Differential-Rotational Probabilities of Modular Addition and Its Approximations

In this paper, we consider differential-rotational cryptanalysis, or RX-analysis, and its application to certain classes of ARX-cryptosystems. We provide exact analytical expressions for the RX-differential probabilities with arbitrary rotation values for modular addition. These expressions are described in terms of differential probabilities, which allows comparison of ordinary and RX-differential behaviour. Furthermore, we consider two operations that approximate modular addition, one of which comes from the NORX cipher. For these operations, we also provide exact analytical expressions for the RX-differential probabilities.