In the article we propose a new compression method (to $2\lceil {\log }_2(q)\rceil +3$ bits) for the ${\mathbb{F}}_{q^2}$-points of an elliptic curve $E_b:y^2=x^3+b$ (for $b\in {\mathbb{F}}_{q^2}^{⁎}$) of j-invariant 0. It is based on ${\mathbb{F}}_q$-rationality of some generalized Kummer surface $G{K}_b$. This is the geometric quotient of the Weil restriction $R_b:={\mathrm{R}}_{{\mathbb{F}}_{q^2}/{\mathbb{F}}_q}(E_b)$ under the order 3 automorphism restricted from $E_b$. More precisely, we apply the theory of conic bundles (i.e., conics over the function field ${\mathbb{F}}_q(t)$) to obtain explicit and quite simple formulas of a birational ${\mathbb{F}}_q$-isomorphism between $G{K}_b$ and ${\mathbb{A}}^2$. Our point compression method consists in computation of these formulas. To recover (in the decompression stage) the original point from $E_b({\mathbb{F}}_{q^2})=R_b({\mathbb{F}}_q)$ we find an inverse image of the natural map $R_b\to G{K}_b$ of degree 3, i.e., we extract a cubic root in ${\mathbb{F}}_q$. For $q\not\equiv 1(\mathrm{mod}27)$ this is just a single exponentiation in ${\mathbb{F}}_q$, hence the new method seems to be much faster than the classical one with x-coordinate, which requires two exponentiations in ${\mathbb{F}}_q$.

在本文中，我们提出了一种新的压缩方法（ $2\lceil {\mathrm{日志}}_2（q）\rceil +3$ 位） ${\mathbb{F}}_{q^2}$椭圆曲线的点 ${Ë}_b：{ÿ}^2=X^3+b$ （对于 $b\in {\mathbb{F}}_{q^2}^{⁎}$）的j-不变量0。它基于${\mathbb{F}}_q$广义Kummer曲面的非理性 $G{ķ}_b$。这是Weil限制的几何商${\mathrm{[R}}_b：={\mathrm{[R}}_{{\mathbb{F}}_{q^2}/{\mathbb{F}}_q}（{Ë}_b）$ 在3阶自同构下 ${Ë}_b$。更确切地说，我们应用圆锥束理论（即，圆锥在函数域上${\mathbb{F}}_q（Ť）$）来获得两分式的明确且非常简单的公式 ${\mathbb{F}}_q$之间的同构 $G{ķ}_b$ 和 ${\mathbb{一种}}^2$。我们的点压缩方法在于计算这些公式。恢复（在减压阶段）原始点${Ë}_b（{\mathbb{F}}_{q^2}）={\mathrm{[R}}_b（{\mathbb{F}}_q）$ 我们发现自然图的反像 ${\mathrm{[R}}_b\to G{ķ}_b$ 的度数为3，即我们在 ${\mathbb{F}}_q$。对于$q\not\equiv \mathrm{1个}（\mathrm{模}27）$ 这只是一个指数 ${\mathbb{F}}_q$，因此新方法似乎比带有x坐标的经典方法快得多，后者需要在x坐标上加两个幂${\mathbb{F}}_q$。