Public key exchange protocols based on tropical lower circulant and anti circulant matrices

In recent years, many efficient key exchange protocols have been proposed based on matrices over the tropical semirings. The tropical addition of two elements is the minimum of the elements, while the tropical multiplication is the sum of the two elements. This paper proposes a novel key exchange protocol based on the min-plus semiring ( $\mathbb{Z}\cup \{\mathrm{\infty }\},\oplus ,\otimes$) by introducing anti- $s$- $p$-circulant matrices, which forms a commutative subset of $M_{n\times n}(\mathbb{Z}\cup \{\mathrm{\infty }\})$. We have given further analysis of the protocol in detail using upper or lower- $s$-circulant matrices. Additionally, we prove that the set of all lower- $s$-circulant matrices is a sub-semiring of the tropical semiring $M_{n\times n}(\mathbb{Z}\cup \{\mathrm{\infty }\})$. We discuss the detailed security analysis of the protocol with upper or lower- $s$-circulant matrices and provide cryptographic algorithms for both key exchange protocols with detailed explanations. We compare the protocol based on upper or lower- $s$-circulant matrices and our proposed protocol in terms of time complexity and memory usage. Finally, we analyse the security and show that our protocol is safe against popular attacks of tropical key exchange protocols. The security of these protocols relies on the difficulty of solving tropical non-linear equations.