We investigate the security of Diffie-Hellman key exchange as used in popular Internet protocols and find it to be less secure than widely believed. First, we present Logjam, a novel flaw in TLS that lets a man-in-the-middle downgrade connections to "export-grade" Diffie-Hellman. To carry out this attack, we implement the number field sieve discrete logarithm algorithm. After a week-long precomputation for a specified 512-bit group, we can compute arbitrary discrete logarithms in that group in about a minute. We find that 82% of vulnerable servers use a single 512-bit group, and that 8.4% of Alexa Top Million HTTPS sites are vulnerable to the attack.a In response, major browsers have changed to reject short groups.
We go on to consider Diffie-Hellman with 768- and 1024-bit groups. We estimate that even in the 1024-bit case, the computations are plausible given nation-state resources. A small number of fixed or standardized groups are used by millions of servers; performing precomputation for a single 1024-bit group would allow passive eavesdropping on 18% of popular HTTPS sites, and a second group would allow decryption of traffic to 66% of IPsec VPNs and 26% of SSH servers. A close reading of published NSA leaks shows that the agency's attacks on VPNs are consistent with having achieved such a break. We conclude that moving to stronger key exchange methods should be a priority for the Internet community.
Diffie-Hellman (DH) key exchange is a popular cryptographic algorithm that allows Internet protocols to agree on a shared key and negotiate a secure connection. It is fundamental to protocols such as Hypertext Transport Protocol Secure (HTTPS), Secure Shell (SSH), Internet Protocol Security (IPsec), Simple Mail Transfer Protocol Secure (SMTPS), and other protocols that rely on Transport Layer Security (TLS). Many protocols use Diffie-Hellman to achieve perfect forward secrecy, the property that a compromise of the long-term keys used for authentication does not compromise session keys for past connections. We examine how Diffie-Hellman is commonly implemented and deployed with common protocols and find that, in practice, it frequently offers less security than widely believed.
There are two reasons for this. First, a surprising number of servers use weak Diffie-Hellman parameters or maintain support for obsolete 1990s-era "export-grade" cryptography. More critically, the common practice of using standardized, hard-coded, or widely shared Diffie-Hellman parameters has the effect of dramatically reducing the cost of large-scale attacks, bringing some within range of feasibility.