# More Constructions of Light MDS Transforms Based on Known MDS Circulant Matrices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Notations and Definitions

**Definition**

**1.**

**Lemma**

**1.**

**A**is also expressed as follows:

**Definition**

**2.**

**Definition**

**3.**

## 3. Extended Constructions and Theoretical Aspects of the Known MDS

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Corollary**

**1.**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3.**

**Theorem**

**4.**

**,**and we can obtain the MDS cluster {${A}_{{h}_{i=I,j=J}^{\alpha}}$} of $\mathit{A}$. For every given parameter group $\left(i,\alpha \right):=\left(I,V\right)$. ${A}_{{h}_{i,j}^{\alpha}}$ traverses all $0\le j\le n-1$ values to make the MDS cluster partition of $\mathit{A}$

**,**and we can obtain the MDS cluster {${A}_{{h}_{i=I,j}^{\alpha =V}}$} of $\mathit{A}$. Regarding the qualities, we have Proposition 1.

**Proposition**

**1.**

**Corollary**

**2.**

**Example**

**1.**

**Example**

**2.**

- For the first element ${\mathrm{h}}_{1,0}\left(\mathsf{\alpha}\right)=0\mathrm{x}04$ in Example 1, let its binary bits $\left(00000100\right)$ be the input of map ${\mathrm{P}}_{{\mathrm{i}}_{0}}$; then, the output is $\left(00000600\right)$. For ${\mathscr{H}}_{1}:=\left(36147250\right)$, the same with all non-zero data (only “6”) mapped from ${\mathrm{P}}_{{\mathrm{i}}_{0}}$ to ${\mathbb{F}}_{8}$ are marked as 1, the other positions are marked as 0; then, we can obtain the n-bit position identification value $\left(01000000\right)$ as ${\mathrm{h}}_{3,0}\left(0\mathrm{x}04\right)=\vartheta \left({\mathrm{h}}_{1,0}\left(0\mathrm{x}04\right)\right)=0\mathrm{x}40$.
- For the second element ${h}_{1,0}\left(\alpha \right)=0\mathrm{x}16$ in Example 1, let its binary bits $\left(00010110\right)$ be the input of map ${P}_{{i}_{0}}$; then, the output is $\left(000\mathbf{4}0\mathbf{6}\mathbf{7}0\right)$. For ${\mathscr{H}}_{1}:=\left(3\mathbf{6}1\mathbf{4}\mathbf{7}250\right)$, the same with all non-zero data “4, 6, 7” mapped from ${P}_{{i}_{0}}$ to ${\mathbb{F}}_{8}$ are marked as 1; the other positions are marked as 0. Then, we can obtain the n-bit position identification value $\left(0\mathbf{1}0\mathbf{1}\mathbf{1}000\right)$ as ${h}_{3,0}\left(0\mathrm{x}16\right)=\vartheta \left({h}_{1,0}\left(0\mathrm{x}16\right)\right)=0\times 58$.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, J.-B.; Wu, Y.; Zhou, Y.
More Constructions of Light MDS Transforms Based on Known MDS Circulant Matrices. *Information* **2022**, *13*, 347.
https://doi.org/10.3390/info13070347

**AMA Style**

Wang J-B, Wu Y, Zhou Y.
More Constructions of Light MDS Transforms Based on Known MDS Circulant Matrices. *Information*. 2022; 13(7):347.
https://doi.org/10.3390/info13070347

**Chicago/Turabian Style**

Wang, Jin-Bo, You Wu, and Yu Zhou.
2022. "More Constructions of Light MDS Transforms Based on Known MDS Circulant Matrices" *Information* 13, no. 7: 347.
https://doi.org/10.3390/info13070347