Constructing new differentially 4-uniform permutations from known ones

•New method to construct differentially 4-uniform permutations from known ones and the inverse function.•New family of differentially 4-uniform permutations from Tan-Qu-Tan-Li functions.•New family of differentially 4-uniform permutations from Gold functions. In this paper we study a new construction of differentially 4-uniform permutations from known ones and the inverse function. We focus on constructing methods of [20]. We split a finite field into its subfield and remainder, and, we choose known differentially 4-uniform permutations over the subfield and the inverse function over the entire field. As a result, we obtain two families of differentially 4-uniform permutations.