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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.
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)
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In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, which means that each integer can have multiple different digit expansions,… (More)