Efficient scalar multiplication in Abelian groups (which is an important operation in public key cryptography) can be performed using digital expansions. Apart from rational integer bases (double-and-add algorithm), imaginary quadratic integer bases are of interest for elliptic curve cryptography , because the Frobenius endomorphism fulfils a quadratic… (More)
Let o be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in o is the sum of pairwise distinct units, if the unit equation u+v = 2 has a non-trivial solution u, v ∈ o *. We generalize this result and give applications to signed double-base digit expansions.
For fixed t ≥ 2, we consider the class of representations of 1 as sum of unit fractions whose denominators are powers of t or equivalently the class of canonical compact t-ary Huffman codes or equivalently rooted t-ary plane " canonical " trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of… (More)
We present an average case analysis of two variants of dual-pivot quicksort, one with a non-algorithmic comparison-optimal partitioning strategy, the other with a closely related algorithmic strategy. For both we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the linear,… (More)
For a fixed integer base $$b\ge 2$$ b ≥ 2 , we consider the number of compositions of 1 into a given number of powers of b and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of b. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on… (More)
The new finite state machine package in the mathematics software system SageMath is presented and illustrated by many examples. Several combinatorial problems, in particular digit problems, are introduced, modeled by automata and transducers and solved using SageMath. In particular, we compute the asymptotic Hamming weight of a non-adjacent-form-like digit… (More)
In this note the precise minimum number of key comparisons any dual-pivot quickselect algorithm (without sampling) needs on average is determined. The result is in the form of exact as well as asymptotic formulae of this number of a comparison-optimal algorithm. It turns out that the main terms of these asymptotic expansions coincide with the main terms of… (More)
2016 All Rights Reserved ii ACKNOWLEDGEMENTS First and foremost, I would like to thank my advisor, Dr. Boris Baltes, for consistent feedback, recommendations, and support (both motivational and monetary). I am especially grateful to my advisor for supporting me on a thesis topic outside of his interests. I would also like to thank my thesis committee… (More)