Generalized quasi-Hadamard transformations on finite groups

In this paper, we introduce a generalization of quasi-Hadamard transformations on a finite group $X$. For $X = {\mathbb{Z}}_{2^m}$, it includes the pseudo-Hadamard transformation used in the Safer block cipher, the Twofish block cipher and Quasi-Hadamard transformations. We get a criterion of their bijectivity. It depends on a class of transformations which include orthomorphisms and complete transformations. Using Kronecker product of matrices, we also define generalized quasi-Hadamard transformations on $X^{2^d}$ for any $d \geq 1 $. 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.