Zero-derivative normal form for quadratic APN permutations

Determine whether every quadratic almost perfect nonlinear permutation over the binary field of odd dimension n≥5 is linearly equivalent to a permutation S whose reduced representative satisfies red(S)'=0.

Background

The paper establishes that every quadratic APN permutation in odd dimension n≥5 is linearly equivalent to an optimal representative whose reduced derivative has no zero and whose nonempty fibres have size two. The authors note that a stronger normal form would be useful: if the derivative of a linearly equivalent permutation S were identically zero, Proposition outNF would produce an optimal representative by a direct input shear, without the global counting argument used for general dimensions.

This stronger property is verified for the Gold classes and for additional families, but the paper does not establish it for every quadratic APN permutation. Thus the unresolved issue is whether the zero-derivative normal form exists throughout the full class covered by the main theorem.

References

Is every quadratic APN permutation in odd dimension $n\ge5$ linearly equivalent to a permutation $S$ with $red(S)'=0$? For such $S$, Proposition~\ref{prop:outNF} gives an optimal map $S\circ A_t$ directly, without the counting argument.

— Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension  (2610.21169 - Fukui, 15 Oct 2026) in Section 7, Concluding remarks, bullet “Zero-derivative normal form”

A description of the inputs at which an optimal output exists is not known.

— Optimal Reduced Representatives of Quadratic APN Permutations in Odd Dimension  (2610.21169 - Fukui, 15 Oct 2026) in Section 7, Concluding remarks, bullet “Same input”