Signal-processing modules working directly on encrypted data provide an elegant solution to application scenarios where valuable signals must be protected from a malicious processing device. In this paper, we investigate the implementation of the discrete Fourier transform (DFT) in the encrypted domain by using the homomorphic properties of the underlying cryptosystem. Several important issues are considered for the direct DFT: the radix-2 and the radix-4 fast Fourier algorithms, including the error analysis and the maximum size of the sequence that can be transformed. We also provide computational complexity analyses and comparisons. The results show that the radix-4 fast Fourier transform is best suited for an encrypted domain implementation in the proposed scenarios.