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

  • Rana Barua
  • Kishan Chand Gupta
  • Sumit Kumar PandeyEmail author
  • Indranil Ghosh Ray
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11356)


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.


Diffusion layer Format preserving encryption Format preserving set 


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Rana Barua
    • 1
  • Kishan Chand Gupta
    • 2
  • Sumit Kumar Pandey
    • 3
    Email author
  • Indranil Ghosh Ray
    • 4
  1. 1.R.C. Bose Centre for Cryptology and SecurityIndian Statistical InstituteKolkataIndia
  2. 2.Applied Statistics UnitIndian Statistical InstituteKolkataIndia
  3. 3.Ashoka UniversitySonepatIndia
  4. 4.Department of Electrical and Electronic EngineeringCity University, LondonLondonUK

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