Paper 2014/974

Non-Linearity and Affine Equivalence of Permutations

P R Mishra, Indivar Gupta, and N Rajesh Pillai

Abstract

In this paper we consider permutations on n symbols as bijections on Z/nZ. Treating permutations this way facilitates us with additional structures such as group, ring defined in the set Z/nZ. We explore some of the properties of permutations arising out of this treatment. We propose two properties viz. affine equivalence and non-linearity for permutations on the lines similar to there description given in the case of functions. We also establish some results which are quite similar to those given for Boolean functions. We also define Mode Transform of a permutation and investigate its relationship with non-linearity. We propose an efficient algorithm using Mode transform for computing non-linearity of a permutation and show that it is O(n^2), as compared to O(n^3) of the direct approach. At the end we discuss these properties in the context of cryptography.