I have a follow up question to Given a private key, is it possible to derive it’s public key?
Are the public and the private keys the 'same' (in the sense that you just choose to make one public) or can you do more with the private key than with the public key?
EDIT - to better state my question:
When the two keys are generated, could I just randomly choose one of them to be the public key?
From http://www.webopedia.com/TERM/P/public_key_cryptography.html:
A cryptographic system that uses two keys -- a public key known to everyone and a private or secret key known only to the recipient of the message. When John wants to send a secure message to Jane, he uses Jane's public key to encrypt the message. Jane then uses her private key to decrypt it.
An important element to the public key system is that the public and private keys are related in such a way that only the public key can be used to encrypt messages and only the corresponding private key can be used to decrypt them. Moreover, it is virtually impossible to deduce the private key if you know the public key.
No. That is the idea of generating a pair of keys in PPK world. You typically encrypt with the public key and decrypt with the private key. So you'd share the public key with your friends and ask them to use it when they send you their bank account number.
It really depends on what you call "private key." In almost every practical sitation, the sender knowing the private key also knows the public key. It provides others with its public key so needs to know it. So in essense, that "private key" will contain "public key."
From a theoretical perspective that applies to RSA cryptosystem, probably the primary difference is how you generate the exponent. For public key, the exponent is selected randomly with some criteria that reduces the possibility of attacks and the private exponent is calculated from it.
If the private exponent was selected randomly, you couldn't be sure that the resulting public exponent would have the characteristics so that it couldn't be cracked easily.
Some paper descriptions present roles of public and private keys as quite symmetrical but you definitely can't swap roles of private and public key in real world.
Common usage:
There is several reasons for that:
you don't want to leave a choice to the user as to which key should be published and which not. The public key is published worldwide and you can consider it as your public identity. The private part is needed when you have to prove to someone else that you have more insight than others about this identity: you can read messages sent to it, you are able to sign messages that can be verifyed by anyone who knows your public id. If what part of public/private key to publish were left to the user you'll end end with users publishing both. But that's not the main reason.
when you have private keys, you really have both keys every common implementation I know offer tools to extract public keys from private files. That's true for pgp, gpg, openssl. It means so called private key files store both private and public keys as described in algorithms. That's by design.
For exemple with openssl the sequence of commands to generate a RSA key pair can be:
openssl genrsa -out private.key 2048
openssl rsa -in private.key -pubout -out public.key
It should be clear enough that the first command generate both keys in the private key file and that the public key is merely extracted from it.
The consequence is that if your private key is ever compromized, both your keys would be compromized. The other way around is secure, you can't deduce the private key if you know the public key neither from the file nor from a mathematical attack.