国防科技大学学报2026，Vol.48Issue(3)：368-384,17.DOI:10.11887/j.issn.1001-2486.25120009
几乎完全非线性函数研究进展
Research progress of almost perfect nonlinear functions
摘要
Abstract
APN(almost perfect nonlinear)functions,renowned for their optimal differential properties,have become a research focus in the field of cryptographic functions.This paper systematically reviewed the research progress of APN functions:first,it summarized the general methods for generating APN function examples;second,it refined the construction techniques of existing infinite families of APN functions and clarifies their specific constructions;third,it introduced the equivalence classification results of APN function examples and infinite families;fourth,it combed through the research conclusions on the cryptographic properties of APN functions,such as permutation property,algebraic degree,and nonlinearity;fifth,it reviewed some applications of APN functions in coding theory and combinatorial design;finally,the research prospects of APN functions were prospected.Currently,the construction of APN functions is still dominated by quadratic ones,and no infinite families of polynomials with higher algebraic degree have been found.Major challenges,such as the"big APN problem",remain unsolved.Future research may focus on constructing APN polynomials with non-classical Walsh spectra,discovering APN polynomials with higher degree,among others,and exploring their applications in coding theory and combinatorial design.关键词
几乎完全非线性函数/构造/等价性/置换/代数次数/非线性度Key words
almost perfect nonlinear functions/constructions/equivalence/permutation/algebraic degree/nonlinearity分类
信息技术与安全科学引用本文复制引用
施晨苗,李康荃,屈龙江..几乎完全非线性函数研究进展[J].国防科技大学学报,2026,48(3):368-384,17.基金项目
国家重点研发计划资助项目(2024YFA1013000) （2024YFA1013000）
国家自然科学基金资助项目(12525115,12571579) （12525115,12571579）
湖南省自然科学基金资助项目(2026JJ40001) （2026JJ40001）
