### Decomposition of Permutations in a Finite Field

Svetla Nikova, 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.

Available format(s)
Category
Secret-key cryptography
Publication info
Preprint. MINOR revision.
Keywords
Carlitz Theoremdecomposition of power functionsthreshold implementationsAPN
Contact author(s)
svetla nikova @ esat kuleuven be
History
2020-11-02: last of 2 revisions
See all versions
Short URL
https://ia.cr/2018/103

CC BY

BibTeX

@misc{cryptoeprint:2018/103,
author = {Svetla Nikova and Ventzislav Nikov and Vincent Rijmen},
title = {Decomposition of Permutations in a Finite Field},
howpublished = {Cryptology ePrint Archive, Paper 2018/103},
year = {2018},
note = {\url{https://eprint.iacr.org/2018/103}},
url = {https://eprint.iacr.org/2018/103}
}
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