What is the fastest algorithm for the following problem?

Given two strings $ x, y \in \Sigma^*$ as input, decide whether there exists strings $ a, b, c \in \Sigma^*$ , such that $ x=abc$ and $ y=cba$ .

By calculating all the length of the longest common prefix of $ s = x$ y$ and all suffixes of $ s$ , we can compute all candidates for $ a$ in $ O(n)$ time.

For each candidate of $ a$ with length $ k$ , we now just need to check whether $ x_k, x_{k + 1}, \ldots, x_{n}$ is a rotation of $ y_0, y_1, \ldots, y_{n – k}$ . This can of course be done in $ O(n)$ .

However checking the rotations naively results in a worst-case $ O(n^2)$ algorithm.

It seems that using the result from either https://arxiv.org/pdf/1601.08051.pdf or https://arxiv.org/pdf/1311.6235.pdf would make the algorithm run in expected $ O(n)$ time.

Is there a simpler way of speeding up the rotation checking, where it is still faster than $ O(n^2)$ ? Is there a way of making it deterministic, so that it still runs in $ O(n)$ time?