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近年来,旋转对称布尔函数引起了密码学家的广泛关注.这类布尔函数可以极大地提高密码算法的运算效率,节省资源开销,因此在密码学与编码理论中有着广泛的应用.关于旋转对称函数密码学性质的研究成为该领域的热点问题.Bent函数是一类Walsh谱均匀的偶变元布尔函数,这类函数不仅具有最高的非线性度,而且具有最优的扩散性.因此Bent函数可以很好地抵抗线性攻击和差分攻击.这些性质使得Bent函数在分组密码S盒的构造、Bent序列的构造、编码理论Kerdock码的构造、组合设计中差集的构造等领域中都有重要的应用.许多密码算法的非线性部件都是通过修改Bent函数得到.然而公开领域中构造旋转对称Bent函数的方法还不多.本文研究了旋转对称Bent函数的构造,给出了一类三次旋转对称布尔函数为Bent函数的充要条件.利用该条件可以非常方便地判断一类给定的旋转对称函数是否为Bent函数.而且本文构造的旋转对称Bent函数的代数表达式非常简单,因此这类函数在密码算法的设计中具有较强的优势.
In recent years, the rotational symmetric Boolean function has aroused widespread concern of cryptographers. Such Boolean functions can greatly improve the computational efficiency of the cryptographic algorithm and save resource overhead, so it has a wide range of applications in cryptography and coding theory. The study on the nature of the function cryptography has become a hot topic in this field.The Bent function is a kind of even-even Boolean function of Walsh spectrum, which not only has the highest nonlinearity but also has the best diffusivity, so the Bent function Which can well resist linear attack and differential attack. These properties make Bent function important in the fields of packet S-box construction, Bent sequence construction, encoding theory Kerdock code construction, and the construction of diversity in combined design However, there are not many methods to construct rotational symmetric Bent function in the public domain.In this paper, we study the construction of rotational symmetric Bent function and give a class of cubic rotational symmetry Boolean functions are necessary and sufficient conditions for the Bent function. Using this condition, it is easy to determine a given rotation pair The function is a function of Bent. Bent functions and algebraic expressions we construct a very simple rotation symmetry, therefore these functions has a strong advantage in the design of cryptographic algorithms.