Abstract
Type-II Generalized Feistel Schemes are one of the most popular versions of Generalized Feistel Schemes. Their round function consists in applying a classical Feistel transformation to p sub-blocks of two consecutive words and then shifting the k = 2p words cyclically. The low implementation costs it offers are balanced by a low diffusion, limiting its efficiency. Diffusion of such structures may however be improved by replacing the cyclic shift with a different permutation without any additional implementation cost. In this paper, we study ways to determine permutations with the fastest diffusion called optimal permutations.To do so, two ideas are used. First, we study the natural equivalence classes of permutations that preserve cryptographic properties; second, we use the representation of permutations as coloured trees.For both heuristic and historical reasons, we focus first on even-odd permutations, that is, those permutations for which images of even numbers are odd. We derive from their structure an upper bound on the number of their equivalence classes together with a strategy to perform exhaustive searches on classes. We performed those exhaustive searches for sizes k ≤ 24, while previous exhaustive searches on all permutations were limited to k ≤ 16. For sizes beyond the reach of this method, we use tree representations to find permutations with good intermediate diffusion properties. This heuristic leads to an optimal even-odd permutation for k = 26 and best-known results for sizes k = 64 and k = 128.Finally, we transpose these methods to all permutations. Using a new strategy to exhaust equivalence classes, we perform exhaustive searches on classes for sizes k ≤ 20 whose results confirmed the initial heuristic: there always exist optimal permutations that are even-odd and furthermore for k = 18 all optimal permutations are even-odd permutations.
Highlights
Since its first appearance in 1973 with the cipher Lucifer, later evolving into Data Encryption Standard, DES [DES77], the Feistel network has become one of the main flavour of block ciphers
As already suggested in [SM10], we focus in Section 4 on the so-called even-odd permutations and derive from a rigorous analysis of their equivalence classes an upper bound on the number of such equivalence classes together with a strategy to exhaustively run through all of them which allowed us to record all optimal ones for sizes k ≤ 24
For higher values of k, beyond the reach of practical exhaustive searches, we can compute from their structure a lower bound on the minimal diffusion round of even-odd permutations
Summary
Since its first appearance in 1973 with the cipher Lucifer, later evolving into Data Encryption Standard, DES [DES77], the Feistel network has become one of the main flavour of block ciphers. A natural generalisation of Feistel Networks called Generalized Feistel Structures reproduces these routines, splitting the message into k ≥ 2 blocks, where k is called the partition number Among those Generalized Feistel Structures, the so-called Type-II Feistel Ciphers introduced in [ZMI89] consist in iterations of a round function of the form:. Recent Generalized Feistel Structures use a small partition number to balance implementation sizes and speed Such structures come with a serious drawback, Type-II Feistel Ciphers have a low diffusion: k rounds are needed to ensure that an input difference diffuses to all output blocks. They recorded even-odd permutations with the lowest diffusion round for sizes k ≤ 16 One of those permutations is used in the block cipher TWINE [SMMK12] or the cryptographic permutation Simpira [GM16].
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