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

## TL;DR

This paper introduces three novel methods to characterize multi-output correlation-immune Boolean functions, improving computational efficiency and aiding in the design of functions resistant to side-channel attacks.

## Contribution

The paper proposes three new characterization techniques for t-th order correlation-immune multi-output Boolean functions, including a generalized Boolean function approach and Fourier spectral methods, reducing computational complexity.

## Key findings

- Walsh transform characterization reduces complexity from $(2^m-1)\sum_{j=1}^t \binom{n}{j}$ to $m\sum_{j=1}^t \binom{n}{j}$.
- Fourier spectral methods are more efficient for symmetric multi-output CI functions.
- New characterizations support the design of functions for side-channel attack resistance.

## Abstract

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.