As a method for answer aggregation in crowdsourced data management, rank aggregation aims to combine different agents' answers or preferences over the given alternatives into an aggregate ranking which agrees the most with the preferences. However, since the aggregation procedure relies on a data curator, the privacy within the agents' preference data could be compromised when the curator is untrusted. Existing works that guarantee differential privacy in rank aggregation all assume that the data curator is trusted. In this paper, we formulate and address the problem of locally differentially private rank aggregation, in which the agents have no trust in the data curator. By leveraging the approximate rank aggregation algorithm KwikSort, the Randomized Response mechanism, and the Laplace mechanism, we propose an effective and efficient protocol LDP-KwikSort. Theoretical and empirical results show that the solution LDP-KwikSort:RR can achieve the acceptable trade-off between the utility of aggregate ranking and the privacy protection of agents' pairwise preferences.