# Project Description

Optimising structures or algorithms is a permanent goal in various branches of mathematics and their applications. Apart from the optimal solutions, we are interested also in their qualitative and quantitative behaviour for large values of the parameters in order to get a better understanding of their nature. Furthermore, we are then able to compare our solutions with other approaches and strategies. This leads to the quest for a precise asymptotic and probabilistic analysis of discrete structures and algorithms. In this vast field, we are especially interested in mathematical problems in graph theory and in applications in cryptography.

In the past decades a variety of different graph theoretical indices were investigated intensively, some of them because molecules can be modelled as undirected graphs and certain physicochemical properties depend on the structure of these graphs, quantified by various graph theoretical indices. It is a natural question to determine the range of graph theoretical indices and to identify those graphs maximising or minimising the indices over a certain class of graphs, e.g. trees. Thus, it is our intention to determine extremal graphs for indices related to the Wiener index (sum of pairwise distances), the Merrifield-Simmons index (number of independent sets) and the Hosoya index (number of matchings) under various constraints. The description of those extremal graphs sometimes involves other concepts, one example being special digital expansions of the parameters, e.g. the order of the graph.

On the other hand, digital expansions are also a key ingredient for our study of problems motivated by cryptography. Public-key (asymmetric) cryptography relies on the efficient computation of one-way functions, one common example being computing scalar multiples $nP$ of an element $P$ in some Abelian group. The classic procedure is to represent $n$ by a digital expansion and to compute the scalar multiple $nP$ by Horner's scheme, resulting in the so-called double-and-add method. Apart from the multiplicative group of a finite field, the point group of an elliptic (or hyperelliptic) curve is widely used. The additional structure of these groups allows many variations of the basic double-and-add method. In particular, digital expansions to non-rational bases are of interest. We intend to derive optimal expansions in various contexts and perform precise asymptotic and probabilistic analyses of the resulting algorithms in the sense of D. E. Knuth.