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Nonadjacent Radix-τ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

  • Ian F. Blake (a1), V. Kumar Murty (a2) and Guangwu Xu (a3)

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 $\mathbb{Q}\left( \sqrt{-3} \right)$ and $\mathbb{Q}\left( \sqrt{-7} \right)$ . The (window) nonadjacent form of $\tau $ -expansion of integers in $\mathbb{Q}\left( \sqrt{-7} \right)$ was first investigated by Solinas. For integers in $\mathbb{Q}\left( \sqrt{-3} \right)$ , 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.

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References

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[1] Blake, I., Seroussi, G., and Smart, N., Elliptic Curves in Cryptography. Cambridge University Press, 1999.
[2] Blake, I., Murty, V. K., and Xu, G., A note on window τ-NAF algorithm. Inform. Process. Lett. 95(2005), no. 5, 496502.
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Nonadjacent Radix-τ Expansions of Integers in Euclidean Imaginary Quadratic Number Fields

  • Ian F. Blake (a1), V. Kumar Murty (a2) and Guangwu Xu (a3)

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