In the context of a private cryptographic key suspected to be exposed, the best immediate action is to revoke the certificate associated with that key. Revoking the certificate ensures that it cannot be used to establish new secure sessions, which prevents attackers from using the potentially compromised key to impersonate or decrypt communications. The revocation process typically involves updating the Certificate Revocation List (CRL) or leveraging the Online Certificate Status Protocol (OCSP), both of which are used by clients to check the validity of certificates.