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Block ciphers use substitution boxes (S-boxes) whose aim is to create confusion into the cryptosystems. Functions used as S-boxes should have low differential uniformity, high nonlinearity and algebraic degree larger than 3 (preferably strictly larger). They should be fastly computable; from this viewpoint, it is better when they are in even number of variables. In addition, the functions should be bijections in a substitution-permutation network. Almost perfect nonlinear (APN) functions have the lowest differential uniformity 2 and the existence of APN bijections over for even is a big open problem. In the present paper, we focus on constructing differentially 4-uniform bijections suitable for designing S-boxes for block ciphers. Based on the idea of permuting the inverse function, we design a construction providing a large number of differentially 4-uniform bijections with maximum algebraic degree and high …