Abstract
Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.
Highlights
Accepted: 6 September 2021In some methods, the inversion of an integer matrix is an avoidable operation
Babai’s Round-off Method (RoM) is an approximation method that was proposed by Babai [1]
RoM is used in the decryption algorithm of the lattice-based Goldreich–Goldwasser–Halevi encryption scheme (GGH crypto-system) [3] and its upgraded version [4]
Summary
The inversion of an integer matrix is an avoidable operation. For example, Babai’s Round-off Method (RoM) is an approximation method that was proposed by Babai [1]. Mensions, n, the inversion of the basis B is efficiently performed, and storing all entries, including the non-integer entries in the matrix B−1 , requires a low storage capacity These tasks could become cumbersome once the dimension n becomes larger, such as n ≥ 500. The inverse of the almost-unimodular matrix is proven to consist of only a single non-integer entry, regardless of how large the dimension n is. With this property, the almost-unimodular matrix Gmight be useful in any method or technique where the inversion operation of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1
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