Equivalence of Generalised Feistel Networks

Abstract

This paper focuses on equivalences between Generalised Feistel Networks (GFN) of type-II. We introduce a new definition of equivalence which captures the concept that two GFNs are identical up to re-labelling of the inputs/outputs, and give a procedure to test this equivalence relation. Such two GFNs are therefore cryptographically equivalent for several classes of attacks. It induces a reduction o the space of possible GFNs: the set of the (k!)² possible even-odd GFNs with 2k branches can be partitioned into k! different classes.
This result can be very useful when looking for an optimal GFN regarding specific computationally intensive properties, such as the minimal number of active S-boxes in a differential trail. We also show that in several previous papers, many GFN candidates are redundant as they belong to only a few classes. Because of this reduction of candidates, we are also able to suggest better permutations than the one of WARP: they reach 64 active S-boxes in one round less and still have the same diffusion round that WARP. Finally, we also point out a new family of permutations with good diffusion properties.