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I don't understand what an active s-box is (in reference of design criteria 8 of the DES).

Because according to DES, all 8 s-boxes are used every round. But this is not what active means.

So my question: What is an active S-Box, and what does it mean when an s-box is active (or when two or three s-boxes are active)?

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A linear transformation in a block cipher is also considered an substitution, just not a non linear one. When they talk about active s-boxes, they are talking about not specifically about the nonlinear s-box, but a level of input being substituted with a different output. Combining these across multiple rounds results in what they call 'active' s-boxes.

Specifically, they are talking in regards to linear/differential characteristics over a given round count. When the differences for the characteristic are either both zero or both nonzero, the characteristic is considered consistent. Any s-box where the input/output masks for this characteristic are nonzero is considered active.

Any substitution that passes through a linear transformation can then be branched into the next substitution. In AES, there is the mixcolumns transformation between the nonlinear s-box of one round and the next. This linear transformation has a branch number of 5, resulting in 5 active s-boxes across 2 rounds.

This is probably not the best description, it may be better to read the initial papers on linear and differential cryptanalysis and come up with a better definition, since mine may not be applicable to a Feistel cipher.

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