2

Suppose Alice wants to check if Bob has a certain keypair. Alice could send some random payload to Bob and let Bob sign that payload with his private key. Alice is now able to check if the signature matches because of the known public key.

What is this payload called?

4

While there may be other names, calling it a challenge seems appropriate. From the challenge-response Wikipedia page:

In computer security, challenge-response authentication is a family of protocols in which one party presents a question ("challenge") and another party must provide a valid answer ("response") to be authenticated.

Later on the article goes on to describe the symmetric encryption version of the situation you mention:

One way this is done involves using the password as the encryption key to transmit some randomly generated information as the challenge, whereupon the other end must return as its response a similarly encrypted value which is some predetermined function of the originally offered information, thus proving that it was able to decrypt the challenge. For instance, in Kerberos, the challenge is an encrypted integer N, while the response is the encrypted integer N + 1, proving that the other end was able to decrypt the integer N.

0

This is typically known as a proof of possession. The proof may be achieved through a challenge response protocol, but not always, eg PKI certificate requests can include a proof of possession of the private key as part of the request.

Proof of possession also typically occurs intrinsically as part of key agreement protocols, however in cases where you are not trying to negotiate say a session key, and just want to know that Alice knows the key, it is called proof of possession.

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