This thesis examines the Mordell-Weil group for application in cryptography. This approach has recently been proposed by Gerhard Frey. The use of the Mordell-Weil group for discrete logarithm schemes is a variant of elliptic curve cryptosystems. We extended the original idea by Frey with the goal of a performance improvement. The arithmetic complexity using the Mordell-Weil group will be compared to ordinary elliptic curve cryptosystems. The main goals of this thesis are (1) to investigate the algorithmic complexity of Mordell-Weil cryptosystems relative to elliptic curve cryptosystems; (2) the appropriate selection of the group parameters for a successful adaptation to different platforms; (3) a C++ library which makes it possible to easily use this algebra for cryptographic systems based on groups; and (4) to obtain software performance measures for the new cryptosystem. Point multiplication, the crucial operation for elliptic curve cryptosystems, is more than 20% less complex in the Mordell-Weil group than in an ordinary elliptic curve while preserving the same level of security. We show how to further improve the system such that it is particularly suited to 32-bit and 16-bit hardware platforms. The speed-up of the Mordell-Weil group approach comes at the cost of a slightly larger bit-size that is needed to represent a curve point and a more costly curve generation.

Rational Catalan combinatorics connects various Catalan numbers to the representation theory of rational Cherednik algebras for Coxeter and complex reflection groups. Lewis, Reiner, and Stanton seek a theory of rational Catalan combinatorics for the general linear group over a finite field. The finite general linear group is a modular reflection group that behaves like a finite Coxeter group. They conjecture a Hilbert series for a space of invariants under the action of this group using (q,t)-binomial coefficients. They consider the finite general linear group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. Often conjectures about reflection groups are solved by considering the local case of a group fixing one hyperplane and then extending via the theory of hyperplane arrangements to the full group. The Lewis, Reiner and Stanton conjecture had not previously been formulated for groups fixing a hyperplane. We formulate and prove their conjecture in this local case.