This paper studies the statistical characterization of detecting an adversary\nwho wants to harm some computation such as machine learning models or\naggregation by altering the output of a differentially private mechanism in\naddition to discovering some information about the underlying dataset. An\nadversary who is able to modify the published information from a differentially\nprivate mechanism aims to maximize the possible damage to the system while\nremaining undetected. We present a trade-off between the privacy parameter of\nthe system, the sensitivity and the attacker's advantage (the bias) through\ndetermining the threshold for the best critical region of the hypothesis\ntesting problem for deciding whether or not the adversary's attack is detected.\nSuch trade-offs are provided for Laplace mechanisms using one-sided and\ntwo-sided hypothesis tests. Corresponding error probabilities are analytically\nderived and ROC curves are presented for various levels of the sensitivity, the\nabsolute mean of the attack and the privacy parameter. Subsequently, we provide\nan interval for the bias induced by the adversary so that the defender detects\nthe attack. Finally, we adapt the Kullback-Leibler differential privacy to\nadversarial classification.\n