Let F denote the finite field with 256 elements, and let n be a positive integer relatively prime to 255. Let A be a random bijective F_2 linear transformation from F to F. Let S(x)=A*(x^n). Then S is a permutation of F. In this visualization, the quantity n is denoted as 'power'.

If j is an element of F, then let f_j(x)=S(j+x)-S(j). Then the animation shows the spectrum of \sum_j f_j where each f_j is considered as a permutation matrix and where we take the sum over all elements of F. Each of these spectra contains two outliers (one of these outliers occurs since the permutations f_j all leave 0 fixed), so we remove these two outliers in the animation.

When n=255-2^i for some i less than 8, the function S closely resembles the AES S-box as the AES S-box is of the form S(x)=A*(x^(-1))+b for a well-chosen matrix A and vector b . In this case, the spectrum of S forms a circular pattern because the sum \sum_j f_j is nearly orthogonal after scaling. When n=255-2^i, we also observe that all of the eigenvalues of \sum_j f_j except for the two outliers have magnitude around 16. This distribution of the eigenvalues is beneficial for the security of the AES S-box. Consider the Markov chain on F^2 where we transition from (x,y) to (j+S(x),j+S(y)) where j is randomly selected from F. Since the eigenvalues of \sum_j f_j (except for the outliers) are all around 16, this Markov chain will have a very fast mixing time, and such a fast mixing time is what we want for cryptography. We also observe that for the AES S-box, we get improved cryptographic security by using S(x)=A*x^(-1) for some invertible matrix A instead of S(x)=x^(-1) since for S(x)=x^(-1), the eigenvalues of \sum_j f_j will have magnitude up to 18 after we remove the outliers.

If n=2^i for some i, then the mapping S is linear, and in this case, the function S cannot be used for any cryptographic purpose.

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