A new construction of differentially 4-uniform (n, n-1)-functions

In this paper, a new way to construct differentially 4-uniform (n; n-1)-functions is presented. As APN (n; n)-functions, these functions offer the best resistance against differential cryptanalysis and they can be used as substitution boxes in block ciphers with a Feistel structure. Constructing such functions is assumed to be as difficult as constructing APN (n; n)-functions. A function in our family of functions can be viewed as the concatenation of two APN (n - 1; n - 1)-functions satisfying some necessary conditions. Then, we study the special case of this construction in which the two APN functions differ by an affne function. Within this construction, we propose a family in which one of the APN functions is a Gold function which gives the quadratic differentially 4-uniform (n, n-1)-function (formula presented) where (formula presented) and x_nF₂ with gcd(i,n-1)=1. We study the nonlinearity of this function in the case i = 1 because in this case we can use results from Carlitz which are unknown in the general case. We also give the Walsh spec- trum of this function and prove that it is CCZ-inequivalent to functions of the form L ₒ F where L is an affne surjective (n, n-1)-function and F is a known APN (n; n)-function for n ≤ 8, or the Inverse APN (n; n)-function for every n ≥ 5 odd, or any AB (n, n)-function for every n > 3 odd, or any Gold APN (n; n)-function for every n > 4 even.