On Boolean functions derived from linear maps over $$\mathbb {Z}_4$$ and their application to secret sharing

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.