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→Practical Considerations
* The algorithms used in public-key cryptography are very slow. Therefore, it's common to use public key cryptography to exchange a key for a faster, symmetrical cypher, and then use that cypher for the actual data stream.
* ''Signing'' a message means authenticating it by taking a checksum (hash) of the message and encrypting that. This is faster than encrypting the entire message.
* Public key cryptography is very susceptible to the manipulation of public keys by an attacker interposed between the two parties at the time that the public keys are shared (e.g., when someone is downloading your public key from your website). This is called a ''Man in the Middle'' (MIM) attack. To protect against this, keys can be signed by a mutually-trusted third party (called a Certification Authority, or CA) -- this is how your browser communicates securely with your bank.
* Public key cryptography is susceptible to theft of the private key (e.g., hacking your system, or stealing your laptop). The private key is therefore often protected with a passphrase.
* Cryptographic systems can be broken by guessing the key. The easiest way to do this is to sequentially try all possible key values. As computers become faster, more keys can be tested in a reasonable length of time, so keys must be made longer. Each bit added to the length of the key doubles the number of possible key values, doubling the average length of time that it will take to guess the key.
