How were the number of rounds for different key sizes of AES selected?

Question

The number of AES rounds increases with the key length. Why increase the number of rounds at all, and how were these round counts chosen?

Answer 1

A quote from The Design of Rijndael (Section 3.5 "The Number of Rounds")

Two rounds of Rijndael provide 'full diffusion' in the following sense: every state bit depends on all state bits two rounds ago, or a change in one state bit is likely to affect half of the state bits after two rounds. Adding four rounds can be seen as adding a 'full diffusion step' at the beginning and at the end of the cipher.

And also

For Rijndael versions with a longer key, the number of rounds was raised by one for every additional 32 bits in the cipher key.

Unfortunately no derivation of this magic 1:32 ratio is given.

Answer 2

There are two reasons:

• More rounds means more security against cryptanalysis, simply, since there is more confusion and diffusion.
• For a secure block cipher, there should be no attack faster than exhaustive key search (i.e. brute force). As exhaustive key search takes a lot longer for a larger key size, a theoretical attacker can afford more work to "break" the larger cipher. Thus we also increase the round number a bit to increase the security level of our cipher accordingly.
• For a larger key size (as well as a larger block size), we need more rounds so that every key bit affects every ciphertext bit in a similar way, i.e. without measurable differences which would allow any cryptanalysis.

The 10 rounds for AES-128 seem to be about the lower level of what is (approximately) 128-bit-secure, and 10 rounds for a AES-256-like-cipher would have way below 256 bits of security.