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An increasing number of real-time applications, such as aircraft control and medical electronics systems, require high quality of security to assure confidentiality, authenticity and integrity of information. However, most existing algorithms for scheduling independent tasks in real-time systems do not adequately consider security requirements of real-time tasks. In recognition of this problem we propose a novel dynamic scheduling algorithm with security awareness, which is capable of achieving high quality of security for real-time tasks while improving resource utilization. We have conducted extensive simulation experiments to quantitatively evaluate the performance of our approach. Specifically, experimental results show that compared with three heuristic algorithms, the proposed algorithm can consistently improve overall system performance in terms of quality of security and system guarantee ratio under a wide range of workload characteristics.