几乎完全非线性函数研究进展OA

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.