A simplified and generalized treatment of DES related ciphers

TL;DR

This paper studies generalized DES-like ciphers replacing XOR with arbitrary group operations, demonstrating their structural properties and limitations through theoretical proofs.

Contribution

It introduces a simplified two-round DES variant with all components and proves its encryption permutations do not form a group, extending to n-round Feistel permutations over groups.

Findings

The two-round simplified DES is not a group under composition.

The set of n-round Feistel permutations over groups does not form a group for n ≤ 6.

The set of permutations does not generate the alternating group.

Abstract

This work is a study of DES-like ciphers where the bitwise exclusive-or (XOR) operation in the underlying Feistel network is replaced by an arbitrary group operation. We construct a two round simplified version of DES that contains all the DES components and show that its set of encryption permutations is not a group under functional composition, it is not a pure cipher and its set of encryption permutations does not generate the alternating group. We present a non-computational proof that for n\leq6 the set of n-round Feistel permutations over an arbitrary group do not constitute a group under functional composition.