On the generalization of $g$-circulant MDS matrices
Atif Ahmad Khan, Shakir Ali, Bhupendra Singh
TL;DR
This work introduces consta-$g$-circulant matrices over $\mathbb{F}_q$ defined by $h(x)=x^m-\lambda+\sum_{i=0}^{m-1}h_i x^i$ to extend circulant and $g$-circulant constructions for MDS diffusion matrices. It establishes a tight counting framework via CRT, giving an explicit upper bound $(\left\lfloor (m-1)/\operatorname{ord}(\lambda)\right\rfloor+1)\,q^m$ and a product-based formula for the number of invertible instances when $x^m-\lambda=\prod f_i(x)^{e_i}$; it also derives necessary conditions for MDS-ness and fully characterizes MDS cases for $3\times3$ and $4\times4$ matrices. The paper then generalizes to consta-$\theta_g$-circulant matrices with skew polynomial rings, providing analogous invertibility and MDS criteria, including explicit involutory examples. Finally, constructive algorithms for small orders and illustrative examples demonstrate practical guidance for diffusion-layer design and efficient cryptographic implementations.
Abstract
A matrix $M$ over the finite field $ \mathbb{F}_q $ is called \emph{maximum distance separable} (MDS) if all of its square submatrices are non-singular. These MDS matrices are very important in cryptography and coding theory because they provide strong data protection and help spread information efficiently. In this paper, we introduce a new type of matrix called a \emph{consta-$g$-circulant matrix}, which extends the idea of $g$-circulant matrices. These matrices come from a linear transformation defined by the polynomial $ h(x) = x^m - λ+ \sum_{i=0}^{m-1} h_i x^i $ over $ \mathbb{F}_q $. We find the upper bound of such matrices exist and give conditions to check when they are invertible. This helps us know when they are MDS matrices. If the polynomial $ x^m - λ$ factors as $ x^m - λ= \prod_{i=1}^{t} f_i(x)^{e_i}, $ where each \( f_i(x) \) is irreducible, then the number of invertible consta-$g$-circulant matrices is $ N \cdot \prod_{i=1}^{t} \left( q^{°f_i} - 1 \right), $ where $r$ is the multiplicative order of $λ$, and \( N \) is the number of integers \( k \) such that $ 0 \leq k < \left\lfloor \frac{m - 1}{r} \right\rfloor + 1 \quad \text{and} \quad \gcd(1 + rk, m) = 1. $ This formula help us to reduce the number of cases to check whether such matrices is MDS. Moreover, we give complete characterization of $g$-circulant MDS matrices of order 3 and 4. Additionally, inspired by skew polynomial rings, we construct a new variant of $g$-circulant matrix. In the last, we provide some examples related to our findings.