Finite fields and applications in coding theory and cryptography

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The proposed research area is finite fields and applications in coding theory and cryptography. My recent research has centered on the theoretical study of discrete objects/structures and their properties over finite fields, as well as on their applications to other branches of mathematics and information theory. These objects include polynomials and sequences over finite fields, which have a large number of applications in coding theory, communications and cryptography. This is a fascinating and vibrant area of research in the intersection of discrete math, number theory, theoretical computer science and information theory. Many open problems and conjectures over finite fields arise from useful problems in information theory. It is my long term vision to play a significant and lasting contribution to this area of research. The combinatorial properties of polynomials such as permutations and value set sizes, arithmetic properties of polynomials such as irreducibility, primitivity, divisibility and factorization, as well as the pseudo-randomness of sequences, are central topics of fundamental research. For example, there has been an increasing demand for further studies of objects such as permutation polynomials, irreducible polynomials, primitive polynomials, and feedback shift register sequences due to their applications in block ciphers and stream ciphers, as well as signal sets in wireless communications. Indeed, 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; improving the complexity of list decoding algorithm for Reed-Solomon codes requires the further study of polynomials with prescribed ranges; the design of reliable stream ciphers requires good pseudo-random sequences; the implementation of linear feedback shift register (LFSR) sequences requires the understanding of existence of primitive polynomials with certain low weight (i.e., 3 or 5 nonzero coefficients) over the binary field. My long term goal is thus two-fold: 1) to better understand the combinatorial and arithmetic properties of these fundamental objects over finite fields, and their construction, distribution and enumeration; 2) to better understand the interplay among different objects and properties over finite fields and find genuine applications such as constructing good codes and S-boxes. My scientific approach requires not only extensively theoretical efforts, but also massive computational experiments. This quest involves a combination of knowledge from combinatorics, number theory, algebra, computer science, and information theory. Positive solutions to some of these problems would not only have significant impact on the research community but also have direct technology advance.