On the Classification of Dillon's APN Hexanomials
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917285122277376 |
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| author | Bartoli, Daniele Grimaldi, Giovanni Giuseppe Stanica, Pantelimon |
| author_facet | Bartoli, Daniele Grimaldi, Giovanni Giuseppe Stanica, Pantelimon |
| contents | We systematically analyze a class of hexanomial functions over finite fields of characteristic $2$ proposed by Dillon (2006) as candidates for almost perfect nonlinear (APN) functions, significantly extending earlier partial-APN results. For functions over $\mathbb{F}_{q^2}$, where $q=2^n$, of the form \[ F(x)=x(Ax^2+Bx^q+Cx^{2q})+x^2(Dx^q+Ex^{2q})+x^{3q}, \] we derive necessary conditions on the coefficients $A,B,C,D,E$ for APNness using algebraic number theory and algebraic-geometry methods over finite fields.
Our main contribution is a comprehensive case-by-case analysis that excludes large classes of Dillon hexanomials via vanishing patterns of key coefficient polynomials. We identify algebraic obstructions -- including absolutely irreducible components of associated varieties and degree incompatibilities in polynomial factorizations -- that prevent these functions from attaining optimal differential uniformity. These results substantially narrow the search space for new APN functions in this family and provide a framework applicable to other APN candidates.
We complement the theory with extensive computations: exhaustive searches over $\mathbb{F}_{2^2}$ and $\mathbb{F}_{2^4}$, and random sampling over $\mathbb{F}_{2^6}$ and $\mathbb{F}_{2^8}$, yielding hundreds of APN hexanomials. Complete CCZ-equivalence testing shows that, although many examples occur, they fall into few distinct classes. For $q\in\{2,4\}$, all examples are CCZ-equivalent to the Budaghyan--Carlet family, while in larger dimensions none appear equivalent to that family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01003 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Classification of Dillon's APN Hexanomials Bartoli, Daniele Grimaldi, Giovanni Giuseppe Stanica, Pantelimon Number Theory Information Theory Algebraic Geometry 11G20, 11T06, 12E20, 14Q10 We systematically analyze a class of hexanomial functions over finite fields of characteristic $2$ proposed by Dillon (2006) as candidates for almost perfect nonlinear (APN) functions, significantly extending earlier partial-APN results. For functions over $\mathbb{F}_{q^2}$, where $q=2^n$, of the form \[ F(x)=x(Ax^2+Bx^q+Cx^{2q})+x^2(Dx^q+Ex^{2q})+x^{3q}, \] we derive necessary conditions on the coefficients $A,B,C,D,E$ for APNness using algebraic number theory and algebraic-geometry methods over finite fields. Our main contribution is a comprehensive case-by-case analysis that excludes large classes of Dillon hexanomials via vanishing patterns of key coefficient polynomials. We identify algebraic obstructions -- including absolutely irreducible components of associated varieties and degree incompatibilities in polynomial factorizations -- that prevent these functions from attaining optimal differential uniformity. These results substantially narrow the search space for new APN functions in this family and provide a framework applicable to other APN candidates. We complement the theory with extensive computations: exhaustive searches over $\mathbb{F}_{2^2}$ and $\mathbb{F}_{2^4}$, and random sampling over $\mathbb{F}_{2^6}$ and $\mathbb{F}_{2^8}$, yielding hundreds of APN hexanomials. Complete CCZ-equivalence testing shows that, although many examples occur, they fall into few distinct classes. For $q\in\{2,4\}$, all examples are CCZ-equivalent to the Budaghyan--Carlet family, while in larger dimensions none appear equivalent to that family. |
| title | On the Classification of Dillon's APN Hexanomials |
| topic | Number Theory Information Theory Algebraic Geometry 11G20, 11T06, 12E20, 14Q10 |
| url | https://arxiv.org/abs/2511.01003 |