Malignant breast tumors and benign masses appear in mammograms with different shape characteristics: the former usually have rough, spiculated, or microlobulated contours, whereas the latter commonly have smooth, round, oval, or macrolobulated contours. Features that characterize shape roughness and complexity can assist in distinguishing between malignant tumors and benign masses. Signatures of contours may be used to analyze their shapes. We propose to use a signature based on the turning angle function of contours of breast masses to derive features that capture the characteristics of shape roughness as described above. We propose methods to derive an index of the presence of convex regions (XRTA), an index of the presence of concave regions (VRTA), an index of convexity (CXTA), and two measures of fractal dimension (FDTA and FDdTA) from the turning angle function. The methods were tested with a set of 111 contours of 65 benign masses and 46 malignant tumors with different parameters. The best classification accuracies in discriminating between benign masses and malignant tumors, obtained for XRTA, VRTA, CXTA, FDTA, and FDdTA in terms of the area under the receiver operating characteristics curve, were 0.92, 0.92, 0.93, 0.93, and, 0.92, respectively.