### Journals Information

**
Mathematics and Statistics Vol. 12(1), pp. 63 - 68 DOI: 10.13189/ms.2024.120109 Reprint (PDF) (287Kb) **

## On The Existence of MDS Matrices over

**Defita ^{1}^{,2}^{,*}, Intan Muchtadi-Alamsyah ^{2}, Aleams Barra ^{2}**

^{1}Doctoral Program in Mathematics, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Indonesia

^{2}Algebra Research Group, Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, Indonesia

**ABSTRACT**

An MDS (maximum distance separable) matrix is a square matrix where all its submatrices are non-singular. The MDS matrices are used in some cryptographic systems' encryption and decryption processes. The matrix used in the process of decryption is the inverse matrix used in the encryption. Therefore, choosing a matrix which inverse is easy to find, is more efficient. Orthogonal and involutory matrices are two kinds of matrices in which inverses are easy to find. On the other hand, in terms of storage memory, a circulant matrix is more advantageous than any square matrix. In 2019, Cauchois and Loidreau proved that there is no involutory circulant MDS matrices of order 2m for m≥2 over a field of characteristics prime number p≥2. In 2022, Adhiguna et al. stated that there is no orthogonal circulant MDS matrix of even order and of order divisible by p > 2 over a field with characteristic p. This research concerns about observing the existence of MDS matrices over ring , that is, the ring where and q is the power of p. This paper shows that there is no involutory circulant MDS and no orthogonal circulant MDS matrices of certain order over . Furthermore, we can generalize these results for ring .

**KEYWORDS**

Finite Ring, Circulant Matrix, Orthogonal Matrix, Involutory Matrix, MDS Matrix

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Defita , Intan Muchtadi-Alamsyah , Aleams Barra , "On The Existence of MDS Matrices over ," Mathematics and Statistics, Vol. 12, No. 1, pp. 63 - 68, 2024. DOI: 10.13189/ms.2024.120109.

(b). APA Format:

Defita , Intan Muchtadi-Alamsyah , Aleams Barra (2024). On The Existence of MDS Matrices over . Mathematics and Statistics, 12(1), 63 - 68. DOI: 10.13189/ms.2024.120109.