伽罗瓦环上APN置换的约化多项式提升与有效非APN界
Reduced polynomial lifts of APN permutations over Galois rings and effective non-APN bounds
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- University of Perugia（佩鲁贾大学）
- Naval Postgraduate School（海军研究生院）
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中文总结 AI 辅助
本文阐明了伽罗瓦环上的APN提升猜想，证明了有效非APN界，给出了偶次数推论，并证明三次置换多项式满足修正后的提升猜想。
中文摘要 AI 辅助
我们阐明了Rønjom和Sandrib（CCDS，2026）提出的关于伽罗瓦环上的APN提升猜想。$\boldsymbol{\textbf{F}}_q$上的函数有多个多项式表示，其形式导数可能不同，因此该猜想必须使用次数小于$q$的唯一约化表示；若不进行这种归一化，该猜想不成立。随后，伽罗瓦环上的标准置换多项式判据给出了精确的约化结果：若APN置换的约化表示$f$满足对所有$x \boldsymbol{\textbf{F}}_{2^m}$有$f'(x) \neq 0$，则$f$可提升为$\boldsymbol{\textbf{GR}}(2^k,m)$（$k>1$）的置换，反之亦然。因此，修正后的提升猜想等价于有限域上的临界点猜想。接下来，我们利用Janwa–Wilson–Rodier曲面（该曲面通过对角线排列外的有理点编码APN条件）证明了一个有效的不存在性结果：对于所有不属Gold指数$2^r+1$和Kasami–Welch指数$2^{2r}-2^r+1$的奇数次数$d \boldsymbol{\textbf{5}}$，Hernando–McGuire和Aubry–McGuire–Rodier的结果在无穷远超平面截面中提供了一个绝对不可约因子，这对应于该曲面上一个定义在基域上且不包含在对角线排列中的绝对不可约分量。随后，显式的Cafure–Matera估计给出了一个可计算数$\boldsymbol{\textbf{APNmzero}}(d)$，使得当$m \boldsymbol{\textbf{APNmzero}}(d)$时，$\boldsymbol{\textbf{F}}_{2^m}$上次数为$d$的多项式均不是APN。定性的最终非APN结果由Aubry–McGuire–Rodier给出；我们的贡献在于提供了显式阈值。差分表的一个直接恒等式还给出了偶次数的推论：若$g$具有上述奇数次数，则$ax+g(x^2)+c$（$a \neq 0$）与$g$具有相同的差分均匀性，因此在相同的显式范围内也不是APN。最后，我们直接证明了每个三次置换多项式都有一个有理临界点，从而满足修正后的提升猜想。
英文摘要
We study coefficientwise permutation lifts of reduced polynomial representatives of almost perfect nonlinear (APN) permutations from finite fields to Galois rings. The standard permutation criterion shows that a permutation polynomial over $\mathbb F_{2^m}$ admits such a lift if and only if its formal derivative is nowhere zero on the field. In that case, every coefficientwise lift permutes $\operatorname{GR}(2^k,m)$ for every $k>1$. It is natural to conjecture that the reduced representative of an APN permutation admits no such lift. However, Zhang and Haobo supplied a counterexample to our earlier critical-point conjecture, which also refutes this lifting claim. Their degree-$24$ reduced APN permutation over $\mathbb F_{32}$ has a nowhere-zero formal derivative. We record this example and give an exact linear-image and trace-dual criterion for the absence of rational critical points in quadratic functions. We also obtain explicit non-APN bounds from Janwa-Wilson-Rodier surfaces. For every odd degree $d\ge5$ outside the Gold and Kasami-Welch exponent families, results of Hernando-McGuire and Aubry-McGuire-Rodier yield an absolutely irreducible component defined over the ground field. Applying the Cafure-Matera point estimate gives an explicit threshold above which no polynomial of degree $d$ over $\mathbb F_{2^m}$ is APN. This makes the known qualitative nonexistence result effective. An identity between difference tables extends the bound to polynomials $ax+g(x^2)+c$, where $a\ne0$ and $g$ has an odd degree in the stated range. Finally, every cubic permutation polynomial has a rational critical point and therefore admits no coefficientwise permutation lift.
