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Theory of 2-rotation symmetric cubic Boolean functions

机译：2旋转对称三次布尔函数理论

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A Boolean function in variables is -rotation symmetric if it is invariant under even powers of the cyclic permutation of the variables, but not under the first power (ordinary rotation symmetry); for brevity, we call such a function a -function. A -function is said to be monomial rotation symmetric (MRS) if it is generated by applying powers of to a single monomial. This paper develops the theory of cubic MRS -functions in variables generated by a monomial with and and not both odd (notation ); it turns out that such functions with and both odd are essentially the same as ordinary cubic MRS functions in only variables. A complete description of affine equivalence for these cubic MRS -functions is given, including a simple necessary and sufficient condition for two such functions to be affine equivalent. It is proved that affine equivalence for these functions is the same as affine equivalence under permutations which preserve -rotation symmetry. This is the first time that all affine equivalence classes have been explicitly determined for any large general family of Boolean functions with degree 2. An exact count of the equivalence classes is given and their number is proved to be very small, in fact for any It is proved that the sequence of Hamming weights satisfies a linear recursion with integer coefficients. A similar result for ordinary cubic MRS functions was proved recently (papers by Bileschi, Cusick and Padgett, and by Brown and Cusick), but this paper uses a new method for the -functions proof. Unlike the ordinary MRS function case, both the orders of the recursions for the -functions and the precise values of the roots of the corresponding recursion polynomials can be given explicitly. Finally, a precise value for the weights of the -functions is proved, using a 2011 formula of Cusick and Padgett. These weights are connected to powers of members of the well known Lucas sequence, and so the weights can be found without computing initial values for the recursions.
机译：如果变量的布尔函数在变量的循环排列的偶次幂下是不变的，则它是-旋转对称的，但在一次幂下不是（通常旋转对称）；为简便起见，我们将此类函数称为-function。如果-函数是通过向单个单项式函数施加幂生成的，则称其为单项式旋转对称（MRS）。本文发展了单项式产生的变量的三次MRS函数的理论，该变量具有奇和非奇（记号）；结果发现，具有和都为奇数的此类函数与仅在变量中的普通三次MRS函数本质上相同。给出了这些三次MRS函数仿射等价的完整描述，包括使两个这样的函数成为仿射等价的简单必要和充分条件。证明了这些函数的仿射等价与在保持旋转对称性的置换下的仿射等价相同。这是首次为具有2度的任何大型布尔函数族明确地确定所有仿射等价类。给出了等价类的精确计数，事实证明它们的数目非常小，实际上对于任何证明了汉明权重的序列满足整数系数的线性递归。最近证明了普通三次MRS函数的相似结果（Bileschi，Cusick和Padgett的论文，以及Brown和Cusick的论文），但是本文使用了一种新的函数证明方法。与普通的MRS函数情况不同，可以显式给出-函数的递归顺序和相应递归多项式的根的精确值。最后，使用Cusick和Padgett的2011年公式证明了-函数权重的精确值。这些权重与众所周知的卢卡斯序列的成员的幂相关，因此可以找到权重而无需计算递归的初始值。

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