On the Double Differential Uniformity of Vectorial Boolean Functions

We introduce the double differential distribution table (DDDT) and the double differential uniformity of a vectorial Boolean function to study the security of an S-box to differential attacks. We study several properties of the DDDT and the double differential uniformity and present their explicit values for three of the most practical vectorial Boolean functions: the inverse function, the Gold function, and the Bracken-Leander function. The double differential uniformity is an extension of the differential uniformity and the Feistel boomerang uniformity. It can be used as a distinguisher, and a new criterion for the security of an S-box derived from a vectorial Boolean function against differential attacks.