Uniform linearity bound for APN functions

Establish whether there exists an absolute constant C<1 such that every APN function F:𝔽₂ⁿ→𝔽₂ⁿ for n≥4 satisfies linearity 𝓛(F)≤C·2ⁿ, equivalently whether the nonlinearity satisfies NL(F)≥η·2ⁿ for some absolute constant η>0.

Background

The paper proves additive improvements to the known upper bounds on the linearity of APN functions: 𝓛(F)≤2ⁿ−10 in even dimensions and 𝓛(F)≤2ⁿ−6 in odd dimensions for n≥5. However, these bounds do not establish a dimension-independent constant-factor gap below 2ⁿ.

The unresolved issue is whether APN functions can have linearity approaching the trivial upper scale 2ⁿ, or whether all APN functions possess nonlinearity bounded below by a fixed positive proportion of the domain size. The paper notes that a conditional constant-factor bound was previously known under a multiplicity assumption on the maximum Walsh coefficient, while the stated question asks for such a bound without that assumption.

References

The additive improvement in Theorem~\ref{thm:small} leaves open the following question. Does there exist an absolute constant $C<1$ such that $(F)\le CN$ for every $n\ge4$ and every APN function $F\colon_2n\to_2n$?

— Linearity bounds for APN functions  (2609.35689 - Beierle, 28 Sep 2026) in Introduction, immediately following Theorem 1

It is conjectured that $\sum_{x\in_2n}F(x)=0$ for every APN function on $_2n$ with $n\ge3$; see.

— Linearity bounds for APN functions  (2609.35689 - Beierle, 28 Sep 2026) in Section 1, immediately following Corollary 1.4