Resumen
Maximum distance separable (MDS) codes have the maximum branch number in cryptography, and they are generally used in diffusion layers of symmetric ciphers. The diffusion layer of the Advanced Encryption Standard (AES) uses the circulant MDS matrix with the row element of {2;3;1;1}
{
2
;
3
;
1
;
1
}
in ??28
F
2
8
. It is the simplest MDS matrix in ??42??
F
2
n
4
, recorded as ??=????????(2;3;1;1)
A
=
C
i
r
c
(
2
;
3
;
1
;
1
)
. In this paper, we study the more extensive MDS constructions of ??
A
in ??42??
F
2
n
4
. By transforming the element multiplication operation in the finite field into the bit-level operation, we propose a multivariable operation definition based on simple operations, such as cyclic shift, shift, and XOR. We apply this multivariable operation to more lightweight MDS constructions of ??
A
and discuss the classification of the MDS clusters. We also give an example of the MDS cluster of ??
A
. Without changing the structure, elements, and the implementation cost of the known MDS matrix, the number of existing MDS transformations is expanded to ??2/2
n
2
/
2
times that of its original. The constructions in this paper provide rich component materials for the design of lightweight cryptographic algorithms.