- If I encrypt a message with public key, I can only decrypt with private key.
True, except that public/private keys must match, and there might be several equivalent private keys (as in many definitions of RSA). We should thus say:
In a secure asymmetric encryption system, a message encrypted with a public key can only be decrypted with a matching private key.
- If I encrypt a message with private key, I can only decrypt with public key.
Encrypt with private key and decrypt with public key is infeasible in many cryptosystems, including those based on the Discrete Logarithm Problem, where public and private keys are fundamentally different. Encrypt with private key is also impossible in many implementations of RSA, and with many asymmetric encryption programs like PGP/GPG.
In an RSA encryption system, it is indeed can be mathematically defined how to to apply the encryption operation with the public exponent $e$ in the public key $(N,e)$ replaced by the private exponent $d$ from the private key. But if this heresy is performed, then anybody can decrypt with the original public key (which is public, see 6), and that goes against the primary goal of encryption. Don't do that.
In cryptosystems that perform signature rather than encryption, it is possible to sign a message with the private key, but that's not encryption. RSA can do both signature and encryption, but usual secure RSA-based signature and encryption cryptosystems have more differences than exchanging the role of public and private key.
- I can generate public key (derived key) from private key (base key).
Not in general. Cryptosystems based on the DLP indeed derive the public key from the private key (and the definition of the group, either public and fixed, or part of the private key). But that applies less to RSA.
- In RSA as in the original article, the private key can be considered to be $(N,d)$, where $d$ is chosen essentially at random. It is impossible to find the public key $(N,e)$ from the private key $(N,d)$ alone (for proper choice of $N$): it is also needed the factors of $N$, which are not necessarily part of the private key.
- In modern RSA, it is typically first chosen a small public exponent $e$, then drawn secret random prime factors $p,q$, forming $N=p\,q$, with $(N,e)$ the public key. Only then is the private exponent $d$ and the rest of the private key derived from $e,p,q$. So part of the private key is derived from data including a part of the public key.
What always holds it that a public/private key pair is derived from secret base data (output by a Random Number Generator), but the details of that derivation depend on the cryptosystem.
- I can not generate private key from public key.
True. Generating a private key matching a given public key would be a total break of any asymmetric cryptosystem.
- Private key name comes (from) how I handle the key: I keep it in secret, so I call it private key.
True. Notice that secret key is preferably reserved to symmetric cryptography, where the secret key is shared between cooperating parties; and private key is used for asymmetric cryptography, and assumed a secret that is not shared with anyone unless otherwise specified.
- Public key name comes (from) how (I) handle the key: I publish it, anybody can use it, so this is the public key.
True. There typically are additional differences beyond naming. That's the case with DLP, and often in RSA.
- If I publish the base key, I can call it public key.
No. The base key should not be published, if the base key is the private key, or is the secret base data that was used to generate a public/private key pair. Doing so would make the key pair useless, by revealing the private key matching the public key. Private keys must be kept secret, because knowledge of a private key allows to decipher, or forge signature.
- If I keep in secret the derived key, I can call it private key.
Not in general. In particular, not with DLP, where the "derived key" is bound to be the public key.