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How can I generate some public and private keys on both sides with low overhead?

We have an Authority Center (CA) generating some public and private keys for each client (more than one pair for each client). One method for sending keys to client is sharing a secret key between CA ...
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Can I ensure that a public key derived from a private key is unique?

In asymmetric cryptography, is it possible to ensure that public key derived from a given private key is unique, no matter the algorithm?