Geodesic
On differential equivalence of quadratic APN functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 21-24.

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A vectorial Boolean function $F\colon\mathbb F_2^n\to\mathbb F_2^n$ is called almost perfect nonlinear (APN) if the equation $F(x)+F(x+a)=b$ has at most $2$ solutions for all vectors $a,b\in\mathbb F_2^n$, where $a$ is nonzero. For a given $F$, an associated Boolean function $\gamma_F(a,b)$ in $2n$ variables is defined so that it takes value $1$ iff $a$ is nonzero and the equation $F(x)+F(x+a)=b$ has solutions. We introduce the notion of differentially equivalent functions as vectorial functions that have equal associated Boolean functions. The problem to describe the differential equivalence class of a given APN function is very interesting since the answer can potentially lead to some new constructions of APN functions. We start analyzing this problem with the consideration of affine functions $A$ such that a quadratic APN function $F$ and $F+A$ are differentially equivalent functions. We completely describe these affine functions $A$ for an arbitrary APN Gold function $F$. Computational results for known quadratic APN functions in small number of variables $(2,\dots,8)$ are presented.
Keywords: vectorial Boolean functions, almost perfect nonlinear functions, differential equivalence. 
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     author = {A. A. Gorodilova},
     title = {On differential equivalence of quadratic {APN} functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {21--24},
     publisher = {mathdoc},
     number = {9},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2016_9_a7/}
}
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A. A. Gorodilova. On differential equivalence of quadratic APN functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 21-24. http://geodesic.mathdoc.fr/item/PDMA_2016_9_a7/

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