Decomposition of permutations in a finite field

We describe a method to decompose any power permutation, as a sequence of power permutations of lower algebraic degree. As a result we obtain decompositions of the inversion in GF(2n) for small n from 3 up to 16, as well as for the APN functions, when n = 5. More precisely, we find decompositions into quadratic power permutations for any n not multiple of 4 and decompositions into cubic power permutations for n multiple of 4. Finally, we use the Theorem of Carlitz to prove that for 3 ≤ n ≤ 16 any n-bit permutation can be decomposed in quadratic and cubic permutations.

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