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Gowers U_3 norm of some classes of bent Boolean functions

机译：一些弯曲布尔函数类的Gowers U_3范数

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摘要

The Gowers norm of a Boolean function is a measure of its resistance to quadratic approximations. It is known that smaller the Gowers norm for a Boolean function larger is its resistance to quadratic approximations. Here, we compute Gowers norms for some classes of Maiorana-McFarland bent functions. In particular, we explicitly determine the value of the Gowers norm of Maiorana-McFarland bent functions obtained by using APN permutations. We prove that this value is always smaller than the Gowers norms of Maiorana-McFarland bent functions obtained by using differentially -uniform permutations, for all . We also compute the Gowers norms for a class of cubic monomial functions, not necessarily bent, and show that for , these norm values are less than that of Maiorana-McFarland bent functions. Further, we computationally show that there exist 6-variable functions in this class which are not bent but achieve the maximum second-order nonlinearity for 6 variables.
机译：布尔函数的高尔斯范数是其对二次逼近的度量。众所周知，布尔函数的Gowers范数越小，其对二次逼近的抵抗力就越大。在这里，我们为某些类别的Maiorana-McFarland弯曲函数计算Gowers范数。特别是，我们明确确定了使用APN置换获得的Maiorana-McFarland弯曲函数的Gowers范数的值。我们证明，对于所有情况，该值始终小于通过使用微分均匀置换获得的Maiorana-McFarland弯曲函数的Gowers规范。我们还为一类立方单项函数（不一定是弯曲的）计算了Gowers范数，并表明对于，这些范数值小于Maiorana-McFarland弯曲函数的范数。此外，我们通过计算表明，此类中存在6个变量函数，这些函数不弯曲但可以实现6个变量的最大二阶非线性。

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