Provable Security against Differential and Linear Cryptanalysis for the SPN Structure

In the SPN (Substitution-Permutation Network) structure, it is very important to design a diffusion layer to construct a secure block cipher against differential cryptanalysis and linear cryptanalysis. The purpose of this work is to prove that the SPN structure with a maximal diffusion layer provides a provable security against differential cryptanalysis and linear cryptanalysis in the sense that the probability of each differential (respectively linear hull) is bounded by p (respectively q), where p (respectively q) is the maximum differential (respectively liner hull) probability of n S-boxes used in the substitution layer. We will also give a provable security for the SPN structure with a semi-maximal diffusion layer against differential cryptanalysis and linear cryptanalysis.