In the present work we address the robustness of the Polynomial Learning With Errors problem extending previous results in Blanco-Chacón et al. (Communications in Mathematics 31(2), 115–135 2023) and in Elias et al. (2015). In particular, we produce two kinds of new distinguishing attacks: a) we generalize Blanco-Chacón et al. (Communications in Mathematics 31(2), 115–135 2023) to the case where the defining polynomial has a root of degree up to 4, and b) we widen and refine the most general attack in Elias et al. (2015) to the non-split case and determine further dangerous instances previously not detected. Finally, we exploit our results in order to show vulnerabilities of some cryptographically relevant polynomials.
