Correlation-immune (CI) multi-output Boolean functions have the property of keeping the same output distribution when some input variables are fixed. Recently, a new application of CI functions has appeared in the system of resisting side-channel attacks (SCA). In this paper, three new methods are proposed to characterize the $t$ th-order CI multi-output Boolean functions ($n$-input and $m$-output). The first characterization is to regard the multi-output Boolean functions as the corresponding generalized Boolean functions. It is shown that a generalized Boolean functions $f_g$ is a $t$ th-order CI function if and only if the Walsh transform of $f_g$ defined here vanishes at all points with Hamming weights between $1$ and $t$. Compared to the previous Walsh transforms of component functions, our first method can reduce the computational complexity from $(2^m-1)\sum^t_{j=1}\binom{n}{j}$ to $m\sum^t_{j=1}\binom{n}{j}$. The last two methods are generalized from Fourier spectral characterizations. Especially, Fourier spectral characterizations are more efficient to characterize the symmetric multi-output CI Boolean functions.

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多输出相关免疫布尔函数的新特征

相关免疫 (CI) 多输出布尔函数具有在某些输入变量固定时保持相同输出分布的特性。最近，在抵抗侧信道攻击（SCA）的系统中出现了CI功能的新应用。在本文中，提出了三种新方法来表征$t$ th-order CI 多输出布尔函数（$n$-input 和$m$-output）。第一个特征是将多输出布尔函数视为相应的广义布尔函数。结果表明，广义布尔函数 $f_g$ 是 $t$ 阶 CI 函数，当且仅当此处定义的 $f_g$ 的 Walsh 变换在所有点的汉明权在 $1$ 和 $t$ 之间消失。与之前的分量函数的 Walsh 变换相比，我们的第一种方法可以将计算复杂度从 $(2^m-1)\sum^t_{j=1}\binom{n}{j}$ 降低到 $m\sum^t_{j=1}\binom{ n}{j}$。最后两种方法是从傅立叶光谱表征中推广出来的。特别是，傅立叶谱表征更有效地表征对称多输出 CI 布尔函数。