Semigroup action based on skew polynomial evaluation with applications to Cryptography

TL;DR

This paper introduces a novel algebraic action based on skew polynomial evaluation, leveraging non-commutative properties to develop a secure cryptographic key exchange protocol.

Contribution

It presents a new skew polynomial action and constructs a subset to control non-commutativity, enabling a secure cryptographic protocol.

Findings

Constructed a skew polynomial action based on polynomial valuation.

Developed a subset controlling non-commutativity in the skew polynomial ring.

Proposed a public key exchange protocol secure in the Canetti-Krawczyk model.

Abstract

Through this work we introduce an action of the skew polynomial ring $\mathbb{F}_{q}\left[X; \sigma, \delta\right]$ over $\mathbb{F}_{q}$ based on its polynomial valuation and the concept of left skew product of functions. This lead us to explore the construction of a certain subset $\mathcal{T}(X)\subset\mathbb{F}_{q}\left[X; \sigma, \delta\right]$ that allow us to control the non-commutativity of this ring, and exploit this fact in order to build a public key exchange protocol that is secure in Canetti and Krawczyk model.