Taking for example, suppose there are now many butterfly diagrams stacked to form a multi-layer network. Obviously, when each computing unit is linear, the entire network can be simplified using a matrix operation, such as FFT. It can be expressed as X=Fx, where F is a special Vandermonde matrix. When layer = n, $X=F^nx$.
Further more, when each computing unit is simple nonlinear, such as square or left right permutation, it is also easy to analyze and obtain the final expression.
But when computing units become more complex, such as Sha-256 and AES, it seems extremely difficult to simplify it. There is a method called Walsh spectral analysis in AES, but it seems like it's just an approximation. Due to all known information and one-to-one correspondence between input and output, there must be a final expression. But what is the cost of finding this expression? Can this final expression reduce the total computational cost?
