# Does CCA-security implies CPA-security

While reading Katz & Lindell's textbook (2nd edition) if found three main security definitions: ING-EAV-Security, ING-CPA-Security and ING-CCA-Security. (From this forum I know there are more, but these are the most basic ones). I also read, that ING-CPA-Security implies ING-EAV-Security and that ING-EAV-Security is weaker than ING-CPA-Security which is weaker than ING-CCA-Security.

My question: Does ING-CCA-Security also implie ING-CPA-Security?

• Hint: How do the CCA definition(s) compare to the CPA ones, in particular the things an adversary can do? Mar 9, 2021 at 14:01
• I know what you mean: The CPA Experiment is included in the CCA experiment, therefore CCA should implie CPA, but because I couldn't find it in the book, I was/am unsure about it Mar 9, 2021 at 14:06
• In that case, could you perhaps find a proof yourself? That is, a reduction that if an adversary wins the CPA experiment against a scheme, it will also win the CCA one? Mar 9, 2021 at 14:11

My Sketch on the reduction proof:

1. Basic Assumption: I got a Cryptosystem #1 that is CCA secure

2. Based on this Cryptosystem I can build a CPA secure Cryptosystem #2, and I assume, that an adversary can break it with non negligible probability

3. Therefore I can build a Reduction: R can simulate the CPA game with A and just gives the messages of A to his challenger and the response back to A. R then outputs the result of A.

4. Because A has a non negligible probability in breaking #2, R also has a non negligible probability in breaking #1, which gives a contradiction to the basic assumption

5. Therefore CCA implies CPA

I hope the basic idea is correct.

• And, a good exercise is to write it in a more formal way. Mar 9, 2021 at 18:13