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Why can't we use an MDC (“Manipulation Detection Code”, as also handled in this Q&A) to provide origin authentication when using public-key encryption?

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Public key encryption uses a public key of the receiver; anybody can encrypt. So origin authentication would only work if you'd also have a shared secret key (in which case the whole public key encryption becomes kind of useless) or a private key (in which case you'd probably use a signature or an authenticated key agreement protocol).

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