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Cryptanalysis of a theorem: decomposing the only known solution to the big APN problem. (English) Zbl 1391.94789
Robshaw, Matthew (ed.) et al., Advances in cryptology – CRYPTO 2016. 36th annual international cryptology conference, Santa Barbara, CA, USA, August 14–18, 2016. Proceedings. Part II. Berlin: Springer (ISBN 978-3-662-53007-8/pbk; 978-3-662-53008-5/ebook). Lecture Notes in Computer Science 9815, 93-122 (2016).
Summary: 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 2\(n\)-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.
For the entire collection see [Zbl 1344.94002].

MSC:
94A60 Cryptography
Software:
SageMath
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