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.

## 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.

## References

1. ↑ Grigoriev, Dima; Shpilrain, Vladimir (2014). "Tropical Cryptography". Communications in Algebra 42 (6): 2624–2632. doi:10.1080/00927872.2013.766827. ISSN 0092-7872.

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