Suppose I have a secret value $a$ which maps to a public point on an elliptic curve $A = a \cdot G$, where $G$ is a generator of the elliptic curve of primer order $q$.
Can I prove to someone that the least significant bit of $a$ is $0$ using a non-interactive proof that does not expose the value of $a$? If I can, what would be the most efficient (in terms of proof size) way to do it?
Edit: this can probably be done using bit commitments (something similar to Confidential Transactions), but in this case, proof sizes are likely to be close to 10KB. Is there a way to do this more efficiently? Maybe somehow relying on the fact that $a$ is even?
Edit2: I've been reading a bit about Division Polynomials and I'm wondering if it is possible to use them somehow to prove that $a$ is even.