Equivalent key size between HMAC and AES?
Well, there are two potential key recovery attacks against HMAC (assuming a reasonable hash function):


Brute force the key; that is, take a valid (Message, MAC) pair, and try every possible key, and look for a key that gives that MAC for that Message
Brute force the internal hashing state immediately after processing the IPAD/OPAD; here, you would take a valid (Message, MAC) pair, and try every possible key, and look for a key that gives that MAC for that Message


Now, for first attack would take $O(2^k)$ time (for a key size of $k$); the second attack would take $O(2^{2n})$ time (if the hash function had an internal state size of $n$ bits).

With realistic key sizes and hash functions, $2^k &lt; 2^{2n}$; hence the best attack is an exhaustive search of the key space; this gives us the same approximate strength as AES for a given key size.