Depending on the parity of n and the regularity of a bent function f from \({{\mathbb F}_{p}^{n}}\) to \({\mathbb F}_{p}\), f can be affine on a subspace of dimension at most n/2, (n − 1)/2 or n/2 − 1. We point out that many p-ary bent functions take on this bound, and it seems not easy to find examples for which one can show a different behaviour. This resembles the situation for Boolean bent functions of which many are (weakly) n/2-normal, i.e. affine on a n/2-dimensional subspace. However applying an algorithm by Canteaut et.al., some Boolean bent functions were shown to be not n/2-normal. We develop an algorithm for testing normality for functions from \({{\mathbb F}_{p}^{n}}\) to \({\mathbb F}_{p}\). Applying the algorithm, for some bent functions in small dimension we show that they do not take on the bound on normality. Applying direct sum of functions this yields bent functions with this property in infinitely many dimensions.

中文翻译：

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关于p元弯曲函数的正态性。

取决于n的奇偶性和从\（{{\ mathbb F} _ {p} ^ {n}} \）到\（{\ mathbb F} _ {p} \）的折弯函数f的正则性，f最多可以仿射到一个尺寸为n / 2，（n − 1）/ 2或n / 2 − 1的子空间上。我们指出，许多p-弯曲函数都处于该边界上，因此似乎不容易找到可以表现出不同行为的示例。这类似于布尔弯曲函数的情况，该函数的许多（弱）为n / 2-法线，即仿射在n上/二维子空间。但是，使用Canteaut等人的算法，某些布尔弯曲函数显示为非n / 2-normal。我们开发了一种算法来测试从\（{{\ mathbb F} _ {p} ^ {n}} \）到\（{\ mathbb F} _ {p} \）函数的正态性的算法。应用该算法，对于一些小尺寸的折弯函数，我们表明它们不承担正态性的限制。应用函数的直接总和，可以在无限多个维度上产生具有此属性的弯曲函数。