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I'm trying to examine some instances where an X.509 certificate uses public and private keys for a paper I'm going to write. However, the only thing I have explicitly seen so far is in one of the components where the subject's public key is identified, along with the algorithm and its parameters.

How else does this certificate use keys?

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A certificate does not "use" keys, it contains keys. That's its job. The certificate contains the public key of an "entity" (a human being, a job function, a server...) and an identity (which designates the said entity in an hopefully unambiguous way). Besides, the certificate is digitally signed by a certification authority, and that signature is to be verified with the CA's public key (which is not in the certificate we are talking about; rather, it is to be found in another certificate which has been issued to the CA itself).

All these public keys have corresponding private keys, which are, naturally, not in the certificate themselves (since they are private); each private key is mathematically linked with the corresponding public key, in a way which depends upon the algorithm type.

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