Finding algorithm: desire RSA's uniqueness and ECDSA's space efficiency
Yes!  There are verifiably pseudorandom functions or VRFs made out of elliptic curves including EC-VRF, related to ECDSA, and VXEdDSA, related to EdDSA.  In addition to getting a unique output for each message, the output is a pseudorandom function of the message, and you get a proof of uniqueness.

More references on VRFs.