Format Preserving Sets: On Diffusion Layers of Format Preserving Encryption Schemes

Abstract

Format preserving encryption refers to a set of techniques for encrypting data such that the ciphertext has the same format as the plaintext. Here, we consider the design of diffusion layers only which can be defined by, in general, a linear transformation. In this paper, we study and explore the format preserving diffusion layers, in particular, the relationship between the $n \times n$ diffusion matrix M over the field $\mathbb {F}_{q}$ and the format preserving set $\mathbb {S} \subseteq \mathbb {F}_{q}$ such that whenever $\mathbf {v} \in \mathbb {S}^n$, $M\mathbf {v} \in \mathbb {S}^n$. It is proved in this paper that if such a set $\mathbb {S}$ with respect to a certain type of matrix M contains $\bar{0} \in \mathbb {F}_q$, then it is always a vector space over the smallest field containing entries of M. Moreover, some more interesting results are found when this condition, $\bar{0} \in \mathbb {S}$, is relaxed. We illustrate our results by a credit card example where plaintext and ciphertext both come from the set $\{0,\cdots ,9\}$. We further show that only certain type of $4 \times 4$ matrices over the field $\mathbb {F}_{2^4}$ can be constructed which yield a format preserving set of cardinality 10 which is suited for our credit card example. However, to the best of our knowledge, such matrices do not have any cryptographic significance. Thus, it is impossible to construct any cryptographically significant $4 \times 4$ matrices over the field $\mathbb {F}_{2^4}$ in the diffusion layer which yields a format preserving set of cardinality 10.