MDS matrices are used in symmetric cryptography to hinder differential and linear cryptanalysis. This article proposes and examines a new deterministic method that accelerates circulant matrix MDS testing and the search for circulant MDS matrices. The method is to ascertain the MDS property via computing the determinants of only those submatrices that lie in a suitable subset of square submatrices constructed in advance. It is shown that for 8x8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 8 \times 8 $$\end{document} circulant matrices, this new method reduces thirteenfold the MDS confirmation time and searches for MDS matrices 8 times faster compared to the general method employing all square submatrices. The article also proves that the constructed set can be arranged in a manner that comprises all the submatrices needed for the Laplace expansion of the determinant of any submatrix within the subset. Experiments show that the Laplace expansion allows a further two to seven times speed-up of the MDS testing. Via proposed techniques, several circulant MDS matrices were found including 8x8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{8} \times \varvec{8}$$\end{document} matrices over GF(28)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textbf {GF}}\varvec{(2<^>8)} $$\end{document} and 16x16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varvec{16} \times \varvec{16} $$\end{document} matrices over GF(222),GF(223)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textbf {GF}}\varvec{(2<^>{22}),} {\textbf {GF}}\varvec{(2<^>{23})} $$\end{document} and GF(224)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textbf {GF}}\varvec{(2<^>{24})} $$\end{document} with many multiplicative identity element entries, a few different elements of the low Hamming weight and efficient inverses. Besides that, empirical probability mass functions were found for the random variables representing the least dimension of singular submatrices of 16x16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varvec{16} \times \varvec{16} $$\end{document} circulant matrices of two chosen forms over GF(2m),m is an element of{8,& ctdot;,24}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\textbf {GF}}\varvec{(2<^>m), m \in \{8, \dots , 24\}} $$\end{document}.
