New Characterizations for the Multi-output Correlation-Immune Boolean Functions

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.