Impossible Differential Cryptanalysis on Lai-Massey Scheme

The Lai-Massey scheme, proposed by Vaudenay, is a modified structure in the International Data Encryption Algorithm cipher. A family of block ciphers, named FOX, were built on the Lai-Massey scheme. Impossible differential cryptanalysis is a powerful technique used to recover the secret key of block ciphers. This paper studies the impossible differential cryptanalysis of the Lai-Massey scheme with affine orthomorphism for the first time. Firstly, we prove that there always exist 4-round impossible differentials of a Lai-Massey cipher having a bijective F-function. Such 4-round impossible differentials can be used to help find 4-round impossible differentials of FOX64 and FOX128. Moreover, we give some sufficient conditions to characterize the existence of 5-, 6-, and 7-round impossible differentials of Lai-Massey ciphers having a substitution-permutation (SP) F-function, and we observe that if Lai-Massey ciphers having an SP F-function use the same diffusion layer and orthomorphism as a FOX64, then there are indeed 5- and 6-round impossible differentials. These results indicate that both the diffusion layer and orthomorphism should be chosen carefully so as to make the Lai-Massey cipher secure against impossible differential cryptanalysis.