Cryptology ePrint Archive: Report 2018/103

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 2 Nov 2020

Contact author: svetla nikova at esat kuleuven be

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Version: 20201102:091520 (All versions of this report)

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