Construction of MDS Matrices Based on the Primitive Elements of the Finite Field

With the advent of the information age, the protected objects and application environments vary greatly. Using the same standard algorithm with fixed logic is difficult to meet the increasingly diversified cryptographic application requirements of cloud computing, Internet of things and 5g network environment. Dynamic variable cryptographic algorithm can improve the security and adaptability of cryptographic algorithm, which is the focus of cryptographic algorithm research and design at present. The key to the research and design of dynamic variable cryptographic algorithms is to find dynamic variable cryptographic components with good cryptographic performance. MDS (Maximum Distance Separable) is often used in the diffusion layer design of block ciphers because of its optimal diffusion characteristics. Traditional MDS transformation generally adopts MDS matrices in the finite field. While using them in dynamic variable cryptographic algorithms, it is difficult to realize parametric application because of their logical differences and implementation imbalance. Aiming at the lightweight MDS variable design requirements of dynamic variable cryptographic algorithm, this paper first proposes a MDS parametric construction method based on block cyclic shift and XOR operation. Compared with the traditional MDS construction in the finite field, we obtain a class of MDS component clusters with larger scale which is convenient for software and hardware implementation. The main design idea is to convert the multiplication operation in the finite field into cyclic shift operation, and introduce parameters to participate in the MDS operation in the way of direct XOR. Based on the MDS matrix in AES diffusion layer, two kinds of variable MDS diffusion layers are given by this method. We adopt the consistent structure based on cyclic shift and XOR operation, and each parameter corresponds to a different MDS transformation. The free transformation of operations in different finite fields of traditional MDS matrix can be realized by adjusting the parameters, which can be used for the parametric design of diffusion layer of dynamic variable cryptographic algorithm.