with less noise. It provides examples of functions with low sensitivity, such as histograms and distance functions, and demonstrates how they can be analyzed privately with minimal noise. The paper concludes with a separation result showing that non-interactive mechanisms require very large databases to answer certain types of queries, while interactive mechanisms can handle these queries more efficiently.