Diffusion properties of generalized quasi-Hadamard transformations on finite Abelian groups

In this paper, we introduce a generalization of quasi-Hadamard transformations on a finite abelian group $X$. For $X = \mathbb{Z}_{2^m}$, it includes the pseudo-Hadamard transformation employed in block ciphers Safer and Twofish, and the quasi-Hadamard transformations proposed by H. Lipmaa. For bijective generalized quasi-Hadamard transformations, we describe diffusion properties of imprimitivity systems of regular permutation representations of additive groups $\mathbb{Z}_{2^m}^2$ and $\mathbb{Z}_{2^{2m}}$. We describe a set of generalized quasi-Hadamard transformations having the best diffusion properties of the imprimitivity systems. We also give conditions such that some generalized quasi-Hadamard transformations have bad diffusion properties.