After swapping random indexes of trivial involution $S[x] = ¬x$, I found the following involution S-Box with a non-linearity of $98$ and a differential uniformity of $6$.

Is there an upper limit to maximum non-linearity attainable for an involution S-Box?

For example: the Anubis cipher S-Box has a non-linearity of $96$ and a differential uniformity of $8$. Will the Anubis cipher become stronger if I use the S-Box I found?

byte[] S = new byte[256]
  • $\begingroup$ are you sure the nonlinearity of the Anubis s-box is 96? my test says it is 94 $\endgroup$ – Richie Frame Jun 4 '16 at 9:51
  • $\begingroup$ Should consider trying the original Anubis s-box. The current S-box is the product of 2 4-bit s-boxes, to make hardware implementations easier. Presumably the first s-box would be more secure as there is a limit to security of a random 4-bit s-box as opposed to a random 8-bit s-box. $\endgroup$ – user3201068 Nov 24 '16 at 4:53

Assuming your differential uniformity and non-linearity figures are correct, then yes, your s-box is slightly stronger against basic differential and linear cryptanalysis. Although Anubis already was essentially immune to basic differential and linear cryptanalysis. However your s-box would need to be evaluated against other forms of cryptanalysis (e.g. interpolation, invariant subspace, etc) in order to determine if it was "better" in a more general sense. It may also be less efficient in terms of hardware implementation than the current Anubis s-box (to invoke another, no-less crucial sense of "better").

Note: The AES s-box is affine-equivalent to an involution (as will be obvious if you think about the multiplicative inversion function), and has a higher non-linearity and lower differential uniformity. Non-linearity and differential uniformity are the same for all affine-equivalent permutations. So that should help address your question about the maximum non-linearity of an involution.

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  • $\begingroup$ His figures are correct. Autocorrelation is 96, SSI is 251392, SAC and 2nd order SAC relative error is 21.9%, AVAL rel error is 4.69%, 2nd order AVAL rel error is 7.42%, BIC is 0.2288, 2nd order BIC is 0.28139 $\endgroup$ – Richie Frame May 31 '16 at 20:22
  • $\begingroup$ "SSI is 251392". what do you mean by SSI. $\endgroup$ – vinu Jun 1 '16 at 4:05
  • $\begingroup$ @vinu - you need to include an "@Richie Frame" at the start of your comment to ensure he sees it. $\endgroup$ – J.D. Jun 1 '16 at 9:26
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    $\begingroup$ @EllaRose - I don't have a link, but the intuition is this - if two functions $f(x)$ and $g(x)$ are affine equivalent to each other, that means $f(x) = a(g(b(x)))$ where $a(x)$ and $b(x)$ are affine functions. In other words $f(x)$ and $g(x)$ are the same s-box except one is composed with some affine functions. Differences propagate through affine functions with probability 1, so those affine functions cannot change the differential probabilities of the core 'shared' s-box. Hence the two functions will have the same max differential probability (same uniformity). $\endgroup$ – J.D. Mar 3 '17 at 0:42
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    $\begingroup$ @EllaRose - P.S. if that short explanation doesn't suffice I think the topic of how affine equivalence affects differential uniformity could make for a good question . . . $\endgroup$ – J.D. Mar 3 '17 at 0:53

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