The Gray map converts a symbol in \(\mathbb {Z}_4\) to a pair of binary symbols. Therefore, under the Gray map, a linear function from \(\mathbb {Z}_4^n\) to \(\mathbb {Z}_4\) gives rise to a pair of boolean functions from \(\mathbb {F}_2^{2n}\) to \(\mathbb {F}_2\). This paper studies such boolean functions. We state and prove a condition for the nonlinearity of such functions and derive closed-form expressions for them. Further, results related to the mutual information between random variables that satisfy such expressions have been derived. These results are then used to construct a couple of nonlinear boolean secret sharing schemes. These schemes are then analyzed for their closeness to ‘perfectness’ and their ability to resist ‘Tompa–Woll’-like attacks.

灰度图将\(\mathbb {Z}_4\)中的符号转换为一对二进制符号。因此，在格雷映射下，从\(\mathbb {Z}_4^n\)到\(\mathbb {Z}_4\)的线性函数产生了一对从\(\mathbb {F} _2^{2n}\)到\(\mathbb {F}_2\) 。本文研究了此类布尔函数。我们陈述并证明此类函数的非线性条件，并推导出它们的封闭式表达式。此外，还导出了与满足这样的表达式的随机变量之间的互信息有关的结果。然后使用这些结果构建几个非线性布尔秘密共享方案。然后分析这些方案是否接近“完美”以及抵抗“Tompa-Woll”类攻击的能力。