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Local Differential Privacy (LDP) protocols enable an untrusted server to perform privacy-preserving, federated data analytics. Various LDP protocols have been developed for different types of data such as categorical data, numerical data, and key-value data. Due to their distributed settings, LDP protocols are fundamentally vulnerable to poisoning attacks, in which fake users manipulate the server's analytics results via sending carefully crafted data to the server. However, existing poisoning attacks focused on LDP protocols for simple data types such as categorical and numerical data, leaving the security of LDP protocols for more advanced data types such as key-value data unexplored. In this work, we aim to bridge the gap by introducing novel poisoning attacks to LDP protocols for key-value data. In such a LDP protocol, a server aims to simultaneously estimate the frequency and mean value of each key among some users, each of whom possesses a set of key-value pairs. Our poisoning attacks aim to simultaneously maximize the frequencies and mean values of some attacker-chosen target keys via sending carefully crafted data from some fake users to the sever. Specifically, since our attacks have two objectives, we formulate them as a two-objective optimization problem. Moreover, we propose a method to approximately solve the two-objective optimization problem, from which we obtain the optimal crafted data the fake users should send to the server. We demonstrate the effectiveness of our attacks to three LDP protocols for key-value data both theoretically and empirically. We also explore two defenses against our attacks, which are effective in some scenarios but have limited effectiveness in other scenarios. Our results highlight the needs for new defenses against our poisoning attacks.