Tropical cryptography

In tropical analysis, tropical cryptography refers to the study of a class of cryptographic protocols built upon tropical algebras. In many cases, tropical cryptographic schemes have arisen from adapting classical (non-tropical) schemes to instead rely on tropical algebras. The case for the use of tropical algebras in cryptography rests on at least two key features of tropical mathematics: in the tropical world, there is no classical multiplication (a computationally expensive operation), and the problem of solving systems of tropical polynomial equations has been shown to be NP-hard.

Basic Definitions
The key mathematical object at the heart of tropical cryptography is the tropical semiring $$(\mathbb{R} \cup \{\infty\},\oplus,\otimes)$$ (also known as the min-plus algebra), or a generalization thereof. The operations are defined as follows for $$x,y \in \mathbb{R} \cup \{\infty\}$$:

$$x \oplus y = \min\{x,y\}$$ $$x \otimes y = x + y$$

It is easily verified that with $$\infty$$ as the additive identity, these binary operations on $$\mathbb{R} \cup \{\infty\}$$ form a semiring.