Construction of correlation immune Boolean functions

It is shown in this paper that every correlation immune Boolean function of n variables can be written as f(x) = g(xG ), where 9 is an algebraic non-degenerate Boolean function of k (k :::; n) variables and G is a generating matrix of an tn, k, d] linear code. In this expression the correlation immunity of f(x) must be at least d 1. In this paper we further prove when the correlation immunity exceeds this lower bound. A method which can theoretically search all possible correlation immune functions exhaustively is proposed. Constructions of higher order correlation immune functions as well as algebraic non-degenerate correlation immune functions are discussed in particular. It is also shown that many cryptographic properties of 9 can be inherited by the correlation immune function f(x) = g(xGT ) which enables us to construct correlation immune functions with other cryptographic properties.