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Recently, lattice-reduction-aided detectors have been proposed for multiinput multioutput (MIMO) systems to achieve performance with full diversity like the maximum likelihood receiver. However, these lattice-reduction-aided detectors are based on the traditional Lenstra-Lenstra-Lovasz (LLL) reduction algorithm that was originally introduced for reducing real lattice bases, in spite of the fact that the channel matrices are inherently complex-valued. In this paper, we introduce the complex LLL algorithm for direct application to reducing the basis of a complex lattice which is naturally defined by a complex-valued channel matrix. We derive an upper bound on proximity factors, which not only show the full diversity of complex LLL reduction-aided detectors, but also characterize the performance gap relative to the lattice decoder. Our analysis reveals that the complex LLL algorithm can reduce the complexity by nearly 50% compared to the traditional LLL algorithm, and this is confirmed by simulation. Interestingly, our simulation results suggest that the complex LLL algorithm has practically the same bit-error-rate performance as the traditional LLL algorithm, in spite of its lower complexity.