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Providing physical-layer security for mobile users in future broadband wireless networks is of both theoretical and practical importance. In this paper, we formulate an analytical framework for resource allocation in a downlink OFDMA-based broadband network with coexistence of secure users (SU) and normal users (NU). The problem is formulated as joint power and subcarrier allocation with the objective of maximizing average aggregate information rate of all NU's while maintaining an average secrecy rate for each individual SU under a total transmit power constraint for the base station. We solve this problem in an asymptotically optimal manner using dual decomposition. Our analysis shows that an SU becomes a candidate competing for a subcarrier only if its channel gain on this subcarrier is the largest among all and exceeds the second largest by a certain threshold. Furthermore, while the power allocation for NU's follows the conventional water-filling principle, the power allocation for SU's depends on both its own channel gain and the largest channel gain among others. We also design a suboptimal algorithm to reduce the computational cost. Numerical studies are conducted to evaluate the performance of the proposed algorithms in terms of the achievable pair of information rate for NU's and secrecy rate for SU at different power consumptions.