Geodesic
Changing almost perfect nonlinear functions on affine subspaces of small codimensions
Hiroaki Taniguchi1 ; Alexandr Polujan2 ; Alexander Pott2 ; Razi Arshad3
1Yamato University, Department of Education
2Otto-von-Guericke-Universität Magdeburg
3Kingston University, School of Computer Science and Mathematics
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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A function $F\colon \mathbb{F}_2^n \to \mathbb{F}_2^m$ with $m\ge n$ is called almost perfect nonlinear (APN) if, for every nonzero $a \in \mathbb{F}_2^n$ and every $b \in \mathbb{F}_2^m$, the equation $F(x + a) + F(x) = b$ has at most two solutions $x\in\mathbb{F}_2^n$. One of the central problems in the research on APN functions lies in discovering new constructions of these mappings. In this paper, we introduce secondary construction methods for APN functions by modifying given ones on affine subspaces of small codimensions. We provide explicit criteria for determining when such modifications preserve the APN property and show that that some of the newly constructed functions are inequivalent to the original ones.
DOI : 10.37236/13868
Classification : 06E30, 11T06, 94A60

Hiroaki Taniguchi  1   ; Alexandr Polujan  2   ; Alexander Pott  2   ; Razi Arshad  3

1 Yamato University, Department of Education
2 Otto-von-Guericke-Universität Magdeburg
3 Kingston University, School of Computer Science and Mathematics 
@article{10_37236_13868,
     author = {Hiroaki Taniguchi and Alexandr Polujan and Alexander Pott and Razi Arshad},
     title = {Changing almost perfect nonlinear functions on affine subspaces of small codimensions},
     journal = {The electronic journal of combinatorics},
     year = {2025},
     volume = {32},
     number = {4},
     doi = {10.37236/13868},
     zbl = {arXiv:2501.03922},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/13868/}
}
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Hiroaki Taniguchi; Alexandr Polujan; Alexander Pott; Razi Arshad. Changing almost perfect nonlinear functions on affine subspaces of small codimensions. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13868

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