G-Perfect nonlinear functions

Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both Z2-valued and Za2-valued. Our final constructions demonstrate G-perfect nonlinear planar permutations (from Za2 to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.