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Theory of 3-rotation symmetric cubic Boolean functions

机译：三旋转对称三次布尔函数理论

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Rotation symmetric Boolean functions have been extensively studied in the last 15 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about such basic questions as when two such functions are affine equivalent. This question in important in applications, because almost all important properties of Boolean functions (such as Hamming weight, nonlinearity, etc.) are affine invariants, so when searching a set for functions with useful properties, it suffices to consider just one function in each equivalence class. This can greatly reduce computation time. Even for quadratic functions, the analysis of affine equivalence was only completed in 2009. The much more complicated case of cubic functions was completed in the special case of affine equivalence under permutations for monomial rotation symmetric functions in two papers from 2011 and 2014. There has also been recent progress for some special cases for functions of degree > 3. In 2007 it was found that functions satisfying a new notion of k-rotation symmetry for k > 1 (where the case k= 1 is ordinary rotation symmetry) were of substantial in-terest in cryptography and coding theory. Since then several researchers have used these functions for k = 2 and 3 to study such topics as construction of bent functions, nonlinearity and covering radii of various codes. In this paper we develop a detailed theory for the monomial 3-rotation symmetric cubic functions, extending earlier work for the case k = 2 of these functions.
机译：由于旋转对称布尔函数在密码学和编码理论中的重要性，因此在过去的15年中已经对其进行了广泛的研究。直到最近，人们对诸如两个仿射函数何时等效的基本问题还知之甚少。这个问题在应用中很重要，因为布尔函数的几乎所有重要属性（例如汉明权重，非线性等）都是仿射不变量，因此在搜索具有有用属性的函数集时，只需在每个函数中考虑一个函数就足够了等效类。这样可以大大减少计算时间。即使对于二次函数，仿射等价分析也仅在2009年完成。在2011年和2014年的两篇论文中，在单项旋转对称函数置换的仿射等价的特殊情况下完成了三次函数的更为复杂的情况。在度数> 3的某些特殊情况下也取得了最新进展。2007年，发现对于k> 1满足新的k旋转对称性概念的函数（其中k = 1是普通旋转对称性）非常重要。对密码学和编码理论很感兴趣。从那以后，几位研究人员一直使用这些函数（k = 2和3）来研究诸如弯曲函数的构造，非线性和各种代码的覆盖半径之类的主题。在本文中，我们为单项式3旋转对称三次函数开发了详细的理论，并扩展了k = 2时这些函数的早​​期工作。

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