The notion of misuse-resistant authenticated encryption (MRAE) was defined in the symmetric setting by Rogaway and Shrimpton, along with the SIV mode that achieves it. The idea being that modern authenticated encryption cipher modes require a unique nonce and fail more or less catastrophically if a nonce is reused. An MRAE cipher mode maintains a strong definition of security even in this case, although not semantic security.
If we look at digital signatures, then there are also unique values that must be used. This has been a classic vulnerability in ECDSA, e.g., in the Playstation 3 hack or the attacks on Android Bitcoin wallets. The Ed25519 signature scheme is deterministic to avoid these problems, replacing the random value with a hash/PRF over the secret key and the message. RFC 6979 proposes a deterministic variant of ECDSA that is along similar lines. Both seem close in spirit to the SIV construction: replacing a random/unique value with a psuedorandom one generated from the message and a secret key.
But what about asymmetric encryption? For RSA it is common to use either PKCS#1 v1.5 or OAEP padding. Both require random elements for security, and it seems to me that both would not achieve semantic security (IND-CPA) if these repeat as RSA is otherwise deterministic. But it is not clear to me to what extent security is lost in this case. While the deterministic authenticated encryption (DAE) that SIV ensures in this case is not appropriate for public key encryption, does RSA-OAEP achieve the notion of deterministic privacy described in Appendix B of http://web.cs.ucdavis.edu/~rogaway/papers/keywrap.pdf? If not, has anyone ever considered an SIV-like construction for RSA? On the face of it, it would seem you could do the same trick and replace the random seed value in OAEP with the output of a PRF calculated over the message and the private key (and potentially a random nonce too). Would this be secure?