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I know counter mode can be CPA secure, when used with block ciphers modeled as random permutations. If we use pseudo-random permutations, is it still CPA secure? Is there a proof?

How about CBC mode? When used with block ciphers modeled as pseudo-random permutations, will it be CPA secure? Again, is there a proof?

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Standard security proofs for CTR and CBC deal with pseudo-random permutations (or pseudo-random functions). Relevant references can be found in Sections 4 and 5 of Rogaway's survey. However, it is unclear what you mean by a proof with random permutations. Where is the key then?

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