Are there attacks which will return the RSA private key when the attacker knows the public key, plain text, and cipher text when no padding is used in the RSA algorithm? If yes, what are the known attacks, under what conditions have they been successfully implemented, and what other cryptographically-secure asymmetrical encryption algorithms will produce identical cipher texts from the same plain text without the attacker being able to determine the private key through knowledge of the public key, the plain text, and the cipher text?
It is not important that the attacker knows the plain text or the cipher text because they will have access to that information along with the public key. What is important is that the private key can not be obtained and that the public-key encryption results of identical plain texts are the same.
For reference, the asymmetrical encryption usage is a subset of an algorithm which requires the characteristics specified above.
While there is a related question, there was no accepted answer and other responses to that question and similar questions caveat their answers with "when properly used", i.e., when RSA is implemented with padding: Finding Private key in RSA with public key, cipher text and plain text