By considering the role of non-linear approximatioris in linear cryptanalysis we obtain a generalization of Matsui's linear cryptanalytic techniques. This approach allows Ihe cryptanalyst greater flexibility in mounting a linear cryptanalytic attack and we demonstrate the effectiveness of our non-linear techniques with some simple attacks on LOK191. These attacks potentially allow for the recovery of seven additional bits of key information with less than 1/4 of the plaintext that is required using current linear cryptanalytic methods.