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 $(\mathbb {R} \cup \{\infty \},\oplus ,\otimes )$ (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.

References

Grigoriev, Dima; Shpilrain, Vladimir (2014). "Tropical Cryptography". Communications in Algebra. 42 (6): 2624–2632. arXiv:1301.1195. doi:10.1080/00927872.2013.766827. ISSN 0092-7872. S2CID 6744219.

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[GrigorievShpilrain2014-1] Grigoriev, Dima; Shpilrain, Vladimir (2014). "Tropical Cryptography". Communications in Algebra. 42 (6): 2624–2632. arXiv:1301.1195. doi:10.1080/00927872.2013.766827. ISSN 0092-7872. S2CID 6744219.