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This paper is a continuation of the work initiated in [2] by M. Luby and C. Rackoff on Feistel schemes used as pseudorandom permutation generators. The aim of this paper is to study the qualitative improvements of "strong pseudorandomness" of the Luby-Rackoff construction when the number of rounds increase. We prove that for 6 rounds (or more), the success probability of the distinguisher is reduced from O(m2/2n) (for 3 or 4 rounds) to at most O(m4/23n+m2/22n). (Here m denotes the number of cleartext or ciphertext queries obtained by the enemy in a dynamic way, and 2n denotes the number of bits of the cleartexts and ciphertexts). We then introduce two new concepts that are stronger than strong pseudorandomness: "very strong pseudorandomness" and "homogeneous permutations". We explain why we think that those concepts are natural, and we study the values k for which the Luby-Rackoff construction with k rounds satisfy these notions.