<s>
Logjam	B-Protocol
is	O
a	O
security	O
vulnerability	O
in	O
systems	O
that	O
use	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
key	I-Protocol
exchange	I-Protocol
with	O
the	O
same	O
prime	O
number	O
.	O
</s>
<s>
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
key	I-Protocol
exchange	I-Protocol
depends	O
for	O
its	O
security	O
on	O
the	O
presumed	O
difficulty	O
of	O
solving	O
the	O
discrete	O
logarithm	O
problem	O
.	O
</s>
<s>
The	O
authors	O
took	O
advantage	O
of	O
the	O
fact	O
that	O
the	O
number	B-Algorithm
field	I-Algorithm
sieve	I-Algorithm
algorithm	O
,	O
which	O
is	O
generally	O
the	O
most	O
effective	O
method	O
for	O
finding	O
discrete	O
logarithms	O
,	O
consists	O
of	O
four	O
large	O
computational	O
steps	O
,	O
of	O
which	O
the	O
first	O
three	O
depend	O
only	O
on	O
the	O
order	O
of	O
the	O
group	O
G	O
,	O
not	O
on	O
the	O
specific	O
number	O
whose	O
finite	O
log	O
is	O
desired	O
.	O
</s>
<s>
One	O
approach	O
enabled	O
by	O
this	O
vulnerability	O
that	O
the	O
authors	O
demonstrated	O
was	O
using	O
a	O
man-in-the-middle	O
network	O
attacker	O
to	O
downgrade	O
a	O
Transport	B-Protocol
Layer	I-Protocol
Security	I-Protocol
(	O
TLS	B-Protocol
)	O
connection	O
to	O
use	O
512-bit	O
DH	O
export-grade	O
cryptography	O
,	O
allowing	O
them	O
to	O
read	O
the	O
exchanged	O
data	O
and	O
inject	O
data	O
into	O
the	O
connection	O
.	O
</s>
<s>
It	O
affects	O
the	O
HTTPS	B-Protocol
,	O
SMTPS	B-Protocol
,	O
and	O
IMAPS	B-Protocol
protocols	O
,	O
among	O
others	O
.	O
</s>
<s>
The	O
authors	O
needed	O
several	O
thousand	O
CPU	B-Device
cores	O
for	O
a	O
week	O
to	O
precompute	O
data	O
for	O
a	O
single	O
512-bit	O
prime	O
.	O
</s>
<s>
Once	O
that	O
was	O
done	O
,	O
however	O
,	O
individual	O
logarithms	O
could	O
be	O
solved	O
in	O
about	O
a	O
minute	O
using	O
two	O
18-core	O
Intel	B-Device
Xeon	I-Device
CPUs	I-Device
.	O
</s>
<s>
The	O
authors	O
also	O
estimated	O
the	O
feasibility	O
of	O
the	O
attack	O
against	O
1024-bit	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
primes	O
.	O
</s>
<s>
By	O
design	O
,	O
many	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
implementations	O
use	O
the	O
same	O
pre-generated	O
prime	O
for	O
their	O
field	O
.	O
</s>
<s>
The	O
researchers	O
calculated	O
the	O
cost	O
of	O
creating	O
logjam	B-Protocol
precomputation	O
for	O
one	O
1024-bit	O
prime	O
at	O
hundreds	O
of	O
millions	O
of	O
USD	O
,	O
and	O
noted	O
that	O
this	O
was	O
well	O
within	O
range	O
of	O
the	O
FY2012	O
$10.5	O
billion	O
U.S.	O
Consolidated	O
Cryptologic	O
Program	O
(	O
which	O
includes	O
NSA	O
)	O
.	O
</s>
<s>
Because	O
of	O
the	O
reuse	O
of	O
primes	O
,	O
generating	O
precomputation	O
for	O
just	O
one	O
prime	O
would	O
break	O
two-thirds	O
of	O
VPNs	B-Application
and	O
a	O
quarter	O
of	O
all	O
SSH	B-Protocol
servers	O
globally	O
.	O
</s>
<s>
They	O
recommend	O
using	O
primes	O
of	O
2048	O
bits	O
or	O
more	O
as	O
a	O
defense	O
or	O
switching	O
to	O
elliptic-curve	B-Protocol
Diffie	I-Protocol
–	I-Protocol
Hellman	I-Protocol
(	O
ECDH	B-Protocol
)	O
.	O
</s>
<s>
TLS-based	O
services	O
such	O
as	O
web	O
servers	O
offering	O
HTTPS	B-Protocol
can	O
be	O
checked	O
for	O
the	O
vulnerability	O
using	O
scanners	O
such	O
as	O
,	O
the	O
,	O
,	O
,	O
,	O
,	O
or	O
the	O
.	O
</s>
<s>
SSH	B-Protocol
servers	O
can	O
be	O
tested	O
using	O
the	O
or	O
tools	O
.	O
</s>
<s>
On	O
May	O
12	O
,	O
2015	O
,	O
Microsoft	O
released	O
a	O
patch	O
for	O
Internet	B-Application
Explorer	I-Application
.	O
</s>
<s>
On	O
June	O
16	O
,	O
2015	O
,	O
the	O
Tor	B-Application
Project	I-Application
provided	O
a	O
patch	O
for	O
Logjam	B-Protocol
to	O
the	O
Tor	B-Operating_System
Browser	I-Operating_System
.	O
</s>
<s>
On	O
June	O
30	O
,	O
2015	O
,	O
Apple	O
released	O
a	O
patch	O
for	O
both	O
OS	B-Device
X	I-Device
Yosemite	I-Device
and	O
iOS	B-Operating_System
8	I-Operating_System
operating	O
system	O
.	O
</s>
<s>
On	O
June	O
30	O
,	O
2015	O
,	O
the	O
Mozilla	B-Operating_System
project	I-Operating_System
released	O
a	O
fix	O
for	O
the	O
Firefox	B-Application
browser	I-Application
.	O
</s>
<s>
On	O
September	O
1	O
,	O
2015	O
,	O
Google	O
released	O
a	O
fix	O
for	O
the	O
Chrome	B-Application
browser	I-Application
.	O
</s>
<s>
On	O
December	O
6	O
,	O
2017	O
,	O
IETF	O
published	O
called	O
"	O
Increase	O
the	O
Secure	B-Protocol
Shell	I-Protocol
Minimum	O
Recommended	O
Diffie-Hellman	B-Protocol
Modulus	O
Size	O
to	O
2048	O
Bits	O
"	O
.	O
</s>
