# Chosen Cipher Text Security for Double Encryption

If $$E_k$$ is a CCA secure encryption scheme then why is $$E′_{(k_1,k_2)}(M)=E_{k_2}(E_{k_1}(M))$$, $$D′_{(k_1,k_2)}(C)=D_{k_1}(D_{k_2}(C))$$ not CCA secure when $$k_2$$ is known. Is there a way I can make a new cipher with keys $$(k_1, k_2)$$ such that its CCA secure regardless of which key the adversary finds?

My confusion here is how can I use the fact that key $$k_2$$ is known to break CCA security and why would that matter, because don't we treat these schemes as black-box when performing attack? What if I had $$k_1$$ instead, would that make any difference? And I'm quite clueless about the second part about new cipher using $$k_1, k_2$$ and still a secure CCA. Any idea or hint would be greatly appreciated.

• It is related to the bounds. You need to show that the advantage of the adversary os mo longer negligible according to the double the key size. Nov 13, 2020 at 11:41
• What is bounds?
– Alex
Nov 13, 2020 at 20:01
• The negligibility. Nov 13, 2020 at 20:01
• @kelalaka I do understand that advantage should be negligible for the cipher to be secure but how can I show that its not negligible is my question for the given cipher when $k_2$ is known? How to approach is what I'm looking for.
– Alex
Nov 13, 2020 at 22:51
• @kelalaka I'm now able to understand why the mentioned encryption scheme is not CCA secure when the $k_2$ is given!
– Alex
Nov 15, 2020 at 0:58

With the help of this paper, now I'm able to understand why for the mentioned encryption scheme it will be no longer CCA secure when the key $$k_2$$ is known! Do give it a read for those having issues understanding CCA multiple encryption schemes.