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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.

- Clemens Heuberger, Daniel Krenn, Stephan G. Wagner
- ANALCO
- 2013

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)

1IT University of Copenhagen, Denmark, maau@itu.dk 2Institut für Theoretische Informatik, Technische Universität Ilmenau, Germany, martin.dietzfelbinger@tu-ilmenau.de 3Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Austria, clemens.heuberger@aau.at 4Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Austria, math@danielkrenn.at or… (More)

- Clemens Heuberger, Daniel Krenn, Sara Kropf
- ArXiv
- 2014

Article history: Received 9 July 2012 Accepted 23 June 2013 Available online 4 September 2013 Communicated by D. Wan MSC: primary 20D20 secondary 11T06, 13M10, 11C08, 13F20, 20E18

- Daniel Krenn
- 2013

- Daniel Krenn, Stephan G. Wagner
- Algorithmica
- 2015

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)

- Clemens Heuberger, Daniel Krenn, Stephan G. Wagner
- SIAM J. Discrete Math.
- 2015

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 a variant of dual-pivot quicksort. We show that the used algorithmic partitioning strategy is optimal, i.e., it minimizes the expected number of key comparisons. For the analysis, we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the… (More)