The existence of generalized mix functions

To enrich the message space of a cipher and guarantee security, Ristenpart and Rogaway defined mix functions on two sets of equal size. To mix inputs from two sets of different sizes, Stinson generalized the definition of mix functions (called generalized mix functions), and established an existence result for generalized mix functions with 10 undetermined pairs of input sizes. In this paper, we complete the solution to the existence problem for generalized mix functions.