Collecting and analyzing personal data is important in modern information applications. Though the privacy of data providers should be protected, the need to track certain data providers often arises, such as tracing specific patients or adversarial users. Thus, tracking only specific persons without revealing normal users' identities is quite important for operating information systems using personal data. It is difficult to know in advance the rules for specifying the necessity of tracking since the rules are derived by the analysis of collected data. Thus, it would be useful to provide a general way that can employ any data analysis method regardless of the type of data and the nature of the rules. In this paper, we propose a privacy-preserving data analysis construction that allows an authority to detect specific users while other honest users are kept anonymous. By using the cryptographic techniques of group signatures with message-dependent opening (GS-MDO) and public key encryption with non-interactive opening (PKENO), we provide a correspondence table that links a user and data in a secure way, and we can employ any anonymization technique and data analysis method. It is particularly worth noting that no “big brother” exists, meaning that no single entity can identify users who do not provide anomaly data, while bad behaviors are always traceable. We show the result of implementing our construction. Briefly, the overhead of our construction is on the order of 10 ms for a single thread. We also confirm the efficiency of our construction by using a real-world dataset.