**Decomposition of Permutations in a Finite Field**

*Svetla Nikova and Ventzislav Nikov and Vincent Rijmen *

**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 $\mathrm{GF}(2^n)$ 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 $3 \leq n \leq 16$ any $n$-bit permutation can be decomposed in quadratic and cubic permutations.

**Category / Keywords: **secret-key cryptography / Carlitz Theorem, decomposition of power functions, threshold implementations, APN

**Date: **received 25 Jan 2018, last revised 29 Jan 2018

**Contact author: **svetla nikova at esat kuleuven be

**Available format(s): **PDF | BibTeX Citation

**Version: **20180129:150157 (All versions of this report)

**Short URL: **ia.cr/2018/103

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