We propose a general approach to construct cryptographic significant Booleanfunctions of $(r+1)m$ variables based on the additive decomposition$\mathbb{F}_{2^{rm}}\times\mathbb{F}_{2^m}$ of the finite field$\mathbb{F}_{2^{(r+1)m}}$, where $r$ is odd and $m\geq3$. A class of unbalancedfunctions are constructed first via this approach, which coincides with avariant of the unbalanced class of generalized Tu-Deng functions in the case$r=1$. This class of functions have high algebraic degree, but their algebraicimmunity does not exceeds $m$, which is impossible to be optimal when $r>1$. Bymodifying these unbalanced functions, we obtain a class of balanced functionswhich have optimal algebraic degree and high nonlinearity (shown by a lowerbound we prove). These functions have optimal algebraic immunity provided acombinatorial conjecture on binary strings which generalizes the Tu-Dengconjecture is true. Computer investigations show that, at least for smallvalues of number of variables, functions from this class also behave wellagainst fast algebraic attacks.
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机译：我们提出了一种基于加法分解构造$（r + 1）m $变量的密码有效布尔函数的通用方法\\ mathbb {F} _ {2 ^ {rm}} \ times \ mathbb {F} _ {2 ^ m} $的有限字段$ \ mathbb {F} _ {2 ^ {（r + 1）m}} $，其中$ r $为奇数，$ m \ geq3 $。首先通过这种方法构造一类不平衡函数，这与在情况$ r = 1 $中广义Tu-Deng函数的不平衡类的变量一致。此类函数具有较高的代数程度，但其代数免疫度不超过$ m $，这在$ r> 1 $时不可能是最优的。通过修改这些不平衡函数，我们得到了一类具有最佳代数度和高非线性度的平衡函数（以证明的下界表示）。这些函数具有最佳的代数免疫性，前提是对二进制字符串进行组合猜想，从而证明Tu-Deng猜想是正确的。计算机研究表明，至少对于较小数量的变量，此类的函数在快速代数攻击中也表现良好。
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