finite fields and their applications

点击翻译按钮获取中文摘要

The proposed research is in finite fields and their applications. My recent research has centered on the theoretical study of objects/structures and their properties over finite fields, as well as on their applications to other branches of mathematics, computer science, and information theory. These objects include polynomials/functions/sequences over finite fields, which have a large number of applications in coding theory, combinatorics, communications and cryptography. The permutation behavior and arithmetic properties of polynomials such as factorization, divisibility, irreducibility and primitivity, as well as the pseudo-randomness of sequences, are central topics of fundamental research. Indeed, there has been an increasing demand for further studies of objects such as permutation polynomials (PPs), irreducible polynomials, primitive polynomials, special functions, and feedback shift register (FSR) sequences due to their applications in block ciphers and stream ciphers, as well as signal sets in wireless communications. For example, the design of reliable stream ciphers requires good pseudo-random sequences; the design of good S-boxes (permutations) which are resistant against linear/differential cryptanalysis requires useful special functions such as almost perfect nonlinear (APN) permutations over finite fields. My long term goal is thus two-fold: 1) to better