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 [math]\displaystyle{ (\mathbb{R} \cup \{\infty\},\oplus,\otimes) }[/math] (also known as the min-plus algebra), or a generalization thereof. The operations are defined as follows for [math]\displaystyle{ x,y \in \mathbb{R} \cup \{\infty\} }[/math]:

[math]\displaystyle{ x \oplus y = \min\{x,y\} }[/math]
[math]\displaystyle{ x \otimes y = x + y }[/math]

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