In this paper, we investigate on constructing cryptographically significant Boolean functions with n variables based on decompositions of the multiplicative group of the finite field F of the form F* = U × V, where U and V are cyclic subgroups of F* satisfying (|U|, |V|) = 1. For positive integers s, m and n = 2m, we obtain classes of unbalanced functions with optimal algebraic immunity in the cases |U| = 2 + 1, |V| = (2?1)/(2+1) and |U| = 2?1, |V| = (2?1)/(2?1), respectively, where in the latter case the optimal algebraic immunity is based on correctness of the Tu-Deng conjecture. Functions belonging to both classes can be modified to be balanced ones with (potentially) optimal algebraic immunity and optimal algebraic degree, and computer experiments show that they also have high nonlinearity and good immunity against fast algebraic attacks. As by-products, variants of the Tu-Deng conjecture and combinatorial results on binary strings in analogy to it are also obtained.
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机译:在本文中,我们研究了基于F * = U×V形式F * = U×V的有限磁场F的乘法组的分解构建密码显着的布尔函数,其中U和V是F *令人满意的循环子组(| U |,| v |)= 1.对于正整数S,m和n = 2m,我们在案例中获得了最佳代数免疫的不平衡功能的类| U | = 2 + 1,| v | =(2?1)/(2 + 1)和| U | = 2?1,| v |分别在后一种情况下,分别在后一种情况下,最佳代数免疫基于Tu-Deng猜想的正确性,分别为=(2?1)/(2?1)。可以修改属于两个类的功能,以平衡(可能)最佳代数免疫和最佳代数程度,并且计算机实验表明,它们也具有高的非线性和良好的免疫力免受快速代数攻击。还获得了副产品,还获得了Tu-DENG猜想和组合结果的变体,与其类似的二进制字符串。
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