Abstract
We study the functions with low differential uniformity, and concentrates mainly on the properties of perfect nonlinear (PN) functions, including the properties of the derivative of the components of those functions. Some sufficient and necessary conditions have been explored to judge when a function is a PN function. These conditions may be useful in constructing new PN functions. We also construct some functions with differential 4-uniformity that have rarely been studied in the literature. Some of the constructed functions with differential 4-uniformity have high nonlinearity as well. Finally, a class of functions with differential 4-uniformity which are not extended affine equivalent to any power functions are constructed.
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