Involutory Differentially 4-Uniform Permutations from Known Constructions

Substitution box (S-box) is an important component of block ciphers for providing confusion into the cryptosystems. The functions used as S-boxes should have low differential uniformity, high nonlinearity and high algebraic degree. Due to the lack of knowledge on the existence of APN permutations over F22k , which have the lowest differential uniformity, when k > 3, they are often constructed from differentially 4-uniform permutations. Up to now, many infinite families of such functions have been constructed. Besides, the less cost of hardware implementation of S-boxes is also an important criterion in the design of block ciphers. If the S-box is an involution, which means that the compositional inverse of the permutation is itself, then the implementation cost for its inverse is saved. The same hardware circuit can be used for both encryption and decryption, which is an advantage in hardware implementation. In this paper, we investigate all the differentially 4-uniform permutations that are known in the literature and determine whether they can be involutory. We found that some involutory differentially 4-uniform permutations with high nonlinearity and algebraic degree can be given from these known constructions.