Structural Weaknesses of Permutations with a Low Differential Uniformity and Generalized Crooked Functions

Any permutation with a low differential uniformity is shown to be such that its inverse has a derivative with a large image set. An attack exploiting this structural property is then presented against a recent hash function proposal, named Maraca, submitted to the SHA-3 competition. Moreover, the attack can be made much more efficient when the image sets of the derivatives of the inverse permutation are affine subspaces. This cryptanalytic approach leads to some generalizations of the notion of crooked functions, and to the study of their properties.