Abstract: To prevent differential attack on the cipher, cryptographic functions are required to have low differential uniformity. Perfect nonlinear (PN) functions, almost perfect nonlinear (APN) functions and differentially 4-uniform permutations are the most important cryptographic functions with low differential uniformity. Here we survey the recent main research results about cryptographic functions with low differential uniformity such as PN functions, APN functions and differentially 4-uniform permutations. First, we recall the connections between PN functions and the mathematical objects such as the semifield, which survey the known constructions of PN functions and the pseudo-planar functions. Second, the properties and judgement of APN functions are analyzed. We also list the known constructions of APN functions and recall the inequivalent results between them. Third, we summarize the known results on the constructions of differentially 4-uniform permutations and discuss their equivalence. Then, we recall the applications of low differential uniformity functions in the design of actual ciphers. Lastly, we propose some research problems on cryptographic functions with low differential uniformity.

Key words: perfect nonlinear (PN) function, almost perfect nonlinear (APN) function, differentially 4-uniform permutation, low differential uniformity function, S-box