co-CDH ⇒ co-CDH without an isomorphism

Fully-black-box reduction · standard model · folklore · security loss: none

Statement

Let $\mathsf{GrGen}$ output a bilinear group pair with an efficiently computable isomorphism $\psi :{\mathbb{G}}_2\to {\mathbb{G}}_1$ for which co-CDH is hard, and let ${\mathsf{GrGen}}^{\prime }$ output the same pair with the generator $g_1=\psi (g_2)$ of ${\mathbb{G}}_1$. Then co-CDH without an isomorphism is hard for ${\mathsf{GrGen}}^{\prime }$, with the same advantage — folklore.

Sketch

The map $(g_2,g_2^a,h)\mapsto (h,\psi (g_2^a),g_2^a)$ sends a uniform co-CDH instance to a uniform instance $(g_1^{\alpha },g_1^{\beta },g_2^{\beta })$ with $g_1^{\alpha }=h$ and $\beta =a$, whose answer $g_1^{\alpha \beta }$ is $h^a$.