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We propose a public-key encryption scheme and key agreement protocols based on a group action on a set. We construct an implementation of these schemes for the action of the class group $\mathcal{CL}(\mathcal{O}_K)$ of an imaginary quadratic field $K$ on the set $\mathcal{ELL}$ p,n $(\mathcal{O}_K)$ of isomorphism classes of elliptic curves over $\mathbb{F}_p$ with $n$ points and the endomorphism ring $\mathcal{O}_K$. This introduces a novel way of using elliptic curves for constructing asymmetric cryptography.

Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves