Some Algebraic Properties of RX-Differential Probabilities of Boolean Mappings

Introduction. ARX- and LRX- cryptosystems are based on extremely simple operations available at the level of computing processor instructions, such as modular addition, bitwise addition, rotations, etc. Due to their simple implementation and extremely high speed, ARX- and LRX- cryptosystems have become an important part of so-called lightweight cryptography – a field dedicated to the development of reliable algorithms for low-resource devices and the Internet of Things. However, their simple structure also makes them vulnerable to attacks, so the creation of such systems requires careful analysis and evaluation of cryptographic security against known attack methods, such as RX-analysis. The purpose of the article is to derive exact analytical expressions for the probabilities of RX-differentials of Boolean mappings with linear shifts. This will enable a more detailed analysis of the cryptographic properties of such mappings. Results. Exact analytical expressions have been obtained for the probabilities of RX-differentials, as well as ordinary differentials and rotation pairs, for binary Boolean mappings with linear shifts. It has been proven that RX-differentials for given mappings and their dual functions have the same probabilities. It is shown that adding constants – a common method of increasing security against rotational cryptanalysis – does not always produce the desired effect. For rotation-invariant mappings with shifts, an unexpected connection between the probabilities of rotation pairs and the probabilities of differentials is demonstrated. Conclusions. The obtained results can be used in analyzing the cryptographic security of ARX-cryptosystems and developing new reliable cryptographic algorithms suitable for low-resource devices. Keywords: symmetric cryptography, ARX-cryptosystems, differential cryptanalysis, rotational cryptanalysis, RX-analysis.