4 research outputs found
On the minimal Hamming weight of a multi-base representation
CITATION: Krenn, D., Suppakitpaisarn, V. & Wagner, S. 2020. On the minimal Hamming weight of a multi-base representation. Journal of Number Theory, 208:168–179, doi:10.1016/j.jnt.2019.07.023.The original publication is available at https://www.sciencedirect.comGiven a finite set of bases b1, b2, ..., br (integers greater
than 1), a multi-base representation of an integer n is a sum
with summands dbα1
1 b
α2
2 ··· bαr r , where the αj are nonnegative
integers and the digits d are taken from a fixed finite set.
We consider multi-base representations with at least two
bases that are multiplicatively independent. Our main result
states that the order of magnitude of the minimal Hamming
weight of an integer n, i.e., the minimal number of nonzero
summands in a representation of n, is log n/(log log n). This
is independent of the number of bases, the bases themselves,
and the digit set.
For the proof, the existing upper bound for prime bases
is generalized to multiplicatively independent bases; for the
required analysis of the natural greedy algorithm, an auxiliary
result in Diophantine approximation is derived. The lower
bound follows by a counting argument and alternatively
by using communication complexity; thereby improving the
existing bounds and closing the gap in the order of magnitude.Austrian Science Fundhttps://www.sciencedirect.com/science/article/pii/S0022314X19302768Publisher's versio
Efficient Arithmetic on Subfield Elliptic Curves over Small Odd Characteristics
In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, non-adjacent form (NAF) and its generalizations (e.g., generalized non-adjacent form (GNAF) and radix- non-adjacent form (NAF) \cite{CL73,TYW04}) are proposed for minimizing the non-zero densities in the representations of the scalars. On the other hand, for subfield elliptic curves, Frobenius-adic expansions of the scalars can be used for improving efficiency (\cite{Sma99+}). Unfortunately, there are only a few methods apply the techniques of NAF or its analogue to Frobenius-adic expansion, namely -adic NAF techniques (\cite{Kob98,Sol00,BMX04} and \cite{GLS01}) for Koblitz curves and hyperelliptic Koblitz curves. In this paper, we try to combine these techniques, namely recoding methods for reducing non-zero density and Frobenius-adic expansion, and propose two new efficient recoding methods of scalars for more general family of subfield elliptic curves over odd characteristics. We also prove that the non-zero densities for the new methods are same as those for original GNAF and NAF. As a result, the speed of the proposed schemes improve between 12.5{\%} and 79{\%} over that for previously known schemes
Theoretical and practical efficiency aspects in cryptography
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