# The Logjam Attack

How Diffie-Hellman Fails in Practice:

We have uncovered several weaknesses in how Diffie-Hellman key exchange has been deployed:

Logjam attack against the TLS protocol.The Logjam attack allows a man-in-the-middle attacker to downgrade vulnerable TLS connections to 512-bit export-grade cryptography. This allows the attacker to read and modify any data passed over the connection. The attack is reminiscent of the FREAK attack, but is due to a flaw in the TLS protocol rather than an implementation vulnerability, and attacks a Diffie-Hellman key exchange rather than an RSA key exchange. The attack affects any server that supportsDHE_EXPORTciphers, and affects all modern web browsers. 8.4% of the Top 1 Million domains were initially vulnerable.Threats from state-level adversaries.Millions of HTTPS, SSH, and VPN servers all use the same prime numbers for Diffie-Hellman key exchange. Practitioners believed this was safe as long as new key exchange messages were generated for every connection. However, the first step in the number field sieve—the most efficient algorithm for breaking a Diffie-Hellman connection—is dependent only on this prime. After this first step, an attacker can quickly break individual connections.

The site says that my Safari 8.0.6 is vulnerable.

Their Imperfect Forward Secrecy paper (PDF):

Our calculations suggest that it is plausibly within NSA’s resources to have performed number field sieve precomputations for at least a small number of 1024-bit Diffie-Hellman groups. This would allow them to break any key exchanges made with those groups in close to real time. If true, this would answer one of the major cryptographic questions raised by the Edward Snowden leaks: How is NSA defeating the encryption for widely used VPN protocols?

The further fact is that in NFS,

you can arrange things so that almost all the discrete-logging effort depends only on the prime number p, and not at all on the specific numbers g and h for which you’re trying to take the discrete log. After this initial “precomputation” step, you then have a massive database that you can use to speed up the “descent” step: the step of solving of g^{a}=h (mod p), for any (g,h) pair that you want.It’s a little like the complexity class P/poly, where a single, hard-to-compute “advice string” unlocks exponentially many inputs once you have it. (Or a bit more precisely, one could say that NFS reveals that exponentiation modulo a prime number is

sort ofa trapdoor one-way function, except that the trapdoor information is subexponential-size, and given the trapdoor, inverting the function is still subexponential-time, but a milder subexponential than before.)The kicker is that, in practice, a large percentage of all clients and servers that use Diffie-Hellman key exchange use the same few prime numbers p. This means that, if you wanted to decrypt a large fraction of all the traffic encrypted with Diffie-Hellman, you wouldn’t need to do NFS over and over: you could just do it for a few p’s and cache the results. This fact can singlehandedly change the outlook for breaking Diffie-Hellman.

This work is the result of an unusual collaboration between a fantastic group of co-authors spread all around the world, including institutions such as the University of Michigan, INRIA Paris-Rocquencourt, INRIA Paris-Nancy, Microsoft Research, Johns Hopkins and the University Of Pennsylvania. It’s rare to see this level of collaboration between groups with so many different areas of expertise, and I hope to see a lot more like it. (Disclosure: I am one of the authors, but others did all the good bits.)

[…]

However, there is a second class of servers that are

capableof supporting 512-bit Diffie-Hellman when clients request it, using a special mode called the ‘export DHE’ ciphersuite. Disgustingly, these servers amount to about 8% of the Alexa top million sites (and a whopping 29% of SMTP/STARTLS mail servers).[…]

Here it is in a nutshell: if the server supports DHE-EXPORT, the attacker can ‘edit’ the negotiation messages sent from the a client -- even if the client doesn’t support export DHE -- replacing the client’s list of supported ciphers with

onlyexport DHE. The server will in turn send back a signed 512-bit export-grade Diffie-Hellman tuple, which the client will blindly accept -- because itdoesn’t realize that the server is negotiating the export version of the ciphersuite.From its perspective this message looks just like ‘standard’ Diffie-Hellman with really crappy parameters.

One of the problems with patching the vulnerability is that it breaks things:

On the plus side, the vulnerability has largely been patched thanks to consultation with tech companies like Google, and updates are available now or coming soon for Chrome, Firefox and other browsers. The bad news is that the fix rendered many sites unreachable, including the main website at the University of Michigan, which is home to many of the researchers that

foundthe security hole.This is a common problem with version downgrade attacks; patching them makes you incompatible with anyone who hasn't patched. And it's the vulnerability the media is focusing on.

Update (2015-10-15): Alex Halderman and Nadia Heninger:

However, the documents do not explain

howthese breakthroughs work, and speculation about possible backdoors or broken algorithms has been rampant in the technical community. Yesterday at ACM CCS, one of the leading security research venues, we and twelve coauthors presented a paper that we think solves this technical mystery.[…]

If a client and server are speaking Diffie-Hellman, they first need to agree on a large prime number with a particular form. There seemed to be no reason why everyone couldn’t just use the same prime, and, in fact, many applications tend to use standardized or hard-coded primes. But there was a very important detail that got lost in translation between the mathematicians and the practitioners: an adversary can perform a single enormous computation to “crack” a particular prime, then easily break any individual connection that uses that prime.

How enormous a computation, you ask? Possibly a technical feat on a scale (relative to the state of computing at the time) not seen since the Enigma cryptanalysis during World War II. Even estimating the difficulty is tricky, due to the complexity of the algorithm involved, but our paper gives some conservative estimates. For the most common strength of Diffie-Hellman (1024 bits), it would cost a few hundred million dollars to build a machine, based on special purpose hardware, that would be able to crack one Diffie-Hellman prime every year.