Substitution boxes (S-boxes) are a crucial component of DES-like block ciphers. This research addresses problems with previous approaches towards constructing S-boxes, and proposes a new definition for the robustness of S-boxes to differential cryptanalysis, which is the most powerful cryptanalytic attack known to date. A novel method based on group Hadamard matrices is developed to systematically generate S-boxes that satisfy a number of critical cryptographic properties. Among the properties are the high nonlinearity, the strict avalanche characteristics, the balancedness, the robustness against differential cryptanalysis, and the immunity to linear cryptanalysis. An example is provided to illustrate the S-box generating method.