# Cryptographic invariant maps

In [BGK+18] in section 4, Boneh et al. write that:

For any choice of ideal classes $$\mathfrak{a}_1,\dots,\mathfrak{a}_n,\mathfrak{a}_1',\dots,\mathfrak{a}_n'$$ in $${Cl}(\mathcal{O})$$, the abelian varieties \begin{align} (\mathfrak{a}_1 \star E) \times \dots \times (\mathfrak{a}_n \star E) \text{ and } (\mathfrak{a}_1' \star E) \times \dots \times (\mathfrak{a}_n' \star E) \end{align} are isomorphic over $$\mathbb{F}_q$$ iff $$\mathfrak{a}_1 \cdots \mathfrak{a}_n = \mathfrak{a}_1' \cdots \mathfrak{a}_n'$$ in $${Cl}(\mathcal{O})$$. In particular: \begin{align} (\mathfrak{a}_1 \star E) \times \dots \times (\mathfrak{a}_n \star E) \cong (\mathfrak{a}_1\cdots\mathfrak{a}_n) \star E \times E^{n-1} \end{align}

There is no proof in the paper and I did not succeed proving this myself. Can someone point me to a proof for these assertions?

• This is proved in the paper in Appendix A.4, at least up to the "if" part, and provides a reference for the "only if" part. See theorem A.1 which proves the second equation you mentioned, and then proves the first from it. Commented Jan 23, 2022 at 22:35