Generalized Cipolla-Lehmer root computation in finite fields

We consider the computation of r-th roots in finite field-s. For the computation of square roots, there are two typical probabilistic methods: the Tonelli-Shanks method and the Cipolla-Lehmer method. The former method can be extended to the case of r-th roots, which is called the Adleman-Manders-Miller(AMM) method. The latter method had been generalized to the case of r-th roots with r prime. In this paper, we extend the Cipolla-Lehmer to the case of r-th root with r prime power and give the expected running time of our algorithm.