The Correlation of a Boolean Function with Its Variables

The correlation of a Boolean function with its variables is closely related to the correlation attack on stream cipher. The Walsh transformation is the main tool to study the correlation of Boolean functions. The Walsh transformation of a Boolean function with r variables has 2r coefficients. Let k denote the number of non-zero coefficients of the Walsh Transformations. The paper studies the functions with 1 ≤k ≤ 8. It is proved that the functions with k = 1 are the linear functions only, there are no functions with k = 2, 3, 5, 6, 7, and finally we construct all functions with k = 4 or 8.