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ARX-based ciphers, constructed by the modular addition, rotation and XORoperations, have been receiving a lot of attention in the design of lightweightsymmetric ciphers. For their differential cryptanalysis, most automatic searchmethods of differential trails adopt the assumption of independence of moduloadditions. However, this assumption does not necessarily hold when the trailincludes consecutive modular additions (CMAs). It has already been found thatin this case some differential trails searched by automatic methods before areactually impossible, but the study is not in depth yet, for example, few efforthas been paid to exploiting the root causes of non-independence between CMAsand accurate calculation of probabilities of the valid trails. In this paper,we devote to solving these two problems. By examing the differential equationsof single and consecutive modular additions, we find that the influence ofnon-independence can be described by relationships between constraints on theintermediate state of two additions. Specifically, constraints of the firstaddition can make some of its output bits non-uniform, and when they meet theconstraints of the second addition, the differential probability of the wholeCMA may be different from the value calculated under the independenceassumption. As a result, we can build SAT models to verify the validity of agiven differential trail of ARX ciphers and #SAT models to calculate the exactprobabilities of the differential propagation through CMAs in the trail,promising a more accurate evaluation of probability of the trail. Our automicmethods and searching tools are applied to search related-key differentialtrails of SPECK and Chaskey including CMAs in the key schedule and the roundfunction respectively.

Towards non-independence of modular additions in searching differential  trails of ARX ciphers: new automatic methods with application to SPECK and  Chaskey