A Study on Tropical Semirings and Their Applications in Cryptography

Abstract

In this thesis, we study the algebraic structure of tropical semirings and their applications newlinein the field of cryptography. In tropical algebra, addition of two elements is the maximum newlineor minimum of the elements, while multiplication is the sum of two elements. This newlinethesis analyse the principal solution of tropical linear systems with some special matrices newlineby normalization method. We suggest the advanced normalization method for solving newlinetropical linear system and analyse the relationship between normalization and advanced newlinenormalization methods. Also we analyse the generalized principal solution and all possible newlinesolutions of tropical linear systems with special matrices by advanced normalization newlinemethod. We propose a method to find the degrees of freedom for any solvable tropical newlinelinear system by the use of advanced normalization method. We also suggest the advanced newlinenormalization method to find the column and row rank of any given tropical matrices. We newlineanalyse the determinant of toeplitz matrices and solutions of the tropical linear system newlinewith toeplitz matrices newline