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


Published:20081201
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
algebraic integer, radix expression, window nonadjacent expansion, algorithm, point multiplication of elliptic curves, cryptography

MSC Classifications: 
11A63, 11R04, 11Y16, 11Y40, 14G50 show english descriptions
Radix representation; digital problems {For metric results, see 11K16} Algebraic numbers; rings of algebraic integers Algorithms; complexity [See also 68Q25] Algebraic number theory computations Applications to coding theory and cryptography [See also 94A60, 94B27, 94B40]
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]
