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A keyed hash produces an output that depends both on the input data, and a key. If it is cryptographically secure, then it satisfies the above properties of a non-keyed hash, and in addition.
There are no known ways faster than bruteforce to find the key used, given a set of input and output pairs produced with that key.
There are no known ways better than random chance to find the correct output for any input under a particular key, without knowledge of the key.
Keyed Hashing is secret key is used as input for hashing along with message like HMAC.
Keyed hashing is usually used to build message authentication codes (MACs), the most common of which is the hashed-based MAC (HMAC).
MACs are basically cryptographic checksums. They are used to detect when an attacker has tampered with a message. Therefore they require a secret key (to be withheld from an attacker) and should be as fast as possible (to reduce overhead).