This paper considers the problem of distributed detection in the presence of a Sybil attack where a malicious sensor node can send multiple falsified decisions using multiple fake identities to a Fusion Center (FC) to degrade its decision-making performance. We study the problem under the Neyman-Pearson (NP) setup. We find that, due to the Sybil attack, the decisions received at the FC become correlated and that the degree of correlation is dependent on the number of fake identities used. The paper characterizes the optimal Sybil attack that blinds the FC, i.e., makes the FC incapable of making an informed decision. We find that if the sum of the local detection and false alarm probabilities of the sensor nodes is 1, the FC can be made blind when at least 50% of the decisions are sent using fake identities. However, if this condition is not met, then all decisions would have to be sent using fake identities in order to blind the FC. The paper also investigates strategic interactions between the FC and the Sybil attacker using Game Theory and proves the existence of a Nash Equilibrium (NE). Numerical results are presented to gain important insights.