Shifted Eisenstein polynomials, irreducible compositions of polynomials and group key exchanges

Abstract

In my dissertation, I have covered multiple different topics. First, we consider the concept of natural density over the integers, and extend it to holomorphy rings over function fields. This allows us to give a function field analogue of Cesaro’s theorem, which gives the “probability” that an m-tuple of random elements of the holomorphy ring is oprime. We also generalize this and consider the density of k × m matrices over holomorphy rings which can be extended to unimodular m × m matrices. In the second part, we determine the natural density of shifted Eisenstein polynomials. This means that we compute the density of integer polynomials f(x) of a fixed degree n for which some shift f(x + i) for an integer i satisfies Eisenstein’s irreducibility criterion. We then also compute the density of affine Eisenstein polynomials. Thirdly, we consider an arbitrary set of monic quadratic polynomials over a finite field and ask ourselves which compositions of copies of them are irreducible. We first give a criterion to decide whether all such compositions are irreducible, and then show that in general, the irreducible compositions have the structure of a regular language. In the final chapter, we study cryptographic protocols for key exchange in ad-hoc groups. We first translate some protocols from the literature to the more general setting of semigroup actions, and then propose our own variants of these protocols, which aim to have improved security or efficiency. Then, we demonstrate a couple of active attacks on certain such protocols which are in some ways more powerful than man-in-the-middle attacks.