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首页> 外文会议>Institute of Electrical and Electronics Engineers International Symposium on Information Theory >Constructing Boolean functions with potentially optimal algebraic immunity based on additive decompositions of finite fields (extended abstract)
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Constructing Boolean functions with potentially optimal algebraic immunity based on additive decompositions of finite fields (extended abstract)

机译：基于有限域的加性分解构造具有潜在最佳代数免疫性的布尔函数（扩展摘要）

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摘要

We propose a general approach to construct cryptographic significant Boolean functions of (r + 1)m variables based on the additive decomposition F2rm × F2m of the finite field F2(r+1)m, where r ≥ 1 is odd and m ≥ 3. A class of unbalanced functions is constructed first via this approach, which coincides with a variant of the unbalanced class of generalized Tu-Deng functions in the case r = 1. Functions belonging to this class have high algebraic degree, but their algebraic immunity does not exceed m, which is impossible to be optimal when r > 1. By modifying these unbalanced functions, we obtain a class of balanced functions which have optimal algebraic degree and high nonlinearity (shown by a lower bound we prove). These functions have optimal algebraic immunity provided a combinatorial conjecture on binary strings which generalizes the Tu-Deng conjecture is true. Computer investigations show that, at least for small values of number of variables, functions from this class also behave well against fast algebraic attacks.
机译：我们提出了一种基于有限域F2（r + 1）m的加法分解F2rm×F2m构造（r + 1）m变量的重要密码布尔函数的通用方法，其中r≥1为奇数，m≥3。首先通过这种方法构造一类不平衡函数，这与在r = 1的情况下广义Tu​​-Deng函数的不平衡类的变体重合。属于此类的函数具有高代数度，但它们的代数免疫力不高超过m，当r> 1时不可能是最优的。通过修改这些不平衡函数，我们获得了一类具有最佳代数度和高非线性度的平衡函数（通过证明的下限表示）。这些函数具有最佳的代数免疫性，提供了对二进制字符串的组合猜想，该猜想可以概括出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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