Semigroup action based on skew polynomial evaluation with applications to Cryptography
Daniel Camazón-Portela, Juan Antonio López-Ramos
TL;DR
This work defines a semigroup action derived from evaluating skew polynomials over a finite field, leveraging the noncommutative structure of $\mathbb{F}_{q}\left[X; \sigma, \delta\right]$ and a left skew product to obtain a tractable yet non-invertible action. By constructing the subset $\mathcal{T}(X)$ and exploiting evaluation identities, the authors design an extended Diffie–Hellman-like public-key protocol whose security rests on novel hardness assumptions: SAP, CGSAP, and DGSA. They formalize these assumptions through attack games and provide a security analysis in the authenticated-links model, with a reduction to decisional generalized semigroup-action problems. The results contribute to post-quantum cryptography by offering a noncommutative, algebraically structured foundation for key exchange and potential encryption schemes.
Abstract
Through this work we introduce an action of the skew polynomial ring $\mathbb{F}_{q}\left[X; σ, δ\right]$ 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 $\mathcal{T}(X)\subset\mathbb{F}_{q}\left[X; σ, δ\right]$ 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.