Searching for perfect diffusion matrices with lightweight coefficients

Diffusion layer is one of the main components of block ciphers and hash functions. MDS matrices are widely used to implement the diffusion layer. In this article, we first study the 4 × 4 MDS diffusion matrices constructed with linear feedback shift registers (LFSRs) of Fibonacci. For better hardware implementation, we focus on the low Hamming weight coefficients which are in the set {1,α,α+1,α2,α2+1,α3} and there are not two identical elements in a row or column. In addition, we introduce a way to calculate the number of XORs. Then, we give the minimum of XORs required to implement a multiplication by a finite element x by using GF(28) defined by different irreducible polynomials and present some new lightweight coefficients that are lighter than the current known ones, such as the ones used in AES. These MDS matrices not only are lighter than AES diffusion matrix but also do not contain two identical elements in a row or column.