2019/06/27 by Sihem Mesnager, Longjiang Qu, Mesnager, Sihem +1 · 1 citation
Computer Science · Engineering · #Coding theory and cryptography #Cryptographic Implementations and Security #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1907.01066
openalex publication_date 2019/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Two-to-one (2-to-1) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of xrh(x(q-1)/d), those from permutation polynomials, from linear translators and from APN functions. Then we present 2-to-1 polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of 2-to-1 mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.