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**2016/956**

**Title:**

Two Simple Composition Theorems with H-coefficients

**Authors:**
Jacques Patarin

**Abstract:**

We will present here 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 obtained with the bounds obtained with the coupling technique. Finally, we will show an interesting difference between 5 and 6 Random Feistel Schemes. With 5 rounds on $2n$ bits $\rightarrow 2n$ bits, when the number of $q$ queries satisfies $\sqrt{2^n} \ll q \ll 2^n$, 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.

**ePrint:**
https://eprint.iacr.org/2016/956

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