We will present two new and simple theorems that show that when we compose permutation generators with independent keys, then the "quality" of CCA security increases. These theorems (Theorems 2 and 5 of this paper) are written in terms of H-coefficients (which are nothing else, up to some normalization factors, than transition probabilities). Then we will use these theorems on the classical analysis of Random Feistel Schemes (i.e. Luby-Rackoff constructions) and we will compare the results with the coupling technique. Finally, we will show an interesting difference between 5 and 6 Random Feistel Schemes. With 5 rounds on 2n bits → 2n bits, when the number of q queries satisfies √ 2 n q 2 n , we have some "holes" in the H-coefficient values, i.e. some H values are much smaller than the average value of H. This property for 5 rounds does not exist anymore on 6 rounds.