  We extend two of the attacks on the PLWE problem presented in (Y. Elias, K.
E. Lauter, E. Ozman, and K. E. Stange, Ring-LWE Cryptography for the Number
Theorist, in Directions in Number Theory, E. E. Eischen, L. Long, R. Pries, and
K. E. Stange, eds., vol. 3 of Association for Women in Mathematics Series,
Cham, 2016, Springer International Publishing, pp. 271-290) to a ring
$R_q=\mathbb{F}_q[x]/(f(x))$ where the irreducible monic polynomial
$f(x)\in\mathbb{Z}[x]$ has an irreducible quadratic factor over
$\mathbb{F}_q[x]$ of the form $x^2+\rho$ with $\rho$ of suitable multiplicative
order in $\mathbb{F}_q$. Our attack exploits the fact that the trace of the
root is zero and has overwhelming success probability as a function of the
number of samples taken as input. An implementation in Maple and some examples
of our attack are also provided.
