Optimality of the Width-ww Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

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 equation. One strategy for improving the efficiency is to increase the digit set (at the prize of additional precomputations). A common choice is the width\nbd-$w$ non-adjacent form (\wNAF): each block of $w$ consecutive digits contains at most one non-zero digit. Heuristically, this ensures a low weight, i.e.\ number of non-zero digits, which translates in few costly curve operations. This paper investigates the following question: Is the \wNAF{}-expansion optimal, where optimality means minimising the weight over all possible expansions with the same digit set? The main characterisation of optimality of \wNAF{}s can be formulated in the following more general setting: We consider an Abelian group together with an endomorphism (e.g., multiplication by a base element in a ring) and a finite digit set. We show that each group element has an optimal \wNAF{}-expansion if and only if this is the case for each sum of two expansions of weight 1. This leads both to an algorithmic criterion and to generic answers for various cases. Imaginary quadratic integers of trace at least 3 (in absolute value) have optimal \wNAF{}s for $w\ge 4$. The same holds for the special case of base $(\pm 3\pm\sqrt{-3})/2$ and $w\ge 2$, which corresponds to Koblitz curves in characteristic three. In the case of $\tau=\pm1\pm i$, optimality depends on the parity of $w$. Computational results for small trace are given

Spacecraft induced error sources

The attitude control and measurement systems aboard the LANDSAT 2 and D satellites are described and associated errors are discussed. Also, the ephemeris errors from various tracking systems are examined. Use of the Global Positioning System and improved attitude control instruments is expected to greatly reduce the errors in LANDSAT D in comparison with previous LANDSATS

Canonical Trees, Compact Prefix-free Codes and Sums of Unit Fractions: A Probabilistic Analysis

For fixed $t\ge 2$, 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 distinct summands (normal distribution), the path length (normal distribution), the width (main term of the expectation and concentration property) and the number of leaves at maximum distance from the root (discrete distribution)

Automata in SageMath---Combinatorics meet Theoretical Computer Science

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 expansion, which was not known before

The height of multiple edge plane trees

Multi-edge trees as introduced in a recent paper of Dziemia\'nczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height. The main part of this paper is concerned with multi-edge trees counted by their number of edges. The distribution of the number of vertices as well as the height are analysed asymptotically

Algorithmic counting of nonequivalent compact Huffman codes

It is known that the following five counting problems lead to the same integer sequence~$f_t(n)$: the number of nonequivalent compact Huffman codes of length~$n$ over an alphabet of $t$ letters, the number of `nonequivalent' canonical rooted $t$-ary trees (level-greedy trees) with $n$~leaves, the number of `proper' words, the number of bounded degree sequences, and the number of ways of writing $1= \frac{1}{t^{x_1}}+ \dots + \frac{1}{t^{x_n}}$ with integers $0 \leq x_1 \leq x_2 \leq \dots \leq x_n$. In this work, we show that one can compute this sequence for \textbf{all} $n<N$ with essentially one power series division. In total we need at most $N^{1+\varepsilon}$ additions and multiplications of integers of $cN$ bits, $c<1$, or $N^{2+\varepsilon}$ bit operations, respectively. This improves an earlier bound by Even and Lempel who needed $O(N^3)$ operations in the integer ring or $O(N^4)$ bit operations, respectively