Security-sensitive applications that access and generate large data sets are emerging in various areas such as bioinformatics and high energy physics. Data grids provide data-intensive applications with a large virtual storage framework with unlimited power. However, conventional scheduling algorithms for data grids are inadequate to meet the security needs of dataintensive applications. To remedy this deficiency, we address in this paper the problem of scheduling dataintensive jobs on data grids subject to security constraints. Using a security- and data-aware technique, SAHA (Security-Aware and Heterogeneity-Aware scheduling strategy) is proposed to improve quality of security for data-intensive applications running on data grids. Results based on real-world traces show that the proposed scheduling scheme dramatically improves security and performance over two existing scheduling algorithms.