Exploring rank‎, ‎determinant of tropical Toeplitz matrices and solvability of tropical Toeplitz linear system‎

Toeplitz matrices exhibit a fascinating characteristic in that their elements are constant along each diagonal. This inherent structure not only gives rise to several intriguing theoretical properties but also holds significant importance due to its strong impact on practical applications. In tropical algebra, the tropical algebraic eigenvalues, ranks, and determinants of Toeplitz matrices can be obtained without computing the coefficients of their characteristic max-polynomials, which is a notable advantage of tropical Toeplitz matrices. In recent years, cryptographic algorithms based on tropical linear and nonlinear Toeplitz systems have been shown to be vulnerable to attacks that rely on solving tropical Toeplitz linear and nonlinear systems. Our research is motivated by the close relationship between Toeplitz and Hankel matrices, as well as by the important role their determinants play in various branches of mathematics. Additionally, our research aims to explore attack techniques for tropical Toeplitz-based cryptographic protocols, which involve solving tropical linear and nonlinear systems. In this paper, we analyze the determinants and various notions of rank for different classes of tropical Toeplitz matrices. We establish conditions under which the determinant is singular or nonsingular. Furthermore, we investigate the conditions under which tropical Toeplitz matrices are balanced or unbalanced. Finally, we identify conditions for the existence and uniqueness of solutions to tropical Toeplitz linear systems.

کلیدواژه ها:

Tropical linear system ، Tropical determinant ، Tropical rank ، Tropical toeplitz matrix