Grant: Discovery Grants Program - Individual

The focus of this research proposal is on the theory and real-world applications of finite fields. These algebraic objects have a long history in mathematics. The theory, applications and algorithms of finite fields have become central to countless areas in communications and security. In particular, they play a central role in cryptography and coding theory. Our plan has four axes of research: (a) the study of iterations of functions over finite fields; (b) the development of algorithms for operations involving polynomials over finite fields in use in post-quantum cryptography (PQC); (c) the construction of efficient codes; and (d) the creation of combinatorial arrays built from finite field theory. (a) The dynamics of iterations of functions over finite fields has attracted a lot of attention in part due to their relation to integer factorization algorithms and discrete logarithm computations. I have been considerably involved in the development of this area. We propose to advance the investigation of the dynamical behavior of families of functions. A practical goal of this proposal is to study iterations of hash functions, S-box functions, and other practically relevant mappings in cryptography. (b) In PQC we aim at systems to be employed in classical computers but that are resistant to attacks involving quantum machines. We plan to focus on operations involving polynomials over finite fields used in PQC cryptosystems. This includes the development of new constant-time algorithms to avoid side-channel attacks based on execution time differences, and the random generation of special types of polynomials needed for the safe execution of these PQC systems. (c) Coding theory is another important real-world application that we prioritize in this proposal. Among the many coding methods and techniques available, LDPC (Low-Density Parity-Check) codes have become dominant in use due to their easy implementation, fast decoding methods, and capacity-achieving capabilities. We have studied the influence of diverse graph theoretic properties in the performance of these codes. We propose to study other graph theoretic properties to further improve their performance. We also aim to thoroughly study the currently used LDPC codes in the 5G wireless communications industry. We believe that much of our accumulated knowledge so far has the potential to improve some of these practically important codes. (d) Finite field methods and techniques are also used in combinatorics for the construction of diverse objects including combinatorial designs, finite geometries and arrays of different kinds. Ordered orthogonal arrays (OOAs) are a generalization of orthogonal arrays. They are closely related to MDS (maximum distance separable) codes, as well as Monte Carlo integration methods. We propose to use several types of partially ordered sets (posets) to construct OOAs using linear feedback shift register sequences over finite fields.