Efficient Computation of Roots in Finite Fields

We present an algorithm to compute rth roots in $$\mathbb{F}_{q^m}$$ with complexity Õ[(log m + r log q) m log q] if (m,q) = 1 and either (q(q−1),r) = 1 or r|(q−1) and ((q−1)/r,r) = 1. This compares well to previously known algorithms, which need O(rm3 log3q) steps.

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