On the generalized linear equivalence of functions over finite fields

In this paper we introduce the concept of generalized linear equivalence between functions defined over finite fields; this can be seen as an extension of the classical criterion of linear equivalence, and it is obtained by means of a particular geometric representation of the functions. After giving the basic definitions, we prove that the known equivalence relations can be seen as particular cases of the proposed generalized relationship and that there exist functions that are generally linearly equivalent but are not such in the classical theory. We also prove that the distributions of values in the Difference Distribution Table (DDT) and in the Linear Approximation Table (LAT) are invariants of the new transformation; this gives us the possibility to find some Almost Perfect Nonlinear (APN) functions that are not linearly equivalent (in the classical sense) to power functions, and to treat them accordingly to the new formulation of the equivalence criterion

A complete formulation of generalized affine equivalence

In this paper we present an extension of the generalized linear equivalence relation, proposed in [7]. This mathematical tool can be helpful for the classification of non-linear functions f : F p m → F p n based on their cryptographic properties. It thus can have relevance in the design criteria for substitution boxes (S-boxes), the latter being commonly used to achieve non-linearity in most symmetric key algorithms. First, we introduce a simple but effective representation of the cryptographic properties of S-box functions when the characteristic of the underlying finite field is odd; following this line, we adapt the linear cryptanalysis technique, providing a generalization of Matsui’s lemma. This is done in order to complete the proof of Theorem 2 in [7], also by considering the broader class of generalized affine transformations. We believe that the present work can be a step towards the extension of known cryptanalytic techniques and concepts to finite fields with odd characteristic