Diierential Cryptanalysis of Feistel Ciphers and Diierentially -uniform Mappings

In this paper we study the round permutations (or S-boxes) which provide to Feistel ciphers the best resistance against diierential crypt-analysis. We prove that a Feistel cipher with any round keys and with at least 5 rounds resists any diierential attack if its round permutation is diierentially-uniform for a small. This improves an earlier result due to Nyberg and Knudsen which only held for independent and uniformly random round keys. We also give some necessary conditions for a mapping to be almost perfect nonlinear (i.e. diierentially 2-uniform).