Higher-dimensional non-classical Walsh-spectrum APN families

Construct more examples in higher dimensions of APN functions with non-classical Walsh spectrum obtained from APN functions with classical Walsh spectrum by modifying them on hyperplanes with suitable linear functions, and determine whether infinite families of APN functions of the form G(x)=x^3+Tr(x)L(x) over F_{2^n} exist with non-classical Walsh spectrum.

Background

The paper shows that, in dimension 6, modifying the Gold APN function x3 on a hyperplane by a suitable linear term Tr(x)L(x) can produce quadratic APN functions with non-classical Walsh spectrum. The authors ask whether this phenomenon extends beyond the exhibited six-dimensional examples.

The unresolved issue is both constructive and structural: one seeks additional higher-dimensional examples and a determination of whether the form x3+Tr(x)L(x) supports infinite families having non-classical Walsh spectra.

References

Find more such examples in higher dimensions and analyze whether infinite families of APN functions with the non-classical Walsh spectrum of the form F2n 3 x + x3 + Tr(x)L(x) exist.

— Changing almost perfect nonlinear functions on affine subspaces of small codimensions  (2501.03922 - Taniguchi et al., 7 Jan 2025) in Section 5, Conclusion and open problems, item 2

More broadly, one may ask whether analogous switching and perturbation methods can produce APN functions with boomerang uniformity one beyond the ternary DO setting.

— On APN Functions with Boomerang Uniformity One over $\mathbb F_{3^n}$: Differential and Boomerang Spectra and CCZ-Inequivalence  (2609.08968 - Koo et al., 8 Sep 2026) in Section 9, Conclusion and further directions