The resilience of a voting system has been a central topic in computational social choice. Many voting rules, like plurality, are shown to be vulnerable as the attacker can target specific voters to manipulate the result. What if a local differential privacy (LDP) mechanism is adopted such that the true preference of a voter is never revealed in pre-election polls? In this case, the attacker can only infer stochastic information about a voter's true preference, and this may cause the manipulation of the electoral result significantly harder. The goal of this paper is to provide a quantitative study on the effect of adopting LDP mechanisms on a voting system. We introduce the metric PoLDP (power of LDP) that quantitatively measures the difference between the attacker's manipulation cost under LDP mechanisms and that without LDP mechanisms. The larger PoLDP is, the more robustness LDP mechanisms can add to a voting system. We give a full characterization of PoLDP for the voting system with plurality rule and provide general guidance towards the application of LDP mechanisms.