Trace-based cryptanalysis of cyclotomic $R_{q,0}\times R_q$-PLWE for the non-split case

We describe a decisional attack against a version of the PLWE problem in which the samples are taken from a certain proper subring of large dimension of the cyclotomic ring ${\mathbb{F}}_q[x]\mathrm{/}({\mathrm{\Phi }}_{p^k}(x))$ with $k>1$ in the case where $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}p)$ but ${\mathrm{\Phi }}_{p^k}(x)$ is not totally split over ${\mathbb{F}}_q$. Our attack uses the fact that the roots of ${\mathrm{\Phi }}_{p^k}(x)$ over suitable extensions of ${\mathbb{F}}_q$ have zero-trace and has overwhelming success probability as a function of the number of input samples. An implementation in Maple and some examples of our attack are also provided.