On construction of multi-maximum distance separable (MDS) matrix generator based on high dimensional Cat matrices

Maximum distance separable (MDS) matrices are employed to create diffusion layers in block ciphers and hash functions. MDS matrices are generated by linear codes to reduce the cost for software or hardware implementations. However, linear codes can only generate a limited number of MDS matrices in a finite field. As a consequence, many block ciphers adopt the same matrices. This paper describes the design of a generator to create a large number of different MDS matrices at the same time by changing parameters of the high-dimensional Cat Map, named a multi-MDS matrix generator. We identify three types of Cat matrices which can be used to construct a multi-MDS matrix generator. In addition, we prove that the proposed MDS matrices with a small number of both different elements and XOR gates are efficient in terms of implementation performance. Finally, we show that the multi-MDS matrix generator inherits the dynamical properties of the high-dimensional Cat map, improving the resistance of diffusion layers against the powerful techniques of cryptanalysis.