I'm wondering if the two algorithms are similar and whether DSA was really just based off RSA?
- Is DSA based on RSA and public key encryption?
- If so how does DSA introduce signing and makes it different from RSA?
DSA is the Digital Signature Algorithm, and exists next to the RSA Signature and ECDSA (Elliptic Curve DSA) as Digital Signature Standard.
Both DSA and RSA are based on public key cryptography, although RSA was rather an asymmetric encryption method. While you can not encrypt using DSA, you can sign using DSA quite fast, although you need longer to verify your signature, and DSA creates smaller signatures than RSA using the same key size.
RSA uses the public key to encrypt messages, the owner of the private key uses his key to decrypt the message
DSA was designed so the owner uses the private key to sign the messages, and anyone can use the public key to verify the message.
Wikipedia provides some good explanation and examples on how to sign with DSA, encrypt with RSA, and how to use RSA Signature to sign.
DSA is not a variant of RSA. If anything, DSA is a variant of the Schnorr signature scheme, modified in order to make signatures shorter, and, more importantly, to make DSA out of reach of the Schnorr patent (which has long expired). DSA itself was promoted by NIST as a US Federal standard because RSA was also covered by a patent at that time (this one has expired too).