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Extending previous results, we study a class of general quadrinomials over the field of size 2<sup>2m</sup> with odd m and characterize conditions under which they are permutations with 4-uniform BCT, a new and important parameter related to boomerang-style attacks. These permutations are known to have the best known nonlinearity. Numerical data also show that the inverse of these functions all have large algebraic degree, making them desirable for applications.

On Permutation Quadrinomials and 4-Uniform BCT