Practical and Provable Security against Differential and Linear Cryptanalysis for Substitution-Permutation Networks

Abstract

We examine the diffusion layers of some block ciphers referred to as substitution‐permutation networks. We investigate the practical and provable security of these diffusion layers against differential and linear cryptanalysis. First, in terms of practical security, we show that the minimum number of differentially active S‐boxes and that of linearly active S‐boxes are generally not identical and propose some special conditions in which those are identical. We also study the optimal diffusion effect for some diffusion layers according to their constraints. Second, we obtain the results that the consecutive two rounds of SPN structure provide provable security against differential and linear cryptanalysis, i.e., we prove that the probability of each differential (resp. linear hull) of the consecutive two rounds of SPN structure with a maximal diffusion layer is bounded by pn (resp. qn) and that of each differential (resp. linear hull) of the SDS function with a semi‐maximal diffusion layer is bounded by pn‐1 (resp. qn‐1), where p and q are maximum differential and linear probabilities of the substitution layer, respectively.

On the Security of Rijndael-Like Structures against Differential and Linear Cryptanalysis

“…Throughout this paper, we define wt(x) = wt( [9]. Therefore, we can define the branch number β l from the view point of linear cryptanalysis as follows: …”

Improving the Upper Bound on the Maximum Differential and the Maximum Linear Hull Probability for SPN Structures and AES

“…Hong et al proved the upper bound on the maximum differential and the maximum linear hull probability for 2 rounds of SPN structures with highly diffusive linear transformation [7]. Kang et al generalized their result for any value of the branch number of the linear transformation [8].…”

Recursive Diffusion Layers for Block Ciphers and Hash Functions

“…A diffusion layer D taking its maximal β d and β l is called a perfect or MDS diffusion layer. Furthermore, a diffusion layer with β d = β l = s is called an almost perfect diffusion layer [10].…”

“…In the last 2 decades, from these two families several structures have been proposed with provable security against DC and LC. Three rounds of Feistel structure [12,9], five rounds of RC6-like structure [6] and SDS (substitution-diffusion-substitution) structure with a perfect or almost perfect diffusion layer are examples of such structures [10].…”