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In chapter 3 of Katz Textbook, there is a theorem which states as follows:

"Any private key encryption scheme that is CPA-secure is also CPA-secure for multiple encryptions."

I can make sense of the theorem as CPA secure scheme requires randomized encryption and hence encrypting multiple messages will also be secure. How can I prove this formally?

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As is stated in the book, the analogous theorem for public-key encryption is proven in Section 12.2.2. The reason why the proof is delayed until then is that it is more complex than all proofs until this point. In particular, it requires the use of a hybrid argument, and this is introduced only in Chapter 8.

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