According to theorem 3.13, the 6 rounds of AES is 0.472-close to pairwise independence. It is also mentioned t-wise independence used to analyze higher order derivates attacks. it is also mentioned 3-wise indepdent permutations have a potential application in strengthening short encryption keys. My questions are related to the fundamentals of t-wise independence permutations.
Q.1 What does it mean to have 2, 3 or 4-wise independence to a block cipher from a mathematical perspective?
Q.2 what does it mean 6 rounds of AES is 0.472-close to pairwise independence?
Q.3 (assumption) Does higher order independence have opposite relation with key size but same security level?