We study the applicability of differential cryptanalysis to the operation of addition modulo $2^n$ used in different cryptosystems. We obtain an analytical formula for expected value of entropy $H_n$ of rows of the difference distribution table of the corresponding mapping. Moreover, the moments of $2^{H_n}$ are studied. In particular, asymptotic inequalities describing the behavior of values $\mathbb{E}2^{qH_n}$ (for $q \in \mathbb{N}$) and $\mathbb{D}2^{H_n}$ as $n \to \infty$ are obtained. We also find a simple analytical formula for the number of table rows with the same distribution. It permits to compute efficiently the statistical characteristics of the entropy.