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Permutations with low differential uniformity, high nonlinearity and high algebraic degree over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">F</mi><msup><mn>2</mn><mrow><mn>2</mn><mi>k</mi></mrow></msup></msub></semantics></math></inline-formula> are preferred substitution boxes in modern block ciphers. In this paper, we study the bijectivity and the difference uniformity of piecewise function with the help of permutation group theory. Based on our results, We found many at least differentially 6-uniform and differentially 4-uniform permutations over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">F</mi><msup><mn>2</mn><mrow><mn>2</mn><mi>k</mi></mrow></msup></msub></semantics></math></inline-formula>, which can be chosen as the substitution boxes.