This paper investigates the security of Diffie-Hellman key exchange as used in popular Internet protocols and find it to be less secure than widely believed. First, they 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, the researchers implement the number field sieve discrete log algorithm. After a week-long precomputation for a specified 512-bit group, they can compute arbitrary discrete logs in that group in about a minute. They found that 82% of vulnerable servers use a single 512-bit group, allowing them to compromise connections to 7% of Alexa Top Million HTTPS sites. They go on to consider Diffie-Hellman with 768- and 1024-bit groups. They 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. They conclude that moving to stronger key exchange methods should be a priority for the Internet community.
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