发表机构
University of Rouen Normandy; Western Norway University of Applied Sciences; Naval Postgraduate School（鲁昂诺曼底大学; 挪威西部应用科学大学; 海军研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出任意素数特征有限域上函数的p元差分-线性轮廓，建立其与DDT行及二阶矩恒等式的关系，并刻画等价性下的行为。
AI 中文摘要
（二元）差分-线性连接表（DLCT）衡量输入差分与应用于相应输出差分的线性掩码之间的依赖关系。对于向量布尔函数，每个DLCT条目是加性自相关值的一半。我们将此关系扩展到任意素数特征有限域上的函数，通过引入一个级别分辨的p元差分-线性轮廓。其条目是导数分量具有每个规定迹值在$\mathbb{F}_p$中的输入的中心化数量。该轮廓的离散傅里叶变换是通过将输出掩码乘以$\mathbb{F}_p$的非零元素而获得的加性自相关族；当p=2时，恢复通常的二元恒等式。对于固定的输入差分，我们证明所有非零输出掩码上的轮廓恰好确定相应的DDT行，并给出显式反演公式。我们建立了一个二阶矩恒等式：一个导数方向上的总轮廓能量是该DDT行与平衡行之间的平方欧氏距离的常数倍。因此，该能量由整行差分谱决定，而不仅仅由差分均匀性决定。由此得出，一个方向上的所有轮廓恰好当导数是平衡时消失；对于奇特征中的平方映射，这给出了平面性的刻画。作为具体的奇特征例子，我们确定了单项式$x^{p^k+1}$的完整轮廓，并推导了逆单项式的精确Kloosterman和公式。最后，我们确定了轮廓在等价下的行为。EA-等价重新索引输入和输出掩码并平移迹级别，而一般的CCZ等价可能混合多个导数方向。然而，对于平方映射，全局非平凡轮廓能量是CCZ不变的。
英文摘要
The (binary) differential-linear connectivity table (DLCT) measures the dependence between an input difference and a linear mask applied to the corresponding output difference. For vectorial Boolean functions, each DLCT entry is one half of an additive autocorrelation value. We extend this relation to functions over finite fields of arbitrary prime characteristic by introducing a level-resolved p-ary differential-linear profile. Its entries are the centered numbers of inputs for which a derivative component has each prescribed trace value in $\mathbb{F}_p$. The discrete Fourier transform of this profile is the family of additive autocorrelations obtained by multiplying the output mask by the nonzero elements of $\mathbb{F}_p$; when p=2, the usual binary identity is recovered. For a fixed input difference, we show that the profiles over all nonzero output masks determine the corresponding DDT row exactly, and we give an explicit inversion formula. We establish a second-moment identity: the total profile energy in one derivative direction is a constant multiple of the squared Euclidean distance between that DDT row and the balanced row. Thus this energy is determined by the full row differential spectrum, not by differential uniformity alone. It follows that all profiles in a direction vanish exactly when the derivative is balanced; for square maps in odd characteristic, this gives a characterization of planarity. As concrete odd-characteristic examples, we determine the complete profile of the monomial $x^{p^k+1}$ and derive an exact Kloosterman-sum formula for the inverse monomial. Finally, we determine the behavior of the profiles under equivalence. EA-equivalence reindexes the input and output masks and translates the trace level, whereas a general CCZ equivalence may mix several derivative directions. Nevertheless, for square maps the global nontrivial profile energy is CCZ-invariant.
