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My bachelor thesis is about TLS and I have a basic question about the algorithms used there.

If I use e.g. a sha1WithRSAEncryption signing algorithm (RFC3279) for my certificate, that means that the public RSA key of the certificate is signed using the SHA-1 hash.

Now does that mean that RSA has to be the key exchange, too or is the public key of the certificate used for authentication only? If I use RSA as key exchange and for my certificate, will I have two separate key pairs?

Sorry for the many (unstructured) questions, most articles explain either authentication or key exchange.

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that means that the public RSA key of the certificate is signed using the SHA-1 hash.

The certificate is signed and not the key. The key is part of the certificate but a certificate needs more, i.e. subject, issuer, when it is valid, how the key can be used ...

Now does that mean that RSA has to be the key exchange, too or is the public key of the certificate used for authentication only?

Since the authentication is done using the certificate and the certificate contains only the RSA key the authentication is done with RSA. The key exchange can be RSA but can also be Diffie-Hellman. The latter one is actually the recommended way. With TLS 1.3 RSA key exchange is no longer possible (i.e. always Diffie-Hellman) but RSA authentication can still be done.

If I use RSA as key exchange and for my certificate, will I have two separate key pairs?

With RSA key exchange the same key is used as for authentication - i.e. the only key which is contained in the certificate.

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