# Square Integer Matrix with a Single Non-Integer Entry in Its Inverse

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Foundation

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

## 3. Construction of the Almost-Unimodular Matrix

**Theorem**

**1.**

**Proof.**

**Definition**

**4.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Example**

**1.**

**Definition**

**5.**

**Lemma**

**2.**

**Proof.**

**Example**

**2.**

**Definition**

**6.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Inversion of the Almost-Unimodular Matrix

**Theorem**

**3.**

**Proof.**

**Example**

**3.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Babai, L. On Lovász’ Lattice Reduction and the Nearest Lattice Point Problem. Combinatorica
**1986**, 6, 1–13. [Google Scholar] [CrossRef] - Hoffstein, J.; Pipher, J.; Silverman, J.H. Lattices and Cryptography. In An Introduction to Mathematical Cryptography; Springer: New York, NY, USA, 2008; pp. 349–422. [Google Scholar]
- Goldreich, O.; Goldwasser, S.; Halevi, S. Public-key Cryptosystems from Lattice Reduction Problems. In Advances in Cryptology–CRYPTO ’97; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 1997; pp. 112–131. [Google Scholar]
- Mandangan, A.; Kamarulhaili, H.; Asbullah, M.A. A Security Upgrade on the GGH Lattice-Based Cryptosystem. Sains Malays.
**2020**, 49, 1471–1478. [Google Scholar] [CrossRef] - Petzoldt, A. Efficient Key Generation for Rainbow BT. In PQCrypto 2020: Post-Quantum Cryptography. Lecture Notes in Computer Science; Springer: Cham, Switzerland, 2020; pp. 92–107. [Google Scholar]
- Galbraith, S.D. Lattices. In Mathematics of Public Key Cryptography; Cambridge University Press: Cambridge, UK, 2012; pp. 337–346. [Google Scholar]
- Hanson, R. Integer Matrices Whose Inverses Contain Only Integers. Two-Year Coll. Math. J.
**1982**, 13, 18–21. [Google Scholar] [CrossRef] - Hanson, R. Self-inverse Integer Matrices. Coll. Math. J.
**1985**, 16, 190–195. [Google Scholar] [CrossRef] - Arora, A.; Ram, S.; Venkateswarlu, A. Unimodular Polynomial Matrices over Finite Fields. J. Algebr. Comb.
**2021**, 53, 1299–1312. [Google Scholar] [CrossRef] - Abramov, S.A.; Khmelnov, D.E. On Unimodular Matrices of Difference Operators. In Computer Algebra in Scientific Computing; CASC 2018. Lecture Notes in Computer Science; Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E., Eds.; Springer: Cham, Switzerland, 2018; Volume 11077, pp. 18–31. [Google Scholar]
- Vafiadis, D.; Karcanias, N. Unimodular Equivalence and Similarity for Linear Systems. Int. J. Control
**2019**, 92, 2091–2098. [Google Scholar] [CrossRef] [Green Version] - Micheli, G.; Weger, V. On Rectangular Unimodular Matrices over the Algebraic Integers. SIAM J. Discret. Math.
**2019**, 33, 425–437. [Google Scholar] [CrossRef] [Green Version] - Goodaire, E. Matrices and Linear Equations. In Linear Algebra: Pure & Applied; World Scientific Publishing Co.: Singapore, 2014; pp. 93–228. [Google Scholar]
- Uhlmann, J.; Wang, J. On Radically Expanding the Landscape of Potential Applications for Automated-Proof Methods. SN Comput. Sci.
**2021**, 2, 1–9. [Google Scholar] [CrossRef] - Mandangan, A.; Kamarulhaili, H.; Asbullah, M.A. On the Smallest-Basis Problem Underlying the GGH Lattice-based Cryptosystem. Malays. J. Math. Sci.
**2019**, 13, 1–11. [Google Scholar] - Mandangan, A.; Kamarulhaili, H.; Asbullah, M.A. Security Threats on the GGH Lattice-Based Cryptosystem. In Embracing Mathematical Diversity; Chapter 13; UPM Press: Seri Kembangan, Malaysia, 2019; pp. 133–158. [Google Scholar]

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**MDPI and ACS Style**

Mandangan, A.; Kamarulhaili, H.; Asbullah, M.A.
Square Integer Matrix with a Single Non-Integer Entry in Its Inverse. *Mathematics* **2021**, *9*, 2226.
https://doi.org/10.3390/math9182226

**AMA Style**

Mandangan A, Kamarulhaili H, Asbullah MA.
Square Integer Matrix with a Single Non-Integer Entry in Its Inverse. *Mathematics*. 2021; 9(18):2226.
https://doi.org/10.3390/math9182226

**Chicago/Turabian Style**

Mandangan, Arif, Hailiza Kamarulhaili, and Muhammad Asyraf Asbullah.
2021. "Square Integer Matrix with a Single Non-Integer Entry in Its Inverse" *Mathematics* 9, no. 18: 2226.
https://doi.org/10.3390/math9182226