On Diffusion Layers of SPN Based Format Preserving Encryption Schemes: Format Preserving Sets Revisited

Abstract

In Inscrypt 2016, Chang et al. proposed a new family of substitution-permutation (SPN) based format preserving encryption algorithms in which a non-MDS (Maximum Distance Separable) matrix was used in its diffusion layer. In the same year in Indocrypt 2016 Gupta et al., in their attempt to provide a reason for choosing non-MDS over MDS matrices, introduced an algebraic structure called format preserving sets (FPS). They formalised the notion of this structure with respect to a matrix both of whose elements are coming from some finite field $\mathbb {F}_q$. Many interesting properties of format preserving sets $\mathbb {S} \subseteq \mathbb {F}_q$ with respect to a matrix $M(\mathbb {F}_q)$ were derived. Nevertheless, a complete characterisation of such sets could not be derived. In this paper, we fill that gap and give a complete characterisation of format preserving sets when the underlying algebraic structure is a finite field. Our results not only generalise and subsume those of Gupta et al., but also obtain some of these results over a more generic algebraic structure viz. ring $\mathcal {R}$. We obtain a complete characterisation of format preserving sets over rings when the sets are closed under addition. Finally, we provide examples of format preserving sets of cardinalities $10^3$ and $26^3$ with respect to $4 \times 4$ MDS matrices over some rings which are not possible over any finite field.