Generalized differential-linear cryptanalysis of block cipher

Differential-linear cryptanalysis of block ciphers was proposed in 1994. It turns out to be more efficient in comparison with (separately) differential and linear cryptanalytic methods, but its scientific substantiation remains the subject of further research. There are several publications devoted to formalization of differential-linear cryptanalysis and clarification of the conditions under which its complexity can be mathematically accurately assessed. However, the problem of the differential-linear cryptanalytic method substantiation remains completely unresolved. This paper  presents first results obtained by the author in the direction of solving this problem. The class of differential-linear attacks on block ciphers is expanded. Namely, both distinguishing attacks and attacks aimed at recovering one bit of information about a key are considered. In this case, no assumptions are made (as in well-known publications) about the possibility of representing the cipher in the form of some two components. Lower bounds of information complexity of these attacks are obtained. The expressions of these bounds depend on the averaged (by keys) values of the elements’ squares of the generalized autocorrelation table of the encryption transformation. In contrast to the known ones, the obtained bounds are not based on any heuristic assumptions about the investigated block ciphers and are valid for a wider class of attacks as compared to the traditional differential-linear attack. Relations between, respectively, differential, linear and differential-linear properties of bijective Boolean mappings are given. In contrast to the well-known works, the matrix form of the relations is used that makes it possible to clarify better their essence and simplify the proofs. A new relation is derived for the elements of the generalized autocorrelation table of the encryption transformation of the product of two block ciphers, which may be useful in further research.