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.
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Barua, R., Gupta, K.C., Pandey, S.K., Ray, I.G. (2018). On Diffusion Layers of SPN Based Format Preserving Encryption Schemes: Format Preserving Sets Revisited. In: Chakraborty, D., Iwata, T. (eds) Progress in Cryptology – INDOCRYPT 2018. INDOCRYPT 2018. Lecture Notes in Computer Science(), vol 11356. Springer, Cham. https://doi.org/10.1007/978-3-030-05378-9_5
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