New point compression method for elliptic F_(q~2)-curves of j-invariant 0

摘要

In the article we propose a new compression method (to 2[log(2)(q)] + 3 bits) for the F-q(2)-points of an elliptic curve E-b: y(2)= x(3) + b(for b epsilon F-q(2)*) of j-invariant 0. It is based on F-q-rationality of some generalized Kummer surface GK(b). This is the geometric quotient of the Weil restriction R-b := R-Fq2/F-q(E-b) under the order 3 automorphism restricted from Eb. More precisely, we apply the theory of conic bundles (i.e., conics over the function field F-q(t)) to obtain explicit and quite simple formulas of a birational F-q-isomorphism between GK(b) and A(2). Our point compression method consists in computation of these formulas. To recover (in the decompression stage) the original point from E-b(F-q2) = R-b(F-q) we find an inverse image of the natural map R-b - GK(b) of degree 3, i.e., we extract a cubic root in F-q. For q not equivalent to 1 (mod 27) this is just a single exponentiation in F-q, hence the new method seems