Resolving a conjecture on quadratic APN functions and a new quadratic (n,n)-function associated to crooked…

Abstract

We say an (n,n)-function F : 𝔽₂ⁿ → 𝔽₂ⁿ is a crooked function if for any nonzero a ∈ 𝔽₂ⁿ, the image of D_aF(x)=F(x)+F(x+a) is an affine hyperplane. The only known examples of crooked functions are all quadratic almost perfect nonlinear (APN), or equivalently, for every known crooked function, D_aF is affine for all a ∈ 𝔽₂ⁿ. The ortho-derivative π_F :𝔽₂ⁿ → 𝔽₂ⁿ of a crooked function F is the function such that π_F(0)=0, and for any nonzero a, the set {0,π_F(a)}^⊥ is the underlying vector space of Im(D_aF). We prove that for n ≥ 4 and a crooked function F, if k is a non-negative integer such that F has 2^k−1 quadratic component functions, π_F has at least 2ⁿ−2^n−k component functions of algebraic degree n−2. In particular, we resolve Gorodilova's conjecture that every component function of π_F has algebraic degree n−2 when F is quadratic APN. As a second main result, for n ≥ 4, we associate to a crooked function F a quadratic function ε_F : 𝔽₂ⁿ → 𝔽₂ⁿ that satisfies a strong geometric-combinatorial condition regarding the sums of F over $2$-dimensional linear subspaces. As a corollary to both of our main results, we prove that for any even n ≥ 4, any quadratic APN (n,n)-function has at least n semi-bent components. Furthermore, we obtain a congruence result on a problem on m-sequences introduced by Johansen, Helleseth, and Kholosha, and we determine the exact algebraic degrees of some Boolean functions associated to the bent and near-bent components of particular classes of plateaued vectorial functions.

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BibTeX RIS

BibTeX

@misc{carlet2026resolving,
  title = {{Resolving a conjecture on quadratic APN functions and a new quadratic $(n,n)$-function associated to crooked functions}},
  author = {Claude Carlet and Darrion Thornburgh},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.23888},
  eprint = {2608.23888},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23888},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - Resolving a conjecture on quadratic APN functions and a new quadratic (n,n)-function associated to crooked functions
AU  - Claude Carlet
AU  - Darrion Thornburgh
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2608.23888
UR  - https://arxiv.org/abs/2608.23888
AB  - We say an (n,n)-function F : 𝔽_2^n → 𝔽_2^n is a crooked function if for any nonzero a ∈ 𝔽_2^n, the image of D_aF(x)=F(x)+F(x+a) is an affine hyperplane. The only known examples of crooked functions are all quadratic almost perfect nonlinear (APN), or equivalently, for every known crooked function, D_aF is affine for all a ∈ 𝔽_2^n. The ortho-derivative π_F :𝔽_2^n → 𝔽_2^n of a crooked function F is the function such that π_F(0)=0, and for any nonzero a, the set {0,π_F(a)}^⊥ is the underlying vector space of Im(D_aF). We prove that for n ≥ 4 and a crooked function F, if k is a non-negative integer such that F has 2^k−1 quadratic component functions, π_F has at least 2^n−2^n−k component functions of algebraic degree n−2. In particular, we resolve Gorodilova's conjecture that every component function of π_F has algebraic degree n−2 when F is quadratic APN. As a second main result, for n ≥ 4, we associate to a crooked function F a quadratic function ε_F : 𝔽_2^n → 𝔽_2^n that satisfies a strong geometric-combinatorial condition regarding the sums of F over $2$-dimensional linear subspaces. As a corollary to both of our main results, we prove that for any even n ≥ 4, any quadratic APN (n,n)-function has at least n semi-bent components. Furthermore, we obtain a congruence result on a problem on m-sequences introduced by Johansen, Helleseth, and Kholosha, and we determine the exact algebraic degrees of some Boolean functions associated to the bent and near-bent components of particular classes of plateaued vectorial functions.
ER  -

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