Research On Weakly APN Functions

Block cipher is an important branch of symmetric cryptography.As a non-linear component that provides confusion in the block cipher algorithm,S-boxes play a key role to the security of the whole cryptographic algorithm.Functions used as S-boxes are required to have low differential uniformity,high nonlinearity and high algebraic degree.Sometimes,these functions are also required to be permutations.Furthermore,for the software and hardware implementation,such functions are usually needed to be defined on fields with even degrees.As the functions with the lowest differential unifor-mity defined on IF2n,almost perfect nonlinear(APN)functions play a very important role in the design and analysis of block ciphers.However,when n is even,it is still an open problem whether there exist APN permutations over F2n for even n>6.The differential uniformity is defined by measuring the size of preimage set of the deriva-tive function.Based on the concept of weakly differential uniformity proposed by the Italian scholars Caranti et al in 2009,the related properties and criteria of weakly APN functions are characterized in terms of the image set of derivative function in this paper.The purpose of this paper is mainly to study the properties and criteria of the weakly APN functions over F2n.Firstly,according to the properties of APN function-s,we propose a necessary condition for the existence of the weakly APN functions by studying the relation of APN functions and weakly APN functions.Based on the proposed condition,we put forward a necessary and sufficient condition for a differen-tially 4-uniform function to be weakly APN functions.Secondly,with the connection between the weakly APN functions and strong l-anti-invariant functions,a sufficient condition for the differentially 4-uniform function to be a weakly APN function is ob-tained.Thirdly,we discuss some properties of the weakly APN permutation.We get a property of weakly APN quadratic permutations,and discuss the algebraic degree of the weakly APN permutations on F26.Finally,as an application of the necessary and sufficient condition for differentially 4-uniform functions to be weakly APN functions proposed before,whether some known differentially 4-uniform functions are weakly APN functions or not is discussed.And we deduce the five classes of weakly APN functions which are differentially 4-uniform functions.