MISTY1 is a block cipher whose design relies on an assertion of provable security against linear and difierential cryptanalysis. Yet, a simplified and round reduced version of MISTY1 that does not alter the security provability can be attacked with higher order difierential cryptanalysis. We managed to explain this attack by deriving the attacking property from the choice of an atomic component of the algorithm, namely one of the two MISTY1 S-boxes. This allowed us to classify the good and the bad S-boxes built with the same principles and to show that none of the S-boxes with optimal linear and difierential properties has an optimal behaviour with respect to higher order difierential cryptanalysis.