Implicit functions over finite fields and their applications to good cryptographic functions and linear codes

The implicit function theory has many applications in continuous functions as a powerful tool. This paper initiates the research on handling functions over finite fields with characteristic even from an implicit viewpoint, and exploring the applications of implicit functions in cryptographic functions and linear error-correcting codes. The implicit function ${}^{S}G$ over finite fields is defined by the zeros of a bivariate polynomial $G(X,Y)$. First, we provide basic concepts and constructions of implicit functions. Second, some strong cryptographic functions are constructed by implicit expressions, including semi-bent (or near-bent) balanced Boolean functions and 4-differentially uniform involution without fixed points. Moreover, we construct some optimal linear codes and minimal codes by using constructed implicitly defined functions. In our proof, some algebra and algebraic curve techniques over finite fields are used. Finally, some problems for future work are provided.