The Generalized Feistel Structure (GFS) generally uses the sub-block-wise cyclic shift in the permutation layer, the layer between the two F function layers. For Type 2 GFS, at FSE 2010, Suzaki and Minematsu showed that a better diffusion property can be obtained if one uses some other sub-block-wise permutation. In this paper, we consider Type 1, Type 3, Source-Heavy (SH), and Target-Heavy (TH) GFSs, and study if their diffusion properties can be improved by changing the sub-block-wise cyclic shift. For Type 1 GFS and Type 3 GFS, we show that better permutations in terms of diffusion exist. For SH and TH GFSs, we show that the diffusion property does not change even if we change the sub-block-wise cyclic shift. We also experimentally derive optimum permutations in terms of diffusion, and evaluate the security of the resulting schemes against saturation, impossible differential, differential, and linear attacks.

HyRAL is a blockcipher whose block size is 128bits, and it supports the key lengths of 128, 129, ..., 256bits. The cipher was proposed for the CRYPTREC project, and previous analyses did not identify any security weaknesses. In this paper, we first consider the longest key version, 256-bit key HyRAL, and present the analysis in terms of equivalent keys. We first show that there are 251.0 equivalent keys (or 250.0 pairs of equivalent keys). Next, we propose an algorithm that derives an instance of equivalent keys with the expected time complexity of 248.8 encryptions and a limited amount of memory. Finally, we implement the proposed algorithm and fully verify its correctness by showing several instances of equivalent keys. We then consider shorter key lengths, and show that there are equivalent keys in 249-, 250-, ..., 255-bit key HyRAL. For each of these key lengths, we present the expected time complexity to derive an instance of equivalent keys.

The Generalized Feistel Structure (GFS) is one of the structures used in designs of blockciphers and hash functions. There are several types of GFSs, and we focus on Type 1 and Type 2 GFSs. The security of these structures are well studied and they are adopted in various practical blockciphers and hash functions. The round function used in GFSs consists of two layers. The first layer uses the nonlinear function. Type 1 GFS uses one nonlinear function in this layer, while Type 2 GFS uses a half of the number of sub-blocks. The second layer is a sub-block-wise permutation, and the cyclic shift is generally used in this layer. In this paper, we formalize Type 1.x GFS, which is the natural extension of Type 1 and Type 2 GFSs with respect to the number of nonlinear functions in one round. Next, for Type 1.x GFS using two nonlinear functions in one round, we propose a permutation which has a good diffusion property. We demonstrate that Type 1.x GFS with this permutation has a better diffusion property than other Type 1.x GFS with the sub-block-wise cyclic shift. We also present experimental results of evaluating the diffusion property and the security against the saturation attack, impossible differential attack, differential attack, and linear attack of Type 1.x GFSs with various permutations.