A large variety of side channel analyses performed on embedded devices involve the linear correlation coefficient as wrong-key distinguisher. This coefficient is actually a sound statistical tool to quantify linear dependencies between univariate variables. However, when those dependencies are non-linear, the correlation coefficient stops being pertinent so that another statistical tool must be investigated. Recent works showed that the
measure is a promising candidate, since it detects any kind of statistical dependency. Substituting it for the correlation coefficient may therefore be considered as a natural extension of the existing attacks. Nevertheless, the first applications published at CHES 2008 have revealed several limitations of the approach and have raised several questions. In this paper, an in-depth analysis of side channel attacks involving the mutual information is conducted. We expose their theoretical foundations and we assess their limitations and assets. Also, we generalize them to higher orders where they seem to be an efficient alternative to the existing attacks. Eventually, we provide simulations and practical experiments that validate our theoretical analyses.