4 research outputs found
Efficient arithmetic on elliptic curves in characteristic 2
International audienceWe present normal forms for elliptic curves over a field of characteristic 2 analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient algorithms for point addition and scalar multiplication on these forms. The resulting algorithms apply to any elliptic curve over a field of characteristic 2 with a 4-torsion point, via an isomorphism with one of the normal forms. We deduce algorithms for duplication in time and for addition of points in time , where is the cost of multiplication, the cost of squaring , and the cost of multiplication by a constant. By a study of the Kummer curves , we develop an algorithm for scalar multiplication with point recovery which computes the multiple of a point P with per bit where is multiplication by a constant that depends on
The geometry of efficient arithmetic on elliptic curves
The arithmetic of elliptic curves, namely polynomial addition and scalar
multiplication, can be described in terms of global sections of line bundles on
and , respectively, with respect to a given projective embedding
of in . By means of a study of the finite dimensional vector
spaces of global sections, we reduce the problem of constructing and finding
efficiently computable polynomial maps defining the addition morphism or
isogenies to linear algebra. We demonstrate the effectiveness of the method by
improving the best known complexity for doubling and tripling, by considering
families of elliptic curves admiting a -torsion or -torsion point
Pre-Computation Scheme of Window NAF for Koblitz Curves Revisited
Let be a Koblitz curve. The window -adic non-adjacent form (window NAF) is currently the standard representation system to perform scalar multiplications on utilizing the Frobenius map . This work focuses on the pre-computation part of scalar multiplication. We first introduce -operations where and is the complex conjugate of . Efficient formulas of -operations are then derived and used in a novel pre-computation scheme. Our pre-computation scheme requires {\bf M}{\bf S}, {\bf M}{\bf S}, {\bf M}{\bf S}, and {\bf M}{\bf S} () and {\bf M}{\bf S}, {\bf M}{\bf S}, {\bf M}{\bf S}, and {\bf M}{\bf S} () for window NAF with widths from to respectively. It is about two times faster, compared to the state-of-the-art technique of pre-computation in the literature. The impact of our new efficient pre-computation is also reflected by the significant improvement of scalar multiplication. Traditionally, window NAF with width at most is used to achieve the best scalar multiplication. Because of the dramatic cost reduction of the proposed pre-computation, we are able to increase the width for window NAF to for a better scalar multiplication. This indicates that the pre-computation part becomes more important in performing scalar multiplication. With our efficient pre-computation and the new window width, our scalar multiplication runs in at least 85.2\% the time of Kohel\u27s work (Eurocrypt\u272017) combining the best previous pre-computation. Our results push the scalar multiplication of Koblitz curves, a very well-studied and long-standing research area, to a significant new stage