Abstract view
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
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]
