Sounds like they have cracked RSA for 256bit and 512bit keys. But don't worry, it doesn't affect Bitcoin, because it doesn't use RSA anywhere, it uses ECDSA. RSA is known to have problems, while ECDSA is considered pretty safe at the moment. So you really should only watch for algorithms that are used in Bitcoin, the fact that some other algorithms get broken don't affect it.

congratulations they broke a ~4 bit security level key in their attempt

but jokes aside in cryptography the size is not the only thing that matters, the estimated "security level" that the keys provide is the important thing. size is only one factor that determines that level. for instance for Elliptic Curve Cryptography (ECC) the security level is half the key size but in Rivest–Shamir–Adleman (RSA) algorithm the security level is a lot lower than just half (calculated like this).
in short you can refer to the table provided in Standards for Efficient Cryptography (SEC 1):


the achievements that this paper is about, is the Factorization they are using and the optimization that has led to which is reducing the time it takes to break RSA by a lot.

People who knows very little about algorithms will get confused with all the different technologies and how difficult it is to crack it. They hear the word quantum technology and they think all the algorithms are hack-able. 

In any way, let's not get all rallied up about the RSA encryption, because it's one of the easiest algorithms to crack.    Cryptography is also a evolving technology, so we will always have stronger encryption being developed all the time. 

When ECDSA gets cracked, we're all done. Bitcoin will be completely destroyed/devaluated and that's just one thing. Who knows how many companies out there use ECDSA. Watching how technology gets developed, we can't say anything about bitcoin's security. #afraiding_of_quantum_computers

Why so much impatience about that? It will mean the end of bitcoin.

Actually, we are not using RSA (Rivest Shamir Adleman) algorithm at all in fact we are using ECDSA (Elliptic Curve Digital Signature Algorithm).
Though RSA and and ECDSA both have asymmetric encryption, they do have their differences.

RSA uses prime factorization method for it's encryption while ECDSA use elliptic curves.
It is already proven that ECDSA are better in security than RSA as of current methods of cracking.