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When to use symmetric keys, and when public+private key pair?

Symmetric keys are much faster than use of public+private key pair. That is, encryption and decryption using symmetric keys is around 1000 times quicker than doing it with public+private key pair.

However public+private keys are more practical with larger groups of people exchanging messages. If they choose symmetric key encryption, they need symmetric key for each pair of members in a group. With public+private key pair, only two keys are needed for each member of this group.

These approaches can be combined.

Session key, is symmetric key encrypted with public key. That is, message is encrypted with session key, then session key is encrypted with public key and both are sent to destination.

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