**Cryptanalysis of a Theorem: Decomposing the Only Known Solution to the Big APN Problem (Full Version)**

*Léo Perrin, Aleksei Udovenko and Alex Biryukov*

**Abstract: **The existence of Almost Perfect Non-linear (APN) permutations operating on an even number of bits has been a long standing open question until Dillon et al., who work for the NSA, provided an example on 6 bits in 2009.
In this paper, we apply methods intended to reverse-engineer S-Boxes with unknown structure to this permutation and find a simple decomposition relying on the cube function over $GF(2^3)$. More precisely, we show that it is a particular case of a permutation structure we introduce, the butterfly. Such butterflies are $2n$-bit mappings with two CCZ-equivalent representations: one is a quadratic non-bijective function and one is a degree $n+1$ permutation. We show that these structures always have differential uniformity at most 4 when $n$ is odd. A particular case of this structure is actually a 3-round Feistel Network with similar differential and linear properties. These functions also share an excellent non-linearity for $n=3,5,7$.
Furthermore, we deduce a bitsliced implementation and significantly reduce the hardware cost of a 6-bit APN permutation using this decomposition, thus simplifying the use of such a permutation as building block for a cryptographic primitive.

**Category / Keywords: **secret-key cryptography / Boolean functions, APN, Butterfly structure, S-Box decomposition, CCZ-equivalence, Feistel Network, Bitsliced implementation

**Original Publication**** (with major differences): **IACR-CRYPTO-2016

**Date: **received 31 May 2016, last revised 13 Jun 2016

**Contact author: **leo perrin at uni lu

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

**Note: **Added a proof of linear-equivalence of the monomial x^5 and the closed butterfly with e=5. Also added a few notes about the cube and kim functions. Added citation for [22].

**Version: **20160613:132851 (All versions of this report)

**Short URL: **ia.cr/2016/539

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