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I've looked up RC4 algorithm and it's seems easy enough but some parts of it seem random to me. In the key schedule part take for example the following part :

j = (j + S[i] + key[i % keylen]) % 256

I can understand most of it:

  • the key[i % keylen] is padding in-case that the key is shorter than 256 bytes, and
  • the % 256 is to ensure that j will be in the range of 0-255 when performing the swap.

But why do we need the j = j + S[i]?

Is there a mathematical explanation to this or is it just to add some randomness? Also regarding to the pseudo random generator part: can someone point me to a reference which offers an explanation for the not so obvious operations?

I've looked for explanations on the internet but couldn't really find anywhere that explain those bits.

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The correction question you should ask about why various operations in RC4 (or, for that matter, any other cipher) are there would be "if I were to remove that, what would the impact be? Would this weaken the cipher in some way?"

At your current state of knowledge, that may be a rather imponderable question, but it is still the correct one. I can try to give you a simple example: if you remove the $S[i]$ from the update of $j$, leaving

j = (j + key[i % keylen]) % 256

Then $j$ will follow a much simpler trajectory through the array as we run the key setup routine. In particular, if the key bytes just happen to sum to a multiple of 256 (which will happen 1/256 of the time for a random key), then $j$ will visit only (at most) keylen bytes of the array; all the other elements of the array will participate in one swap. Now, it's not immediately clear how a bias like this might influence the cipher; however it is quite plausible that it wouldn't be good.

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