Encrypt with private key and decrypt with public key

Question

What I'd like to know is simple. Can I use OpenSSL to encrypt a string "hello" with a private key then send it to everyone who can decrypt it with the public key to retrieve the original string.

I've searched all around and can't really find anything.

Mathematically I can use the private exponent and public modulus to perform an encryption then use the public exponent and public modulus to perform the decryption.

Can I do this with OpenSSL?

I've read that this is considered signing. Then how can I sign with my private key and receive the original data from the signed file with just the public key?

I've read a lot of the responses on the web and they're all vague. Can anyone give me clear solutions?

If it is possible to encrypt with a private key and decrypt with public can anyone give me an example on how to do it with the openssl tool? Or do I have to write my own implementation?

Solution

Encrypting with the private key is not considered signing. Using RSA, there is in fact encryption using the private key, but this is just an implementation detail of RSA, and it is not encrypting the message, but a hash of the message, so no, verifying the signature does not bring back the original plaintext message.

Yes you can sign and verify signatures in OpenSSL -- cf. https://www.openssl.org/docs/crypto/RSA_sign.html

No, you should not use "textbook RSA" with the modulus and exponent to roll your own encryption. Cf. here, for example: https://crypto.stackexchange.com/questions/1448/definition-of-textbook-rsa

No, you should not swap the use of the private and public keys by encrypting data with the private key and decrypting with the public. Cf. for example https://stackoverflow.com/a/2350959/233596.

UPDATE:

This page suggests that you can use the OpenSSL C interface to encrypt with the private key and decrypt with the public key, by way of these function prototypes:

 int RSA_public_encrypt(int flen, unsigned char *from,
    unsigned char *to, RSA *rsa, int padding);
 int RSA_private_decrypt(int flen, unsigned char *from,
    unsigned char *to, RSA *rsa, int padding);
 int RSA_private_encrypt(int flen, unsigned char *from,
    unsigned char *to, RSA *rsa,int padding);
 int RSA_public_decrypt(int flen, unsigned char *from, 
    unsigned char *to, RSA *rsa,int padding);

(I did not actually try using these functions.)

I tried using the openssl rsautl command line:

$ openssl rsautl -in HELLO -out HELLO.encrypt_by_private -inkey private.pem -encrypt

However:

$ openssl rsautl -in  HELLO.encrypt_by_private -pubin -inkey public.pem -decrypt
A private key is needed for this operation

So, I would say that the command line tool will not do it.