Auto-correlations and new bounds on the nonlinearity of boolean functions

Abstract

It is a well known fact that the nonlinearity of a function f on the n-dimensional vector space V n is bounded from above by 2n−1 − 21/2n−1. In cryptographic practice, nonlinear functions are usually constructively obtained in such a way that they support certain mathematical or cryptographic requirements. Hence an important question is how to calculate the nonlinearity of a function when extra information is available. In this paper we address this question in the context of auto-correlations, and derive four (two upper and two lower) bounds on the nonlinearity of a function (see Table 1). Strengths and weaknesses of each bound are also examined. In addition, a few examples are given to demonstrate the usefulness of the bounds in practical applications. We anticipate that these four bounds will be very useful in calculating the nonlinearity of a cryptographic function when certain extra information on the auto-correlations of the function is available.