Let's suppose I have a 8-bit substitution box composed of 256-bytes and invert it.
Does changing one byte in the substitution box will make the inverse result completely different than if I didn't change it?
Or will only the changed byte be different in the inverse S-box?
If the forward Sbox is
$$[S(0),\ldots,S(255)]$$ it is normally a permutation. You cannot change only one byte and keep the permutation property. You need to change at least two bytes. THus the inverse will also be changed only in two bytes. To give an example if you had $$ [S(0),S(1),\ldots,S(255)]=[7,31,21,\ldots,72] $$ then you might do $$ [S(0),\ldots,S(255)]=[21,31,7,\ldots,72] $$ with only the first and third bytes changed. So in the inverse direction now you have $$ S^{-1}(21)=0,S^{-1}(7)=2, $$ while you originally had $$ S^{-1}(21)=2,S^{-1}(7)=0, $$ with everything else unchanged.
I we can inverse an 8-bit substitution box, this implies it is a permutation (example: the AES S-box).
If we change one byte in that table, then it is no longer a permutation, and can no longer be fully inverted: for one element (the new value) there will be two antecedents/premimages, and for one (the old value) there will be none.
If we exchange two (different) bytes in that table, then it remains a permutation, and can be fully inverted. In the table for the inverse, two bytes are exchanged (those which indexes are the values of the two bytes inverted in the direct table).
In both cases, the other 254 elements in the inverse table are unaffected.
