Cryptanalysis of a key exchange protocol based on a modified tropical structure

This article analyzes a key exchange protocol based on a modified tropical structure proposed by Ahmed et al. in 2023. It is shown that the modified tropical semiring is isomorphic to the \(2\times 2\) tropical circular matrix semiring. Therefore, matrices in this modified tropical semiring can be represented as tropical matrices, and the key exchange protocol is actually based on the tropical matrix semiring. Tropical irreducible matrices exhibit almost linear periodic property. Efficient algorithms for calculating the linear period and defect of irreducible matrices are designed. Based on the public information of the protocol, the equivalent private key can be computed and then the shared key is easily obtained. The analysis shows that the key exchange protocol based on this modified tropical structure is not secure.