Multipermutations in Crypto World: Different Faces of the Perfect Diffusion Layer

Diffusion layers, and specially perfect diffusion layers, are very important subject for cryptographic research. Main quest is a perfect diffusion layer with more optimal hardware and/or software implementations (if possible, the last needs to holds also for its inverse). Different structures can be used for representing these layers, but all are interconnected. We start with multipermutations as a tools for obtaining perfect diffusion, and we summarize the interconnections between them, MDS codes, Latin squares and quasigroups, orthogonal arrays and m-arcs. We give a new construction of perfect recursive diffusion layer from r-recursive MDS codes, or recursively r-differentiable quasigroups.