New point compression method for elliptic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">F</mml:mi></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:math>-curves of j-invariant 0

In the article we propose a new compression method (to 2⌈log2⁡(q)⌉+3 bits) for Fq2-points of an elliptic curve Eb:y2=x3+b (for b∈Fq2⁎) j-invariant 0. It is based on Fq-rationality some generalized Kummer surface GKb. This geometric quotient Weil restriction Rb:=RFq2/Fq(Eb) under order 3 automorphism restricted from Eb. More precisely, apply theory conic bundles (i.e., conics over function field Fq(t)) to obtain explicit and quite simple formulas birational Fq-isomorphism between GKb A2. Our point consists in computation these formulas. To recover (in decompression stage) original Eb(Fq2)=Rb(Fq) find inverse image natural map Rb→GKb degree 3, i.e., extract cubic root Fq. For q≢1(mod27) this just single exponentiation Fq, hence seems be much faster than classical one with x-coordinate, which requires two exponentiations