Tropical cryptography

In tropical analysis, tropical cryptography refers to the study of a class of cryptographic protocols built upon tropical algebras.¹ 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.

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

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

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