Abstract
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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Acknowledgements
This work was supported in part by the Research Council KU Leuven: C16/15/058 and OT/13/071, and by the NIST Research Grant 60NANB15D346.
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Nikova, S., Nikov, V. & Rijmen, V. Decomposition of permutations in a finite field. Cryptogr. Commun. 11, 379–384 (2019). https://doi.org/10.1007/s12095-018-0317-2
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DOI: https://doi.org/10.1007/s12095-018-0317-2