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

作     者：Yuan, Mu Qu, Longjiang Li, Kangquan Wang, Xiaoqiang 

作者机构：Natl Univ Def Technol Coll Sci Changsha 410073 Peoples R China Hubei Univ Fac Math & Stat Hubei Key Lab Appl Math Wuhan 430062 Peoples R China 

出 版 物：《FINITE FIELDS AND THEIR APPLICATIONS》 (Finite Fields Appl.)

年 卷 期：2025年第103卷

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：National Key Research and Development Program of China [2022YFA1004900, 2024YFA1013004] Nature Science Foundation of China (NSFC) [12301671, 62032009, 62202476, 62172427, 12001175] Research Fund of the National University of Defense Technology [ZK22-14] 

主　　题：Implicit expression Finite field Boolean function Differential uniformity Linear code 

摘      要：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 SG 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 4differentially 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.