Geodesic
On constructing circuits for transforming the polynomial and normal bases of finite fields from one to the other
Diskretnaya Matematika, Tome 19 (2007) no. 3, pp. 89-101

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It is shown that the transformations of normal and polynomial bases of the field $GF(p^n)$ from one to the other can be performed by a circuit over $GF(p)$ with complexity $O(n^{1.806})$ and depth $O(\log n)$. 
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     author = {I. S. Sergeev},
     title = {On constructing circuits for transforming the polynomial and normal bases of finite fields from one to the other},
     journal = {Diskretnaya Matematika},
     pages = {89--101},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2007_19_3_a7/}
}
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I. S. Sergeev. On constructing circuits for transforming the polynomial and normal bases of finite fields from one to the other. Diskretnaya Matematika, Tome 19 (2007) no. 3, pp. 89-101. http://geodesic.mathdoc.fr/item/DM_2007_19_3_a7/