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In this paper, a novel secure and private over-the-air federated learning (SP-OTA-FL) framework is studied where noise is employed to protect data privacy and system security. Specifically, the privacy leakage of user data and the security level of the system are measured by differential privacy (DP) and mean square error security (MSE-security), respectively. To mitigate the impact of noise on learning accuracy, we propose a channel-weighted post-processing (CWPP) mechanism, which assigns a smaller weight to the gradient of the device with poor channel conditions. Furthermore, employing CWPP can avoid the issue that the signal-to-noise ratio (SNR) of the overall system is limited by the device with the worst channel condition in aligned over-the-air federated learning (OTA-FL). We theoretically analyze the effect of noise on privacy and security protection and also illustrate the adverse impact of noise on learning performance by conducting convergence analysis. Based on these analytical results, we propose device scheduling policies considering privacy and security protection in different cases of channel noise. In particular, we formulate an integer nonlinear fractional programming problem aiming to minimize the negative impact of noise on the learning process. We obtain the closed-form solution to the optimization problem when the model is with high dimension. For the general case, we propose a secure and private algorithm (SPA) based on the branch-and-bound (BnB) method, which can obtain an optimal solution with low complexity. The effectiveness of the proposed CWPP mechanism and the policies for device selection are validated through simulations.