For example, in the question “Is there any reason not to use Single-Key EM with AES and a constant key?” it is proposed to use AES with a fixed key as the permutation in a xor-permute-xor construction. In the paper linked in the answer – “Key-Alternating Ciphers in a Provable Setting: Encryption Using a Small Number of Public Permutations” (PDF) – there is a proposal for a cipher that uses the full AES cipher with a fixed key(s).
Why is the full round AES required? How would the situation change if AES with only half the number of rounds was used? What about AES with only two rounds? If I recall correctly, two rounds of AES is the minimum required for complete diffusion of the input, so I wonder why 2 rounds wouldn't work just as well.
Why would one use an iterated construction, if the
XOR-permute-XOR construction is providing all the security?
What requirements does the permutation need to fulfill, specifically? In the paper “Minimalism in Cryptography: The Even-Mansour Scheme Revisited” (PDF) they appear to just use the term "fixed random permutation" or "publicly known permutation".