  In this paper, we propose a linear precoder for the downlink of a multi-user
MIMO system with multiple users that potentially act as eavesdroppers. The
proposed precoder is based on regularized channel inversion (RCI) with a
regularization parameter $\alpha$ and power allocation vector chosen in such a
way that the achievable secrecy sum-rate is maximized. We consider the
worst-case scenario for the multi-user MIMO system, where the transmitter
assumes users cooperate to eavesdrop on other users. We derive the achievable
secrecy sum-rate and obtain the closed-form expression for the optimal
regularization parameter $\alpha_{\mathrm{LS}}$ of the precoder using
large-system analysis. We show that the RCI precoder with
$\alpha_{\mathrm{LS}}$ outperforms several other linear precoding schemes, and
it achieves a secrecy sum-rate that has same scaling factor as the sum-rate
achieved by the optimum RCI precoder without secrecy requirements. We propose a
power allocation algorithm to maximize the secrecy sum-rate for fixed $\alpha$.
We then extend our algorithm to maximize the secrecy sum-rate by jointly
optimizing $\alpha$ and the power allocation vector. The jointly optimized
precoder outperforms RCI with $\alpha_{\mathrm{LS}}$ and equal power allocation
by up to 20 percent at practical values of the signal-to-noise ratio and for 4
users and 4 transmit antennas.
