Arithmetic of Supersingular Koblitz Curves in Characteristic Three

Prof. Roberto Avanzi (Bochum, Germany)
25/03/2010 - 11:00
AULA 311, Dipartimento di Matematica dell'Università Roma Tre - Largo San Leonardo Murialdo, 1

Roberto Avanzi (Bochum, Germany)
joint work with Clemens Heuberger (Graz, Austria)
and Helmut Prodinger (Stellenbosch, South Africa)

We consider digital expansions of scalars for supersingular Koblitz curves in characteristic three. These are expansions of integers to the algebraic base of TeX Embedding failed!, where TeX Embedding failed! is a zero of a polynomial TeX Embedding failed!. The obvious application of these expansions is to scalar multiplication on Koblitz curves.

A simple connection between TeX Embedding failed!-adic expansions and balanced ternary representations is given.

Windowed non-adjacent representations are considered whereby the digits are elements of minimal norm. We exploit the rotational symmetry of the digit set to reduce the memory requirements of scalar multiplication by a factor of six with respect to previous methods. Furthermore, we give an explicit description of the elements of the digit set, allowing for a very simple and efficient precomputation strategy.

Additionally, we explicitly describe the action of some endomorphisms on the Koblitz curve as a scalar multiplication by an explicitly given integer.