Change:
A single 0 byte which serves as the separator <del> The value Hash(Handshake Context || Certificate) where Hash is the hash function for the handshake. </del> <ins> If the authenticator request is present, the value "Hash(Handshake Context || authenticator request || Certificate)" If an authenticator request is not present, the value "Hash(Handshake Context || Certificate)" where Hash is the hash function for the handshake. </ins> A HMAC over the value Hash(Handshake Context || Certificate || <ins> CertificateVerify) if an authenticator is present, or Hash(Handshake Context || authenticator request || Certificate || </ins> CertificateVerify) where Hash is the hash function for the handshake, and the HMAC is computed using the hash function from the handshake and the Finished MAC Key as a key. <del> The certificates used in the Certificate message MUST conform to the requirements of a Certificate message in the version of TLS negotiated.  This is described in Section 4.2.3 of TLS13 and Sections </del> <ins> The certificates chosen in the Certificate message MUST conform to the requirements of a Certificate message in the version of TLS negotiated.  If an authenticator request is present, the signature algorithms used to choose the algorithm are taken from the "signature_algorithms" in the from the authenticator.  If there is no authenticator request, the signature algorithms are chosen from the "signature_algorithms" extension from the ClientHello used in the connection.  This is described in Section 4.2.3 of TLS13 and Sections </ins> 7.4.2 and 7.4.6 of RFC5246.  Alternative certificate formats such as <del> RFC7250 Raw Public Keys are not supported. The exported authenticator message is the concatenation of messages: </del> <ins> RFC7250 Raw Public Keys are not supported.  The "server_name" RFC6066, "certificate_authorities" (Section 4.2.4. of TLS13), or "oid_filters" (Section 4.2.5. of TLS13) extensions are used to guide certificate selection, with the extensions provided in the authenticator request taking precedence over the extensions provided in the connection handshake. If an authenticator request was provided, the Certificate message MUST contain only extensions present in the authenticator request. Otherwise, the Certificate message MUST contain only extensions present in the ClientHello. The authenticator message is the concatenation of messages: </ins> Certificate || CertificateVerify || Finished <del> A given exported authenticator can be validated by checking the validity of the CertificateVerify message and recomputing the Finished message to see if it matches. </del> <ins> A given authenticator can be validated by checking the validity of the CertificateVerify message given the authenticator request (if used) and recomputing the Finished message to see if it matches. </ins> <del> 4. </del> <ins> 5. </ins> <del> The creation and validation of exported authenticators SHOULD be implemented inside TLS library even if it is possible to implement it at the application layer.  TLS implementations supporting the use of exported authenticators MUST provide application programming interfaces by which clients and servers may request and verify exported authenticator messages. </del> <ins> The creation and validation of both authenticator requests and authenticators SHOULD be implemented inside the TLS library even if it is possible to implement it at the application layer.  TLS implementations supporting the use of exported authenticators MUST provide application programming interfaces by which clients and servers may request and verify exported authenticator messages. </ins> Given an established connection, the application SHOULD be able to <del> call an "authenticate" API which takes as input: </del> <ins> call the following APIs: "request", which takes as input: </ins> certificate_request_context (from 0 to 255 bytes) <del> valid certificate chain for the connection and associated extensions (OCSP, SCT, etc.) </del> <ins> set of extensions to include (this MUST include signature_algorithms) </ins> <del> signer (either the private key associated with the certificate, or interface to perform private key operation) </del> <ins> It returns an authenticator request, which is a sequence of octets that includes a CertificateRequest message. </ins> <del> signature scheme </del> <ins> "get context", which takes as input </ins> <del> The API returns the exported authenticator as output. </del> <ins> authenticator </ins> <del> Given an established connection and an exported authenticator message, the application SHOULD be able to call a "validate" API that takes an exported authenticator as an input.  If the Finished and CertificateVerify messages verify correctly, the API returns the following as output: </del> <ins> It returns the certificate_request_context. </ins> <del> certificate chain and extensions </del> <ins> "authenticate", which takes as input: </ins> <del> certificate_request_context </del> <ins> a set of certificate chains and associated extensions (OCSP, SCT, etc.) </ins> <del> In order for the application layer to be able to choose the certificates and signature schemes to use when constructing an authenticator, a TLS server SHOULD expose an API that returns the content of the signature_algorithms extension of client's ClientHello message. </del> <ins> a signer (either the private key associated with the certificate, or interface to perform private key operation) for each chain </ins> <del> 5. </del> <ins> an optional authenticator request </ins> <del> This document has no IANA actions. </del> <ins> It returns the exported authenticator as output.  It is RECOMMENDED that the logic for selecting the certificates and extensions to include in the exporter is implemented in the TLS library. Implementing this in the TLS library lets the implementer take advantage of existing extension and certificate selection logic. It is also possible to implement this API outside of the TLS library using TLS exporters.  This may be preferable in cases where the application does not have access to a TLS library with these APIs or when TLS is handled independently of the application layer protocol. "validate", which takes as input: * an optional authenticator request * an authenticator It returns the certificate chain and extensions. </ins> 6. <ins> This document has no IANA actions. 7. </ins> The Certificate/Verify/Finished pattern intentionally looks like the TLS 1.3 pattern which now has been analyzed several times.  In the case where the client presents an authenticator to a server, SIGMAC

Discussion:
Add new "authenticator request" message Make Certificate selection more explicit (including extensions) Update API considerations accordingly
I update the draft in response to the comments.
Two small comments: Can you add text explaining why authenticator requests are not encrypted with handshake keys? (Are they encrypted at all?) It might help to include a sequence diagram or two showing intended uses of exported authenticators, e.g., for Mike's draft, and where requests fit into TLS connections.
NAME I'm not sure what purpose encrypting them would serve. The draft does say that "The application layer protocol used to send the authenticator SHOULD use TLS as its underlying transport." In other words, the messages have integrity built-in and any desire for confidentiality is the app's to solve.
NAME I was referring to the following sentence: "This message is not encrypted with a handshake key, which differs from how this message is used in TLS 1.3." Perhaps you could add text explaining why this choice was made?
By the time you use this, handshake keys might be long gone. I think that we could avoid this confusion by stating that this is only the handshake messages and not records that form the authenticator. And maybe add that because there are no records, there is no record protection.