How to extract public key using OpenSSL

openssl genrsa -out mykey.pem 1024

will actually produce a public - private key pair. The pair is stored in the generated mykey.pem file.

openssl rsa -in mykey.pem -pubout > mykey.pub

will extract the public key and print that out. Here is a link to a page that describes this better.

EDIT: Check the examples section here. To just output the public part of a private key:

openssl rsa -in key.pem -pubout -out pubkey.pem

To get a usable public key for SSH purposes, use ssh-keygen:

ssh-keygen -y -f key.pem > key.pub

Public-key crypto is not for encrypting arbitrarily long files. One uses a symmetric cipher (say AES) to do the normal encryption. Each time a new random symmetric key is generated, used, and then encrypted with the RSA cipher (public key). The ciphertext together with the encrypted symmetric key is transferred to the recipient. The recipient decrypts the symmetric key using his private key, and then uses the symmetric key to decrypt the message.

The private key is never shared, only the public key is used to encrypt the random symmetric cipher.