Alice has a private key, $x$, and a public key $P = [x] \cdot G$ in a group of order $n$.
Alice would like to also publish her inverse public key (inverted modulo the group order) $P_{inv} = [x^{-1} \mod n] \cdot G $, for the purposes of a simple homomorphic encryption algorithm that allows people to re-encrypt documents for her. For simplicity, I'm going to assume that that system is otherwise perfectly secure.
Alice is using a pairing-based curve, and that same public key $P$ is used in a number of pairing operations, signing operations, and DH key exchanges that could be observed by an attacker.
Can this be used to make an attack on Alice's private key easier? Now we have 2 equations - instead of one.
The closest related proof I've found is: Variations of Diffie-Hellman Problem, which demonstrated that it's hard to get $P_{inv}$ from $P$.
Does anyone know of a paper that uses $P_{inv}$ or uses published public "inversions"? of keys in this way?