This work studies the power of adversarial attacks against machine learning algorithms that use differentially private mechanisms as their weapon. In our setting, the adversary aims to modify the content of a statistical dataset via insertion of additional data without being detected by using the differential privacy to her/his own benefit. The goal of this study is to evaluate how easy it is to detect such attacks (anomalies) when the adversary makes use of Gaussian and Laplacian perturbation using both statistical and information-theoretic tools. To this end, firstly via hypothesis testing, we characterize statistical thresholds for the adversary in various settings, which balances the privacy budget and the impact of the attack (the modification applied on the original data) in order to avoid being detected. In addition, we establish the privacy-distortion trade-off in the sense of the well-known rate-distortion function for the Gaussian mechanism by using an information-theoretic approach. Accordingly, we derive an upper bound on the variance of the attacker’s additional data as a function of the sensitivity and the original data’s second-order statistics. Lastly, we introduce a new privacy metric based on Chernoff information for anomaly detection under differential privacy as a stronger alternative for the (ϵ,δ)-differential privacy in Gaussian mechanisms. Analytical results are supported by numerical evaluations.