Abstract
The graphs \(\mathcal{G}_F=\{(x,F(x)); x\in \mathbb {F}_2^n\}\) of those (n, n)-functions \(F:\mathbb {F}_2^n\mapsto \mathbb {F}_2^n\) that are almost perfect nonlinear (in brief, APN; an important notion in symmetric cryptography) are, equivalently to their original definition by K. Nyberg, those Sidon sets (an important notion in combinatorics) S in \(({\mathbb {F}}_2^n\times {\mathbb {F}}_2^n,+)\) such that, for every \(x\in {\mathbb {F}}_2^n\), there exists a unique \(y\in {\mathbb {F}}_2^n\) such that \((x,y)\in S\). Any subset of a Sidon set being a Sidon set, an important question is to determine which Sidon sets are maximal relatively to the order of inclusion. In this paper, we study whether the graphs of APN functions are maximal (that is, optimal) Sidon sets. We show that this question is related to the problem of the existence/non-existence of pairs of APN functions lying at distance 1 from each others, and to the related problem of the existence of APN functions of algebraic degree n. We revisit the conjectures that have been made on these latter problems.
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Notes
- 1.
The relationship between nonlinearity and almost perfect nonlinearity is not clear. The question whether all APN functions have a rather large nonlinearity is open.
- 2.
This is a necessary and sufficient condition for APNness.
- 3.
Note that we could also reduce ourselves to \(a=b=0\) but we could not reduce ourselves to \(a=F(0)=b=0\) without loss of generality. This is why we consider G in the sequel.
- 4.
See more in [9], where is recalled that no non-quadratic crooked function is known.
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Acknowledgement
We thank Lilya Budaghyan and Nian Li for useful information. The research of the author is partly supported by the Trond Mohn Foundation and Norwegian Research Council.
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Carlet, C. (2022). On APN Functions Whose Graphs are Maximal Sidon Sets. In: Castañeda, A., Rodríguez-Henríquez, F. (eds) LATIN 2022: Theoretical Informatics. LATIN 2022. Lecture Notes in Computer Science, vol 13568. Springer, Cham. https://doi.org/10.1007/978-3-031-20624-5_15
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