ISSN 1000-1239 CN 11-1777/TP

• 综述 •

### 有限域上低差分函数研究进展

1. (国防科技大学文理学院 长沙 410073) (ljqu_happy@hotmail.com)
• 出版日期: 2018-09-01
• 基金资助:
国家重点研发计划项目(2017YFB0802001)；国家自然科学基金优秀青年科学基金项目(61722213) This work was supported by the National Key Research and Development Program of China (2017YFB0802001) and the National Natural Science Foundation of China for Excellent Young Scientists (61722213).

### Recent Progress in Low Differential Uniformity Functions over Finite Fields

Qu Longjiang, Chen Xi, Niu Tailin, Li Chao

1. (College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073)
• Online: 2018-09-01

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.