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source: http://www.cosic.esat.kuleuven.be/publications/thesis-152.pdf ,page 32

Note that key recovery implies invertibility of the cryptographic primitive (because due to the Kerckhoffs’ principle, the corresponding decryption routine is available). The inverse is not always true. PR-CPA secure ⇒ KR-CPA secure.

What does “invertibility of the cryptographic primitive” mean?

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"Invertibility of the cryptographic primitive" is just a fancy way of saying "you can decrypt things encrypted with that key." All the first sentence means is "if you can recover the key, you can decrypt anything encrypted with that key." As the second sentence points out, there are conceivably attacks where you can decrypt things encrypted with a key but can't recover the key itself; the reverse is not true, because if you have the key you can use it to decrypt things.

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