Generalized cyclotomic mappings of finite fields

Finite fields have been studied using methods from a plethora of areas, including algebra. Generalized cyclotomic mappings (GCMs) are functions which are piecewise rather simple. More precisely, they agree with a monomial function on each coset of a given multiplicative subgroup. Their index is the minimum number of these cosets and thus a measure for the complexity of that function. GCMs are important for several applications, for example, in coding theory and cryptography. In particular, GCMs of small index are very useful for designing check digit systems which detect the most frequent types of errors such as single errors and neighbor transpositions, whereas for cryptographic functions the index has to be large. This project focuses on the dynamics (behavior under iteration) of GCMs as well as the following questions: How well can a low-index GCM interpolate certain functions of cryptographic interest (such as the discrete logarithm over a finite prime field)? How does the GCM index relate to popular measures of pseudorandomness? To what extent can one establish analogues of the Weil bounds and of related character sum bounds for low-index GCMs? We also consider the analogous problems for so-called coset-wise affine mappings, an additive analogue of GCMs. The project is expected to require an interesting mix of methods from various areas, including number theory and group theory.