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Nonadjacent Radix-$\tau$ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

Published:2008-12-01
Printed: Dec 2008
• Ian F. Blake
• V. Kumar Murty
• Guangwu Xu
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Abstract

In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-$\tau$ expansion of integers in the number fields $\Q(\sqrt{-3})$ and $\Q(\sqrt{-7})$. The (window) nonadjacent form of $\tau$-expansion of integers in $\Q(\sqrt{-7})$ was first investigated by Solinas. For integers in $\Q(\sqrt{-3})$, the nonadjacent form and the window nonadjacent form of the $\tau$-expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-$\tau$ expansions for integers in all Euclidean imaginary quadratic number fields.
 Keywords: algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography
 MSC Classifications: 11A63 - Radix representation; digital problems {For metric results, see 11K16} 11R04 - Algebraic numbers; rings of algebraic integers 11Y16 - Algorithms; complexity [See also 68Q25] 11Y40 - Algebraic number theory computations 14G50 - Applications to coding theory and cryptography [See also 94A60, 94B27, 94B40]

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