Funny you would put it like that. You can turn the regular CRC algorithm (which is essentially a polynomial division) into a secure Wegman-Carter authenticator (like polynomial-evaluation MACs): the secret key key2 would be an irreducible CRC polynomial, and the scheme would loosely be PRF(nonce, key1) + CRC(message, key2) (see Shoup [1]).
Wegman-Carter authenticators, unlike hash-based ones, are information-theoretically secure. This means that they resist an attacker with infinite resources. In a way, they're the analog of the one-time pad for authentication. They also predate hash-based authenticators by decades: the first secure construction of this kind of authenticator goes back to 1974 [2]!
[1] http://www.shoup.net/papers/macs.pdf
[2] http://www3.alcatel-lucent.com/bstj/vol53-1974/articles/bstj...