# Full diffusion or Partial diffusion between s-box applications?

In regards to maximizing active s-boxes: is it advantageous to apply the non-linear layer after complete diffusion of the state, rather after then partial diffusion?

Using AES as an example, with the well known mixColumns and shiftRows combination: What if this linear diffusion layer were applied twice in succession between s-box applications, instead of just once?

Doing the math for the regular AES with 2 rounds is simple: \begin{align} 1 + 4 &= 5\\ 4 + 16 &= 20 \\ 5 + 20 &= 25\\ \end{align}

• The first number in each row is the number of different s-boxes at the input of the round.
• The second is the number of differences at the output of the round.
• The third is the sum of the two.
• The first row represents round 1
• The second row represents round 2
• Finally, the two rounds are summed to give the total minimum number of active s-boxes.

So to simulate a double application of the linear diffusion layer, I tried substituting 16 for 4, but was unsure of how to proceed: \begin{align} 1 + 16 &= 17\\ 16 + 256 &= 272\\ 17 + 272 &= 289\\ \end{align}

or should it be:

\begin{align} 1 + 16 &= 17\\ 16 + 32 &= 48\\ 17 + 48 &= 65\\ \end{align}

Or something else entirely? How many active s-boxes over the course of two rounds would such a variant of AES have?

Put succinctly, I guess I really want to know: Is there an cryptanalytic advantage to diffusing the entire state before applying the s-box layer? Or is it better to apply s-boxes more frequently, with "just enough" diffusion between them?