Non-Boolean Almost Perfect nonlinear Functions on Non-Abelian Groups

The study of nonlinear properties of Boolean functions is one of the major tasks in secret-key cryptography. But the adjective "nonlinear" has several meanings: it can be related to the resistance against the famous differential attack [2] and, in this interpretation, actually refers to (almost) perfect nonlinear functions. Moreover nonlinearity is also related to the maximum magnitude of the Fourier spectrum of Boolean functions under the names "bent", "almost bent" or "maximal nonlinear" functions which is itself linked to the resistance against the linear attack [17]. These two ways to define nonlinearity are not independent and even, in many situations, are exactly the same. Most of the studies and results on nonlinearity concerns Boolean functions, or in other words, functions from K to N where K and N are both elementary Abelian 2-groups. Even if this kind of groups seems to be very natural for cryptographic purpose, there is no rule that prevent us to use more complex groups and even nonAbelian ones. In this paper, we will discuss the standard notion of nonlinearity in