International Association for Cryptologic Research (IACR)
Abstract
In present paper, we investigate 4 problems.
Firstly,
it is known that, a matrix is MDS if and only if all sub-matrices of this matrix of degree from 1 to n are full rank. In this paper, we propose a theorem that an orthogonal matrix is MDS if and only if all sub-matrices of this orthogonal matrix of degree from 1 to ⌊2n⌋ are full rank. With this theorem, calculation of constructing orthogonal MDS matrices is reduced largely.
Secondly,
Although it has been proven that the 2d×2d circulant orthogonal matrix does not exist over the finite field, we discover that it also does not exist over a bigger set. Thirdly, previous algorithms have to continually change entries of the matrix to construct a lot of candidates. Unfortunately, in these candidates, only very few candidates are orthogonal matrices. With the matrix polynomial residue ring and the minimum polynomials of lightweight element-matrices, we propose an extremely efficient algorithm for constructing 4×4 circulant orthogonal MDS matrices. In this algorithm, every candidate must be an circulant orthogonal matrix.
Finally, we use this algorithm to construct a lot of lightweight results, and some of them are constructed first time