屈龙江, 陈玺, 牛泰霖, 李超. 有限域上低差分函数研究进展[J]. 计算机研究与发展, 2018, 55(9): 1931-1945.
 引用本文: 屈龙江, 陈玺, 牛泰霖, 李超. 有限域上低差分函数研究进展[J]. 计算机研究与发展, 2018, 55(9): 1931-1945.
Qu Longjiang, Chen Xi, Niu Tailin, Li Chao. Recent Progress in Low Differential Uniformity Functions over Finite Fields[J]. Journal of Computer Research and Development, 2018, 55(9): 1931-1945.
 Citation: Qu Longjiang, Chen Xi, Niu Tailin, Li Chao. Recent Progress in Low Differential Uniformity Functions over Finite Fields[J]. Journal of Computer Research and Development, 2018, 55(9): 1931-1945.

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

• 摘要: 为了抵抗差分密码攻击，密码算法设计希望使用低差分函数.完全非线性函数(perfect nonli-near function, PN函数)、几乎完全非线性函数(almost perfect nonlinear function, APN函数)和4差分置换(differentially 4-uniform permutition)是最重要的几类低差分函数(low differential uniformity function).总结了近年来在PN函数、APN函数和4差分置换等低差分函数研究方面的主要进展.1)回顾了PN函数与半域等数学对象的联系，梳理了PN函数的已有构造以及伪平面函数的构造;2)分析了APN函数的性质与判定，总结了APN函数的已有构造以及它们之间等价性分析方面的结果;3)对于4差分置换，总结了其已有构造及其等价性分析结果;4)介绍了低差分函数在实际密码算法设计中的应用;5)对低差分函数的下一步研究进行了展望.

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.

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