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I am looking for a 4-bit S-box which has only 4s in its difference distribution table (except the top left corner) but I was not able to find one in the literature.

Has such an S-box been already published? If not, is it theoretically possible to reach this property? If so, how to proceed?

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Yes. All $4\times4$ bit S-boxes have been classified. It is possible to reach a maximum differential of 4 as well as optimum nonlinearity.

SERPENT used optimal $4\times 4$ S-boxes with respect to differential cryptanalysis. So does PRESENT, to the best of my knowledge.

Some links

SAGE is a good way to learn about and experiment with S-boxes.

Edit: None of the optimal Sboxes' representatives from Saarinen have the property of all 4's you want. I have no time, but I believe it can be proved using Theorem 1

Theorem 1

in Blondeau-Nyberg which originally appeared in Eurocrypt 94 Chabaud-Vaudenay that such a DDT pattern would yield weak S-boxes against linear cryptanalysis. Here $n=m=4,$ $cor_x(a\cdot x \oplus b\cdot F(x))$ is the correlation bias of the corresponding linear characteristic, and $P[\delta \stackrel{F}{\rightarrow} \Delta]$ is the diferential probability.

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  • $\begingroup$ Although the maximum value in the DDTs of SERPENT and PRESENT is 4, some entries are also equal to 2: I'm looking for an S-box whose DDT contains 4s only. $\endgroup$ – Raoul722 Feb 6 at 21:14
  • $\begingroup$ I don't know if there is an equivalence class that contains such an example. So you want 4's and 0's. $\endgroup$ – kodlu Feb 6 at 23:27
  • $\begingroup$ why don't you go to the table at the end of the Saarinen paper and check the DDT of the 16 equiv class representatives using Sage? $\endgroup$ – kodlu Feb 6 at 23:30

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