AI 中文总结
本文提出两种统一框架下的显式代数构造，生成具有两两不相交Walsh支撑且无非零线性结构的最大基数plateaued函数族，并实现最优或次优的共同代数次数。
AI 中文摘要
具有两两不相交Walsh支撑的Plateaued布尔函数族在密码布尔函数的次级构造中很有用。特别令人感兴趣的是最大基数族，其成员不允许非零线性结构。据我们所知，先前已知的达到这两个性质的一般构造是谱方法（Hodžić等人，IEEE Trans. Inf. Theory 65(9): 5865--5879, 2019）。在那项工作中，通常不提供显式代数正规型，也没有建立为所有族成员指定共同代数次数的一般方法。在本文中，我们在统一框架内提出两种新的显式代数构造，一种基于线性函数，另一种基于具有bent分量的部分线性函数。设$p\geq 2$和$q\geq 0$满足$q<2^p-p-1$，并设$m=p+q$。两种构造都产生最大基数的$2^{q+1}$个$(q+1)$-plateaued布尔函数族，具有两两不相交的Walsh支撑。没有成员允许非零线性结构，并且每个成员都有显式的广义Maiorana--McFarland表示。第一种构造产生$m+p+1$个变量的函数，并实现任何指定的共同代数次数$3\leq d\leq p+1$，前提是$q<\sum_{i=2}^{d-1}\binom{p}{i}$；其最大可达次数$p+1$是最优的。第二种构造产生$n+p+1$个变量的函数，其中$n>m$且$n-m$为偶数，并实现任何指定的共同代数次数$3\leq d\leq p+(n-m)/2$，前提是$q<\sum_{i=2}^{\min\{d-1,p\}}\binom{p}{i}$；其最大可达次数$p+(n-m)/2$是次优的。
英文摘要
Families of plateaued Boolean functions with pairwise disjoint Walsh supports are useful in secondary constructions of cryptographic Boolean functions. Of particular interest are maximum-cardinality families whose members admit no nonzero linear structures. To the best of our knowledge, the previously known general construction attaining both properties is spectral (Hodžić et al., IEEE Trans. Inf. Theory 65(9): 5865--5879, 2019). In that work, explicit algebraic normal forms are not generally provided, and no general method is established for prescribing a common algebraic degree for all family members. In this paper, we present two new explicit algebraic constructions within a unified framework, one based on linear functions and the other on partially linear functions with bent components. Let $p\geq 2$ and $q\geq 0$ satisfy $q<2^p-p-1$, and set $m=p+q$. Both constructions yield maximum-cardinality families of $2^{q+1}$ $(q+1)$-plateaued Boolean functions with pairwise disjoint Walsh supports. No member admits a nonzero linear structure, and every member has an explicit generalized Maiorana--McFarland representation. The first construction produces functions in $m+p+1$ variables and realizes any prescribed common algebraic degree $3\leq d\leq p+1$, provided that $q<\sum_{i=2}^{d-1}\binom{p}{i}$; its maximum attainable degree $p+1$ is optimal. The second construction produces functions in $n+p+1$ variables, where $n>m$ and $n-m$ is even, and realizes any prescribed common algebraic degree $3\leq d\leq p+(n-m)/2$, provided that $q<\sum_{i=2}^{\min\{d-1,p\}}\binom{p}{i}$; its maximum attainable degree $p+(n-m)/2$ is next-to-optimal.
