I read that we do not know if there exists an 8x8 sbox with differential uniformity = 2.
I suppose we cannot compute every possible sbox because there are $64!$ possible s-boxes. Am I right? Is this problem in $NP$?
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There are 256! possible 8x8 S-boxes (i.e., bijective functions from $\{0,1\}^8$ to $\{0,1\}^8$. This is an absolutely enormous number. You couldn't possibly enumerate all of them within the lifetime of the universe. So, yes, this is one reason why it is not straightforward to determine whether there exists such a S-box with differential uniformity 2. It's not a NP-hard problem. NP only applies to asymptotic problems. For a fixed S-box size, like 8x8, it's not an asymptotic problem and so the NP framework technically doesn't apply. |
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