How big an RSA key is considered secure today?

Question

I think 1024 bit RSA keys were considered secure ~5 years ago, but I assume that's not true anymore. Can 2048 or 4096 keys still be relied upon, or have we gained too much computing power in the meanwhile?

Edit: Lets assume an appropriate padding strategy. Also, I'm asking both about security of signatures and security of data encryption.

Answer 1

In the first decade of the 21th century, and counting, on a given $Year$, no RSA key bigger than $(Year-2000)\cdot 32+512$ bits has been openly factored other than by exploitation of a flaw of the key generator. This linear estimate of academic factoring progress should be used neither for long-term predictions (after 2016 or 1024-bits) nor for choosing a key length.

The current factoring record is 768 bits, by the end of 2009, and quoting this:

it is not unreasonable to expect that 1024-bit RSA moduli can be factored well within the next decade by an academic effort.

Update: I emphasize that the above is about attacks by academics. So far, hackers have always been some years behind (see below). On the other hand, it is quite conceivable that well funded government agencies are many years ahead in the factoring game. They have the hardware and CPU time. And there are so many 1024-bit keys around that it is likely a worthwhile technique to be in a position to break these. It is one of the most credible explanation for claims of cryptanalytic breakthrough by the NSA.

By 2012, the main practical threat to systems using 1024-bit RSA to protect commercial assets is usually not factorization of their key (often, that is penetration of the IT infrastructure by other means). With 2048 bits or more we are safe from that factorization threat for perhaps two decades, with fair (but not absolute) confidence. For varying estimates on how big is safe enough, see this site on keylength.

Update 2: Factorization progress is best shown on a graph (to get at the the raw data e.g. to make a better graph, edit this answer)

This also shows the linear approximation at the beginning of this answer, which actually is a conjecture at even odds for the [2000-2016] period that I made privately circa 2002, and committed publicly in 2004 (in French). Also pictured are the three single events that I know of hostile factorization of an RSA key (other than copycats of these events or by exploitation of a flaw of the key generator):

• The Blacknet PGP Key in 1995. Alec Muffett, Paul Leyland, Arjen Lenstra and Jim Gillogly covertly factored a 384-bit RSA key that was used to PGP-encipher "the BlackNet message" spammed over many usenet newsgroup. There was no monetary loss.

• The French "YesCard" circa 1998. An individual factored the 321-bit key then used (even though it was clearly much too short) in issuer certificates for French debit/credit bank Smart Cards. By proxy of a lawyer, he contacted the card issuing authority, trying to monetize his work. In order to prove his point, he made a handful of counterfeit Smart Cards and actually used them in metro tickets vending machine(s). He was caught and got a 10 months suspended sentence (judgment in French). In 2000 the factorization of the same key was posted (in French) and soon after, counterfeit Smart Cards burgeoned. These worked with any PIN, hence the name YesCard (in French). For a while, they caused real monetary loss in vending machines.

• The TI-83 Plus OS Signing Key in 2009. An individual factored the 512-bit key used to sign downloadable firmware in this calculator, easing installation of custom OS, thus making him a hero among enthusiasts of the machine. There was no direct monetary loss, but the manufacturer was apparently less than amused. Following that, many 512-bit keys (including that of other calculators) have been factored.

Update 3: Definitely, 512-bit RSA is no longer providing sizable security. Despite that, reportedly, certificates with this key size have been recently issued by official Certification Authorities, and used to sign malware, possibly by mean of an hostile factorization.

Answer 2

You might want to look at NIST SP800-57, section 5.2. As of 2011, new RSA keys generated by unclassified applications used by the U.S. Federal Government, should have a moduli of at least bit size 2048, equivalent to 112 bits of security. If you are not asking on behalf of the U.S. Federal Government, or a supplier of unclassified software applications to the U.S. Federal Government, other rules might of course apply.

However, at the very least, these figures indicate what the U.S. Federal Government thinks about the computational resources of it's adversaries, and presuming they know what they are talking about and have no interest in deliberately disclosing their own sensitive information, it should give some hint about the state of the art.

Answer 3

The simplest answer would be to look at the keylength.com site, and if you don't trust that, to the linked papers, particularly by NIST and ECRYPT II. Bare in mind that you may have additional restrictions and - if you are brave or stupid - relaxations depending on the use case.