Investigations on [formula omitted]-boomerang uniformity and perfect nonlinearity

Abstract

We defined in Ellingsen et al. (2020) a new multiplicative c-differential, and the corresponding c-differential uniformity and we characterized the known perfect nonlinear functions with respect to this new concept, as well as the inverse function in any characteristic. Here, we extend, via this differential, the boomerang uniformity concept, as introduced at Eurocrypt ’18 by Cid et al. (2018), as a differential distinguisher of S-boxes of block ciphers. and investigate it in the context of perfect nonlinearity and related functions. We first characterize the new concept via Walsh transforms. We further investigate it in the context of perfect nonlinearity and related functions and describe this uniformity for the inverse function in even, respectively, odd characteristic. The methods are combinatorial and number theoretical in nature.