# Does repeated xoring of the (same) key K lower the entropy of K?

If a symmetric encryption key $$K$$ has been generated by some recommended (NIST for example) RBG then we should have a key with high entropy.

But if we xor other bit-strings (e.g. IVs, other keys, etc.) to $$K$$, how does this affect the level of the entropy of the resulting bit-string?

Specifically, how is the level of entropy affected if we apply:

1. $$K \oplus K^*$$, where $$K^*$$ has the same/similar level of entropy as $$K$$.
2. $$K \oplus IV$$, where $$IV$$ has a much lower entropy than $$K$$.
3. $$K \oplus C$$, where $$C$$ is some ciphertext output that is (repeatedly) fed back into the cipher to begin a new round.

In short, I am interested in the effects of entropy levels when a 'strong entropy' bit string is xored with a 'weak entropy' bit string or another 'strong entropy' bit string.

Related to Q3, I wonder how many times an output can be xored with the same key before entropy is lowered (assuming that happens at all). We assume the output $$C$$ is different each time.

It seems this cannot happen because the output $$C$$ of a cipher should necessarily be unpredictable, but that is on the assumption that xoring two high entropy bit strings will always retain a high entropy.

1. Is there a simple mathematical way to show entropy is retained or lost when two bit strings are xored?

If I take a strong key K of length $$n$$ taken from a good source of randomness, it will have $$n$$ bits of entropy and will be independent of anything else (not derived from it).
If we later Xor it with a weak password K* for instance or something else we still have the same $$n$$ bits of entropy.